Classical and quantum cost of measurement strategies for quantum-enhanced auxiliary field Quantum Monte Carlo
Abstract: Quantum-enhanced auxiliary field quantum Monte Carlo (QC-AFQMC) uses output from a quantum computer to increase the accuracy of its classical counterpart. The algorithm requires the estimation of overlaps between walker states and a trial wavefunction prepared on the quantum computer. We study the applicability of this algorithm in terms of the number of measurements required from the quantum computer and the classical costs of post-processing those measurements. We compare the classical post-processing costs of state-of-the-art measurement schemes using classical shadows to determine the overlaps and argue that the overall post-processing cost stemming from overlap estimations scales like $\mathcal{O}(N9)$ per walker throughout the algorithm. With further numerical simulations, we compare the variance behavior of the classical shadows when randomizing over different ensembles, e.g., Cliffords and (particle-number restricted) matchgates beyond their respective bounds, and uncover the existence of covariances between overlap estimations of the AFQMC walkers at different imaginary time steps. Moreover, we include analyses of how the error in the overlap estimation propagates into the AFQMC energy and discuss its scaling when increasing the system size.
- J. Preskill, Quantum computing in the nisq era and beyond, Quantum 2, 79 (2018).
- H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nature Physics 16, 1050 (2020).
- A. Gresch and M. Kliesch, Guaranteed efficient energy estimation of quantum many-body Hamiltonians using shadowgrouping, arXiv e-prints , arXiv:2301.03385 (2023), arXiv:2301.03385 [quant-ph] .
- A. Zhao, N. C. Rubin, and A. Miyake, Fermionic partial tomography via classical shadows, Phys. Rev. Lett. 127, 110504 (2021).
- G. H. Low, Classical shadows of fermions with particle number symmetry, arXiv 10.48550/arXiv.2208.08964 (2022), 2208.08964 .
- R. Blankenbecler, D. Scalapino, and R. Sugar, Monte carlo calculations of coupled boson-fermion systems. i, Physical Review D 24, 2278 (1981).
- M. Motta and S. Zhang, Ab initio computations of molecular systems by the auxiliary-field quantum monte carlo method, WIREs Computational Molecular Science 8, 10.1002/wcms.1364 (2018).
- S. Zhang, Auxiliary-field quantum Monte Carlo for correlated electron systems, in Emergent Phenomena in Correlated Matter, Vol. 3 (Forschungszentrum, Jülich, 2013) p. 15.
- J. W. Negele, Quantum many-particle systems (CRC Press, 2018).
- W. A. Al-Saidi, S. Zhang, and H. Krakauer, Auxiliary-field quantum Monte Carlo calculations of molecular systems with a Gaussian basis, The Journal of Chemical Physics 124, 224101 (2006).
- W. Purwanto and S. Zhang, Correlation effects in the ground state of trapped atomic bose gases, Phys. Rev. A 72, 053610 (2005).
- S. Zhang and H. Krakauer, Quantum monte carlo method using phase-free random walks with slater determinants, Phys. Rev. Lett. 90, 136401 (2003).
- C. Developers, Cirq (2022), See full list of authors on Github: https://github .com/quantumlib/Cirq/graphs/contributors.
- D. C. Liu and J. Nocedal, On the limited memory bfgs method for large scale optimization, Mathematical programming 45, 503 (1989).
- J. Lee and D. R. Reichman, Stochastic resolution-of-the-identity auxiliary-field quantum monte carlo: Scaling reduction without overhead, The Journal of Chemical Physics 153 (2020).
- D. Aharonov, V. Jones, and Z. Landau, A Polynomial Quantum Algorithm for Approximating the Jones Polynomial, arXiv 10.48550/arXiv.quant-ph/0511096 (2005), quant-ph/0511096 .
- A. Yu. Kitaev, Quantum measurements and the Abelian Stabilizer Problem, arXiv 10.48550/arXiv.quant-ph/9511026 (1995), quant-ph/9511026 .
- A. Luongo, Chapter 2 Quantum computing and quantum algorithms |normal-||| Quantum algorithms for data analysis (2023).
- S. Lu, M. C. Banuls, and J. I. Cirac, Algorithms for quantum simulation at finite energies, PRX Quantum 2, 020321 (2021).
- S. Aaronson, Shadow Tomography of Quantum States, arXiv 10.48550/arXiv.1711.01053 (2017), 1711.01053 .
- G. Boyd and B. Koczor, Training variational quantum circuits with CoVaR: covariance root finding with classical shadows, arXiv 10.1103/PhysRevX.12.041022 (2022), 2204.08494 .
- S. Bravyi and D. Maslov, Hadamard-free circuits expose the structure of the Clifford group, arXiv 10.1109/TIT.2021.3081415 (2020), 2003.09412 .
- Qiskit contributors, Qiskit: An open-source framework for quantum computing (2023).
- S. Polla, G.-L. R. Anselmetti, and T. E. O’Brien, Optimizing the information extracted by a single qubit measurement, Physical Review A 108, 012403 (2023).
- L. Lin, Lecture notes on quantum algorithms for scientific computation, arXiv preprint arXiv:2201.08309 (2022).
- B. Efron, Bootstrap Methods: Another Look at the Jackknife, Ann. Stat. 7, 1 (1979).
- M. Wimmer, Algorithm 923: Efficient numerical computation of the pfaffian for dense and banded skew-symmetric matrices, ACM Trans. Math. Softw. 38, 10.1145/2331130.2331138 (2012).
- J. Lee, H. Q. Pham, and D. R. Reichman, Twenty years of auxiliary-field quantum monte carlo in quantum chemistry: An overview and assessment on main group chemistry and bond-breaking, Journal of Chemical Theory and Computation 18, 7024 (2022).
- E. Van Den Berg, A simple method for sampling random clifford operators, in 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) (2021) pp. 54–59.
- T. mpmath development team, mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.3.0) (2023), http://mpmath.org/.
- A. Zhao and A. Miyake, Group-theoretic error mitigation enabled by classical shadows and symmetries (2023), arXiv:2310.03071 [quant-ph] .
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