Rapidly mixing loop representation quantum Monte Carlo for Heisenberg models on star-like bipartite graphs

TL;DR

This paper introduces a new quantum Monte Carlo method for Heisenberg antiferromagnets on star-like bipartite graphs and proves its rapid mixing, advancing understanding of classical simulation efficiency for these quantum systems.

Contribution

It presents the first Markov chain analysis of a practical QMC algorithm with loop representation for Heisenberg models on specific bipartite graphs.

Findings

Proves polynomial-time rapid mixing for the QMC Markov chain on star-like bipartite graphs.

Introduces a ground state variant of the stochastic series expansion QMC method.

Contributes to understanding the computational complexity of simulating Heisenberg antiferromagnets.

Abstract

Quantum Monte Carlo (QMC) methods have proven invaluable in condensed matter physics, particularly for studying ground states and thermal equilibrium properties of quantum Hamiltonians without a sign problem. Over the past decade, significant progress has also been made on their rigorous convergence analysis. Heisenberg antiferromagnets (AFM) with bipartite interaction graphs are a popular target of computational QMC studies due to their physical importance, but despite the apparent empirical efficiency of these simulations it remains an open question whether efficient classical approximation of the ground energy is possible in general. In this work we introduce a ground state variant of the stochastic series expansion QMC method, and for the special class of AFM on interaction graphs with an $O(1)$-bipartite component (star-like), we prove rapid mixing of the associated QMC Markov chain (polynomial time in the number of qubits) by using Jerrum and Sinclair's method of canonical paths. This is the first Markov chain analysis of a practical class of QMC algorithms with the loop representation of Heisenberg models. Our findings contribute to the broader effort to resolve the computational complexity of Heisenberg AFM on general bipartite interaction graphs.