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.

翻译：量子蒙特卡洛（QMC）方法在凝聚态物理中已被证明具有重要价值，尤其适用于研究无符号问题的量子哈密顿量的基态和热平衡性质。过去十年间，其严格收敛性分析也取得了显著进展。具有二分相互作用图的海森堡反铁磁体（AFM）因其物理重要性，成为计算QMC研究的热门对象；然而，尽管这些模拟在经验上表现出明显的高效性，但关于其基态能量能否在一般情况下被经典算法高效近似，仍是一个悬而未决的问题。本文中，我们引入了随机级数展开QMC方法的一个基态变体，并针对相互作用图具有$O(1)$二分分量（星状）的特殊AFM类别，利用Jerrum和Sinclair的规范路径方法，证明了相关QMC马尔可夫链的快速混合性（混合时间关于量子比特数为多项式级）。这是首次对采用海森堡模型环路表示的实用QMC算法类别进行马尔可夫链分析。我们的研究结果为更广泛地解决一般二分相互作用图上Heisenberg AFM的计算复杂性难题做出了贡献。