Robust Extraction of Thermal Observables from State Sampling and Real-Time Dynamics on Quantum Computers

Khaldoon Ghanem; Alexander Schuckert; Henrik Dreyer

doi:10.22331/q-2023-11-03-1163

Robust Extraction of Thermal Observables from State Sampling and Real-Time Dynamics on Quantum Computers

Khaldoon Ghanem, Alexander Schuckert, and Henrik Dreyer

Quantinuum, Leopoldstrasse 180, 80804 Munich, Germany

Published:	2023-11-03, volume 7, page 1163
Eprint:	arXiv:2305.19322v2
Doi:	https://doi.org/10.22331/q-2023-11-03-1163
Citation:	Quantum 7, 1163 (2023).

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Abstract

Simulating properties of quantum materials is one of the most promising applications of quantum computation, both near- and long-term. While real-time dynamics can be straightforwardly implemented, the finite temperature ensemble involves non-unitary operators that render an implementation on a near-term quantum computer extremely challenging. Recently, Lu, Bañuls and Cirac [28] suggested a "time-series quantum Monte Carlo method" which circumvents this problem by extracting finite temperature properties from real-time simulations via Wick's rotation and Monte Carlo sampling of easily preparable states. In this paper, we address the challenges associated with the practical applications of this method, using the two-dimensional transverse field Ising model as a testbed. We demonstrate that estimating Boltzmann weights via Wick's rotation is very sensitive to time-domain truncation and statistical shot noise. To alleviate this problem, we introduce a technique that imposes constraints on the density of states, most notably its non-negativity, and show that this way, we can reliably extract Boltzmann weights from noisy time series. In addition, we show how to reduce the statistical errors of Monte Carlo sampling via a reweighted version of the Wolff cluster algorithm. Our work enables the implementation of the time-series algorithm on present-day quantum computers to study finite temperature properties of many-body quantum systems.

Featured image: Equilibrium properties from quantum dynamics and the noise-induced infrared catastrophe.

Popular summary

Time-series quantum Monte Carlo is a hybrid classical-quantum algorithm for calculating finite temperature properties of quantum systems. While the algorithm has recently been shown to scale favourably on an ideal simulator, we show here that shot noise, accompanying realistic computations, causes the algorithm to become unstable. In this paper, we address this problem and introduce a novel technique for extracting Boltzmann weights from noisy time series, reducing errors by several orders of magnitude. Additionally, we present a sampling algorithm that significantly reduces statistical errors. These advancements make the algorithm practical on present-day quantum computers and a candidate for quantum advantage.

► BibTeX data

@article{Ghanem2023robustextractionof,
  doi = {10.22331/q-2023-11-03-1163},
  url = {https://doi.org/10.22331/q-2023-11-03-1163},
  title = {Robust {E}xtraction of {T}hermal {O}bservables from {S}tate {S}ampling and {R}eal-{T}ime {D}ynamics on {Q}uantum {C}omputers},
  author = {Ghanem, Khaldoon and Schuckert, Alexander and Dreyer, Henrik},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {7},
  pages = {1163},
  month = nov,
  year = {2023}
}

► References

[1] A. W. Sandvik and J. Kurkijärvi, Phys. Rev. B 43, 5950 (1991).
https://doi.org/10.1103/PhysRevB.43.5950

[2] A. W. Sandvik, J. Phys. A: Math. Gen. 25, 3667 (1992).
https://doi.org/10.1088/0305-4470/25/13/017

[3] M. Suzuki, S. Miyashita, and A. Kuroda, Prog. Theor. Phys. 58, 1377 (1977).
https://doi.org/10.1143/PTP.58.1377

[4] J. E. Hirsch, R. L. Sugar, D. J. Scalapino, and R. Blankenbecler, Phys. Rev. B 26, 5033 (1982).
https://doi.org/10.1103/PhysRevB.26.5033

[5] N. S. Blunt, T. W. Rogers, J. S. Spencer, and W. M. C. Foulkes, Phys. Rev. B 89, 245124 (2014).
https://doi.org/10.1103/PhysRevB.89.245124

