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Quantum Zeno Monte Carlo for computing observables (2403.02763v4)

Published 5 Mar 2024 in quant-ph and cond-mat.str-el

Abstract: The recent development of logical quantum processors marks a pivotal transition from the noisy intermediate-scale quantum (NISQ) era to the fault-tolerant quantum computing (FTQC) era. These devices have the potential to address classically challenging problems with polynomial computational time using quantum properties. However, they remain susceptible to noise, necessitating noise resilient algorithms. We introduce Quantum Zeno Monte Carlo (QZMC), a classical-quantum hybrid algorithm that demonstrates resilience to device noise and Trotter errors while showing polynomial computational cost for a gapped system. QZMC computes static and dynamic properties without requiring initial state overlap or variational parameters, offering reduced quantum circuit depth.

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References (49)
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[9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. 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Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Feynman, R. P. Simulating physics with computers. International Journal of Theoretical Physics 21 (1982). [3] Nielsen, M. A. & Chuang, I. L. Quantum computation and quantum information (Cambridge university press, 2010). [4] Kitaev, A. Y. Quantum measurements and the abelian stabilizer problem. arXiv preprint quant-ph/9511026 (1995). [5] Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Nielsen, M. A. & Chuang, I. L. Quantum computation and quantum information (Cambridge university press, 2010). [4] Kitaev, A. Y. Quantum measurements and the abelian stabilizer problem. arXiv preprint quant-ph/9511026 (1995). [5] Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kitaev, A. Y. Quantum measurements and the abelian stabilizer problem. arXiv preprint quant-ph/9511026 (1995). [5] Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. 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Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Nielsen, M. A. & Chuang, I. L. Quantum computation and quantum information (Cambridge university press, 2010). [4] Kitaev, A. Y. Quantum measurements and the abelian stabilizer problem. arXiv preprint quant-ph/9511026 (1995). [5] Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kitaev, A. Y. Quantum measurements and the abelian stabilizer problem. arXiv preprint quant-ph/9511026 (1995). [5] Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). 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[47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. 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Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. 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[42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kitaev, A. Y. Quantum measurements and the abelian stabilizer problem. arXiv preprint quant-ph/9511026 (1995). [5] Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Abrams, D. S. & Lloyd, S. Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors. Physical Review Letters 83, 5162 (1999). [6] Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. 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Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). 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Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shor, P. W. Fault-tolerant quantum computation (1996). [7] Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. 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[38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). 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URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. 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URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. 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[42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). 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Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. 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URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). 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URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. 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Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). 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Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. 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Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. 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URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
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Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. 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Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Gottesman, D. Theory of fault-tolerant quantum computation. Physical Review A 57, 127 (1998). [8] Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2, 79 (2018). URL https://doi.org/10.22331/q-2018-08-06-79. [9] Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Peruzzo, A. et al. A variational eigenvalue solver on a photonic quantum processor. Nature communications 5, 4213 (2014). [10] McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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[11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). McClean, J. R., Romero, J., Babbush, R. & Aspuru-Guzik, A. The theory of variational hybrid quantum-classical algorithms. New Journal of Physics 18, 023023 (2016). [11] Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. 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[18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. 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The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). 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URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. 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Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. 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URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). 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[27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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[40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. 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The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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[47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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[42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. 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[40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. 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URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. 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URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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[17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). 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[36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. 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Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Bluvstein, D. et al. Logical quantum processor based on reconfigurable atom arrays. Nature 626, 58–65 (2024). [12] Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Berry, D. W. et al. How to perform the most accurate possible phase measurements. Phys. Rev. A 80, 052114 (2009). URL https://link.aps.org/doi/10.1103/PhysRevA.80.052114. [13] Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. 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URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. 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[49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). 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Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). 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[47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. 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[40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. 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[27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). 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URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. 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[49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. 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[37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). 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Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). 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Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Higgins, B. L., Berry, D. W., Bartlett, S. D., Wiseman, H. M. & Pryde, G. J. Entanglement-free heisenberg-limited phase estimation. Nature 450, 393–396 (2007). [14] Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. 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[43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). 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URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. 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[42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
  14. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum 3, 010318 (2022). URL https://link.aps.org/doi/10.1103/PRXQuantum.3.010318. [15] Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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[40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. 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The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. 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[42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
  15. Somma, R. D. Quantum eigenvalue estimation via time series analysis. New Journal of Physics 21, 123025 (2019). [16] Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ding, Z. & Lin, L. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. 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Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
