A Simple and Efficient Joint Measurement Strategy for Estimating Fermionic Observables and Hamiltonians (2402.19230v2)
Abstract: We propose a simple scheme to estimate fermionic observables and Hamiltonians relevant in quantum chemistry and correlated fermionic systems. Our approach is based on implementing a measurement that jointly measures noisy versions of any product of two or four Majorana operators in an $N$ mode fermionic system. To realize our measurement we use: (i) a randomization over a set of unitaries that realize products of Majorana fermion operators; (ii) a unitary, sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable post-processing. Our scheme can estimate expectation values of all quadratic and quartic Majorana monomials to $\epsilon$ precision using $\mathcal{O}(N \log(N)/\epsilon2)$ and $\mathcal{O}(N2 \log(N)/\epsilon2)$ measurement rounds respectively, matching the performance offered by fermionic shadow tomography. In certain settings, such as a rectangular lattice of qubits which encode an $N$ mode fermionic system via the Jordan-Wigner transformation, our scheme can be implemented in circuit depth $\mathcal{O}(N{1/2})$ with $\mathcal{O}(N{3/2})$ two-qubit gates, offering an improvement over fermionic and matchgate classical shadows that require depth $\mathcal{O}(N)$ and $\mathcal{O}(N2)$ two-qubit gates. By benchmarking our method on exemplary molecular Hamiltonians and observing performances comparable to fermionic classical shadows, we demonstrate a novel, competitive alternative to existing strategies.
- A. Zhao, N. C. Rubin, and A. Miyake, Phys. Rev. Lett. 127, 0110504 (2021).
- J. Preskill, Quantum 2, 79 (2018).
- G. Gidofalvi and D. A. Mazziotti, J. Chem. Phys. 126, 024105 (2007).
- A. Jena, S. Genin, and M. Mosca, arXiv preprint arXiv:1907.07859 (2019).
- T. C. Yen, V. Verteletskyi, and A. F. Izmaylov, J. Chem. Theory Comput. 16, 2400 (2020).
- V. Verteletskyi, T. C. Yen, and A. F. Izmaylov, J. Chem. Phys. 152, 124114 (2020).
- X. Bonet-Monroig, R. Babbush, and T. E. O’Brien, Phys. Rev. X 10, 031064 (2020).
- H. Y. Huang, R. Kueng, and J. Preskill, Nat. Phys. 16, 1050 (2020).
- H.-Y. Hu, S. Choi, and Y.-Z. You, Phys. Rev. Res. 5, 023027 (2023).
- G. H. Low, arXiv preprint arXiv:2208.08964 (2022).
- B. O’Gorman, arXiv preprint arXiv:2207.14787 (2022).
- D. McNulty, F. B. Maciejewski, and M. Oszmaniec, Phys. Rev. Lett. 130, 100801 (2023).
- T. Heinosaari, D. Reitzner, and P. Stano, Found. Phys. 38, 1133 (2008).
- T. Heinosaari, J. Kiukas, and D. Reitzner, Phys. Rev. A 92, 022115 (2015).
- D. McNulty, S. Calegari, and M. Oszmaniec, Optimal Fermionic Joint Measurements for Estimating Non-Commuting Majorana Observables (2024).
- S. Bravyi, B. M. Terhal, and B. Leemhuis, New J. Phys. 12, 083039 (2010).
- B. M. Terhal and D. P. DiVincenzo, Phys. Rev. A 65, 032325 (2002).
- S. B. Bravyi and A. Y. Kitaev, Ann. Phys. 298, 210 (2002).
- R. Jozsa and A. Miyake, Proc. R. Soc. A: Math. Phys. Eng. Sci. 464, 3089 (2008).
- P. Jordan and E. P. Wigner, Über das paulische äquivalenzverbot (Springer, 1993).
- W. Hoeffding, J. Am. Stat. Assoc. 58, 13 (1963).
- P. Jaming and M. Matolcsi, Acta Math. Hungarica 147, 179 (2015).
- J. T. Seeley, M. J. Richard, and P. J. Love, J. Chem. Phys. 137 (2012).
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
- A. Gresch and M. Kliesch, arXiv preprint arXiv:2301.03385 (2023).
- S. Bravyi and R. König, Comm. Math. Phys. 316, 641 (2012).
- F. d. Melo, P. Ćwikliński, and B. M. Terhal, New J. Phys. 15, 013015 (2013).
- J. L. Bosse, Floyao (2022).
- J. R. McClean et al., Quantum Science and Technology 5, 034014 (2020).
- Qiskit contributors, Qiskit: An open-source framework for quantum computing (2023).
- T. Banica, I. Nechita, and K. Życzkowski, Open Syst. Inf. Dyn. 19, 1250024 (2012).