Error-resilient Monte Carlo quantum simulation of imaginary time

Mingxia Huo1 and Ying Li2

1Department of Physics and Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
2Graduate School of China Academy of Engineering Physics, Beijing 100193, China

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Computing the ground-state properties of quantum many-body systems is a promising application of near-term quantum hardware with a potential impact in many fields. The conventional algorithm quantum phase estimation uses deep circuits and requires fault-tolerant technologies. Many quantum simulation algorithms developed recently work in an inexact and variational manner to exploit shallow circuits. In this work, we combine quantum Monte Carlo with quantum computing and propose an algorithm for simulating the imaginary-time evolution and solving the ground-state problem. By sampling the real-time evolution operator with a random evolution time according to a modified Cauchy-Lorentz distribution, we can compute the expected value of an observable in imaginary-time evolution. Our algorithm approaches the exact solution given a circuit depth increasing polylogarithmically with the desired accuracy. Compared with quantum phase estimation, the Trotter step number, i.e. the circuit depth, can be thousands of times smaller to achieve the same accuracy in the ground-state energy. We verify the resilience to Trotterisation errors caused by the finite circuit depth in the numerical simulation of various models. The results show that Monte Carlo quantum simulation is promising even without a fully fault-tolerant quantum computer.

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[5] Keisuke Matsumoto, Yuta Shingu, Suguru Endo, Shiro Kawabata, Shohei Watabe, Tetsuro Nikuni, Hideaki Hakoshima, and Yuichiro Matsuzaki, "Calculation of Gibbs partition function with imaginary time evolution on near-term quantum computers", Japanese Journal of Applied Physics 61 4, 042002 (2022).

[6] Yu-Rong Shu, Shao-Kai Jian, and Shuai Yin, "Nonequilibrium Dynamics of Deconfined Quantum Critical Point in Imaginary Time", Physical Review Letters 128 2, 020601 (2022).

[7] Yifei Huang, Yuguo Shao, Weiluo Ren, Jinzhao Sun, and Dingshun Lv, "Efficient quantum imaginary time evolution by drifting real time evolution: an approach with low gate and measurement complexity", arXiv:2203.11112, (2022).

[8] Yongdan Yang, Ying Li, Xiaosi Xu, and Xiao Yuan, "A resource-efficient quantum-classical hybrid algorithm for energy gap evaluation", arXiv:2305.07382, (2023).

[9] Qingxing Xie, Yi Song, and Yan Zhao, "Power of Sine Hamiltonian Operator for Estimating the Eigenstate Energies on Quantum Computers", arXiv:2209.14801, (2022).

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