  16. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum 4, 020331 (2023). URL https://link.aps.org/doi/10.1103/PRXQuantum.4.020331. [17] Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Misra, B. & Sudarshan, E. G. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. 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Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. 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Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). 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On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
  17. The zeno’s paradox in quantum theory. Journal of Mathematical Physics 18, 756–763 (1977). [18] Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. 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[49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Somma, R. D., Boixo, S., Barnum, H. & Knill, E. Quantum simulations of classical annealing processes. Phys. Rev. Lett. 101, 130504 (2008). URL https://link.aps.org/doi/10.1103/PhysRevLett.101.130504. [19] Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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[37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). 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Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Poulin, D. & Wocjan, P. Preparing ground states of quantum many-body systems on a quantum computer. Phys. Rev. Lett. 102, 130503 (2009). URL https://link.aps.org/doi/10.1103/PhysRevLett.102.130503. [20] Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Boixo, S., Knill, E., Somma, R. D. et al. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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[41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. 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Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. 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Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. 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URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. 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[36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. 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Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. 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Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. 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Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). 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Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. 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[42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. 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  20. Eigenpath traversal by phase randomization. Quantum Inf. Comput. 9, 833–855 (2009). [21] Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lin, L. & Tong, Y. Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems. Quantum 4, 361 (2020). URL https://doi.org/10.22331/q-2020-11-11-361. [22] van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). van Vleck, J. H. Nonorthogonality and ferromagnetism. Phys. Rev. 49, 232–240 (1936). URL https://link.aps.org/doi/10.1103/PhysRev.49.232. [23] Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kohn, W. Nobel lecture: Electronic structure of matter—wave functions and density functionals. Rev. Mod. Phys. 71, 1253–1266 (1999). URL https://link.aps.org/doi/10.1103/RevModPhys.71.1253. [24] Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). 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Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. 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[49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zeng, P., Sun, J. & Yuan, X. Universal quantum algorithmic cooling on a quantum computer. arXiv preprint arXiv:2109.15304 (2021). [25] Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. 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[35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Huo, M. & Li, Y. Error-resilient Monte Carlo quantum simulation of imaginary time. Quantum 7, 916 (2023). URL https://doi.org/10.22331/q-2023-02-09-916. [26] Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Wang, G., França, D. S., Zhang, R., Zhu, S. & Johnson, P. D. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. 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[45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. 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URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. 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URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. 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Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. 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Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. 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Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. 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Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). 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Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
  26. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum 7, 1167 (2023). [27] Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Sun, J., Vilchez-Estevez, L., Vedral, V., Boothroyd, A. T. & Kim, M. Probing spectral features of quantum many-body systems with quantum simulators. arXiv preprint arXiv:2305.07649 (2023). [28] Lloyd, S. Universal quantum simulators. Science 273, 1073–1078 (1996). [29] Zalka, C. Simulating quantum systems on a quantum computer. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 313–322 (1998). [30] Kroese, D. P., Taimre, T. & Botev, Z. I. Handbook of Monte Carlo Methods (John Wiley & Sons, 2011). [31] Heath, M. T. Scientific computing: an introductory survey, revised second edition (SIAM, 2018). [32] Landau, L. D. & Lifshitz, E. M. Quantum mechanics: non-relativistic theory Vol. 3 (Elsevier, 2013). [33] Stewart, R. F. Small gaussian expansions of slater-type orbitals. The Journal of Chemical Physics 52, 431–438 (1970). [34] O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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[47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). O’Malley, P. J. J. et al. Scalable quantum simulation of molecular energies. Phys. Rev. X 6, 031007 (2016). URL https://link.aps.org/doi/10.1103/PhysRevX.6.031007. [35] Motta, M. et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics 16, 205–210 (2020). [36] Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). 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[42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Seeley, J. T., Richard, M. J. & Love, P. J. The bravyi-kitaev transformation for quantum computation of electronic structure. The Journal of chemical physics 137 (2012). [37] Steudtner, M. & Wehner, S. Fermion-to-qubit mappings with varying resource requirements for quantum simulation. New Journal of Physics 20, 063010 (2018). [38] Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. 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IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Hubbard, J. & Flowers, B. H. Electron correlations in narrow energy bands. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). 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Mathematical and Physical Sciences 276, 238–257 (1963). [39] qiskit. URL https://www.ibm.com/quantum/qiskit. [40] Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. 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URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). 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[47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). 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Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. 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Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. 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Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V. V., Markov, I. L. & Bullock, S. S. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
  40. Minimal universal two-qubit controlled-not-based circuits. Phys. Rev. A 69, 062321 (2004). URL https://link.aps.org/doi/10.1103/PhysRevA.69.062321. [41] Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Shende, V., Bullock, S. & Markov, I. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
  41. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). [42] Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Cross, A. W., Bishop, L. S., Sheldon, S., Nation, P. D. & Gambetta, J. M. Validating quantum computers using randomized model circuits. Phys. Rev. A 100, 032328 (2019). URL https://link.aps.org/doi/10.1103/PhysRevA.100.032328. [43] Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Negele, J. W. & Orland, H. Quantum many-particle systems Advanced book classics (Westview, Boulder, CO, 1988). URL https://cds.cern.ch/record/729852. Addison-Wesley edition. [44] qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). qsim. URL https://quantumai.google/qsim. [45] Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Trotter, H. F. On the product of semi-groups of operators. Proceedings of the American Mathematical Society 10, 545–551 (1959). [46] Layden, D. First-order trotter error from a second-order perspective. Phys. Rev. Lett. 128, 210501 (2022). URL https://link.aps.org/doi/10.1103/PhysRevLett.128.210501. [47] Tilly, J. et al. The variational quantum eigensolver: A review of methods and best practices. Physics Reports 986, 1–128 (2022). The Variational Quantum Eigensolver: a review of methods and best practices. [48] Watrous, J. Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Layden, D. 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Quantum computational complexity. arXiv preprint arXiv:0804.3401 (2008). [49] Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023). Ni, H., Li, H. & Ying, L. On low-depth algorithms for quantum phase estimation. Quantum 7, 1165 (2023).
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