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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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] Jacob Bringewatt and Lucas T. Brady, "Simultaneous stoquasticity", Physical Review A 105 6, 062601 (2022).

[2] M E Stroeks, J Helsen, and B M Terhal, "Spectral estimation for Hamiltonians: a comparison between classical imaginary-time evolution and quantum real-time evolution", New Journal of Physics 24 10, 103024 (2022).

[3] Artem Rakcheev and Andreas M. Läuchli, "Diabatic quantum and classical annealing of the Sherrington-Kirkpatrick model", Physical Review A 107 6, 062602 (2023).

[4] Sergey Bravyi, Anirban Chowdhury, David Gosset, and Pawel Wocjan, "Quantum Hamiltonian complexity in thermal equilibrium", Nature Physics 18 11, 1367 (2022).

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

[8] Elizabeth Crosson, Tameem Albash, Itay Hen, and A. P. Young, "De-Signing Hamiltonians for Quantum Adiabatic Optimization", Quantum 4, 334 (2020).

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