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

## TL;DR

This paper provides a rigorous analysis of the mixing time for path integral Monte Carlo applied to 1D stoquastic Hamiltonians, justifying its effectiveness for certain disordered quantum spin systems at specific temperatures.

## Contribution

It offers the first formal proof of rapid mixing for PIMC in 1D disordered stoquastic models, including long-range and nearest-neighbor interactions.

## Key findings

- Proves PIMC mixes rapidly at inverse temperatures logarithmic in system size.
- Uses canonical paths method to analyze convergence.
- Justifies PIMC's use for approximating partition functions in these models.

## 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.