[6] H. R. Petras, S. K. Ramadugu, F. D. Malone, and J. J. Shepherd, J. Chem. Theory Comput. 16, 1029 (2020).
https://doi.org/10.1021/acs.jctc.9b01080

[7] S. R. White, Phys. Rev. Lett. 102, 190601 (2009).
https://doi.org/10.1103/PhysRevLett.102.190601

[8] E. M. Stoudenmire and S. R. White, New J. Phys. 12, 055026 (2010).
https://doi.org/10.1088/1367-2630/12/5/055026

[9] R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Phys. Rev. D 24, 2278 (1981).
https://doi.org/10.1103/PhysRevD.24.2278

[10] S. R. White, D. J. Scalapino, R. L. Sugar, E. Y. Loh, J. E. Gubernatis, and R. T. Scalettar, Phys. Rev. B 40, 506 (1989).
https://doi.org/10.1103/PhysRevB.40.506

[11] Y. Liu, M. Cho, and B. Rubenstein, J. Chem. Theory Comput. 14, 4722 (2018).
https://doi.org/10.1021/acs.jctc.8b00569

[12] Y.-Y. He, M. Qin, H. Shi, Z.-Y. Lu, and S. Zhang, Phys. Rev. B 99, 045108 (2019).
https://doi.org/10.1103/PhysRevB.99.045108

[13] C. Sun, U. Ray, Z.-H. Cui, M. Stoudenmire, M. Ferrero, and G. K.-L. Chan, Phys. Rev. B 101, 075131 (2020).
https://doi.org/10.1103/PhysRevB.101.075131

[14] P. Henelius and A. W. Sandvik, Phys. Rev. B 62, 1102 (2000).
https://doi.org/10.1103/PhysRevB.62.1102

[15] M. Troyer and U.-J. Wiese, Phys. Rev. Lett. 94, 170201 (2005).
https://doi.org/10.1103/PhysRevLett.94.170201

[16] K. Temme, T. J. Osborne, K. G. Vollbrecht, D. Poulin, and F. Verstraete, Nature 471, 87 (2011).
https://doi.org/10.1038/nature09770

[17] M.-H. Yung and A. Aspuru-Guzik, Proc. Natl. Acad. Sci. 109, 754 (2012).
https://doi.org/10.1073/pnas.1111758109

[18] M. Motta, C. Sun, A. T. K. Tan, M. J. O'Rourke, E. Ye, A. J. Minnich, F. G. S. L. Brandão, and G. K.-L. Chan, Nat. Phys. 16, 205 (2020).
https://doi.org/10.1038/s41567-019-0704-4

[19] J. Cohn, F. Yang, K. Najafi, B. Jones, and J. K. Freericks, Phys. Rev. A 102, 022622 (2020).
https://doi.org/10.1103/PhysRevA.102.022622

[20] S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012).
https://doi.org/10.1103/PhysRevLett.108.240401

[21] L. Coopmans, Y. Kikuchi, and M. Benedetti, PRX Quantum 4, 010305 (2023).
https://doi.org/10.1103/PRXQuantum.4.010305

[22] C. Sun, Finite Temperature Simulations of Strongly Correlated Systems, Ph.D. thesis, California Institute of Technology (2023).
https://doi.org/10.7907/dchn-p020

[23] D. Poulin and P. Wocjan, Phys. Rev. Lett. 103, 220502 (2009).
https://doi.org/10.1103/PhysRevLett.103.220502

[24] A. N. Chowdhury and R. D. Somma, Quantum Inf. Comput. 17, 41–64 (2017).
https://doi.org/10.26421/QIC17.1-2-3

[25] S. Bravyi, A. Chowdhury, D. Gosset, and P. Wocjan, Nat. Phys. 18, 1367 (2022).
https://doi.org/10.1038/s41567-022-01742-5

[26] R. N. Tazhigulov, S.-N. Sun, R. Haghshenas, H. Zhai, A. T. Tan, N. C. Rubin, R. Babbush, A. J. Minnich, and G. K.-L. Chan, PRX Quantum 3, 040318 (2022).
https://doi.org/10.1103/PRXQuantum.3.040318

