Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians

Elizabeth Crosson1 and Aram W. Harrow2

1Center for Quantum Information and Control, University of New Mexico
2Center for Theoretical Physics, Massachusetts Institute of Technology

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Abstract

Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution.
In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transverse Ising models (TIM) with long-range algebraically decaying interactions as well as disordered XY spin chains with nearest-neighbor interactions. By bounding the convergence time to the equilibrium distribution we rigorously justify the use of PIMC to approximate partition functions and expectations of observables for these models at inverse temperatures that scale at most logarithmically with the number of qubits.
The mixing time analysis is based on the canonical paths method applied to the single-site Metropolis Markov chain for the Gibbs distribution of 2D classical spin models with couplings related to the interactions in the quantum Hamiltonian. Since the system has strongly nonisotropic couplings that grow with system size, it does not fall into the known cases where 2D classical spin models are known to mix rapidly.

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Cited by

[1] Elizabeth Crosson and Samuel Slezak, "Classical Simulation of High Temperature Quantum Ising Models", arXiv:2002.02232.

[2] Humberto Munoz-Bauza, Huo Chen, and Daniel Lidar, "A double-slit proposal for quantum annealing", npj Quantum Information 5, 51 (2019).

[3] Elizabeth Crosson, Tameem Albash, Itay Hen, and A. P. Young, "De-Signing Hamiltonians for Quantum Adiabatic Optimization", arXiv:2004.07681.

[4] Thiago Bergamaschi, "Simulated Quantum Annealing is Efficient on the Spike Hamiltonian", arXiv:2011.15094.

[5] Jacob Bringewatt and Michael Jarret, "Effective Gaps Are Not Effective: Quasipolynomial Classical Simulation of Obstructed Stoquastic Hamiltonians", Physical Review Letters 125 17, 170504 (2020).

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