[27] W. J. Huggins, B. A. O'Gorman, N. C. Rubin, D. R. Reichman, R. Babbush, and J. Lee, Nature 603, 416 (2022).
https://doi.org/10.1038/s41586-021-04351-z

[28] S. Lu, M. C. Bañuls, and J. I. Cirac, PRX Quantum 2, 020321 (2021).
https://doi.org/10.1103/PRXQuantum.2.020321

[29] Y. Yang, J. I. Cirac, and M. C. Bañuls, Phys. Rev. B 106, 024307 (2022).
https://doi.org/10.1103/PhysRevB.106.024307

[30] A. Schuckert, A. Bohrdt, E. Crane, and M. Knap, Phys. Rev. B 107, L140410 (2023).
https://doi.org/10.1103/PhysRevB.107.L140410

[31] C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).
https://doi.org/10.1137/1.9781611971217

[32] U. Wolff, Phys. Rev. Lett. 62, 361 (1989).
https://doi.org/10.1103/PhysRevLett.62.361

[33] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys. 73, 33 (2001).
https://doi.org/10.1103/RevModPhys.73.33

[34] J. Gubernatis, N. Kawashima, and P. Werner, Quantum Monte Carlo Methods: Algorithms for Lattice Models (Cambridge University Press, 2016).
https://doi.org/10.1017/CBO9780511902581

[35] R. N. Silver, D. S. Sivia, and J. E. Gubernatis, Phys. Rev. B 41, 2380 (1990).
https://doi.org/10.1103/PhysRevB.41.2380

[36] M. Jarrell and J. E. Gubernatis, Phys. Rep. 269, 133 (1996).
https://doi.org/10.1016/0370-1573(95)00074-7

[37] K. Vafayi and O. Gunnarsson, Phys. Rev. B 76, 035115 (2007).
https://doi.org/10.1103/PhysRevB.76.035115

[38] K. Ghanem and E. Koch, Phys. Rev. B 101, 085111 (2020a).
https://doi.org/10.1103/PhysRevB.101.085111

[39] K. Ghanem and E. Koch, Phys. Rev. B 102, 035114 (2020b).
https://doi.org/10.1103/PhysRevB.102.035114

[40] K. Ghanem and E. Koch, Phys. Rev. B 107, 085129 (2023).
https://doi.org/10.1103/PhysRevB.107.085129

[41] P. Pfeuty, Ann. Phys. 57, 79 (1970).
https://doi.org/10.1016/0003-4916(70)90270-8

[42] Z. Friedman, Phys. Rev. B 17, 1429 (1978).
https://doi.org/10.1103/PhysRevB.17.1429

[43] G. H. Wannier, Rev. Mod. Phys. 17, 50 (1945).
https://doi.org/10.1103/RevModPhys.17.50

[44] S. Morita and S. Suzuki, Journal of Statistical Physics 162, 123 (2016).
https://doi.org/10.1007/s10955-015-1400-0

[45] C. Shannon, Proc. IRE 37, 10 (1949).
https://doi.org/10.1109/JRPROC.1949.232969

[46] C. W. Groetsch, Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Chapman & Hall/CRC Research Notes in Mathematics Series (Pitman Advanced Pub. Program, 1984).

[47] V. A. Morozov, Criteria for selection of regularization parameter, in Methods for Solving Incorrectly Posed Problems (Springer New York, New York, NY, 1984) pp. 32–64.
https://doi.org/10.1007/978-1-4612-5280-1_2

[48] D. Kandel, R. Ben-Av, and E. Domany, Phys. Rev. Lett. 65, 941 (1990).
https://doi.org/10.1103/PhysRevLett.65.941

[49] G. M. Zhang and C. Z. Yang, Phys. Rev. B 50, 12546 (1994).
https://doi.org/10.1103/PhysRevB.50.12546

Cited by

[1] Chao Yin and Andrew Lucas, "Heisenberg-limited metrology with perturbing interactions", arXiv:2308.10929, (2023).

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