Classical Markov Chain Monte Carlo methods have been essential for simulating statistical physical systems and have proven well applicable to other systems with complex degrees of freedom. Motivated by the statistical physics origins, Chen, Kastoryano, and Gily\'en [CKG23] proposed a continuous-time quantum thermodynamic analog to Glauber dynamic that is (i) exactly detailed balanced, (ii) efficiently implementable, and (iii) quasi-local for geometrically local systems. Physically, their construction gives a smooth variant of the Davies' generator derived from weak system-bath interaction. In this work, we give an efficiently implementable discrete-time quantum counterpart to Metropolis sampling that also enjoys the desirable features (i)-(iii). Also, we give an alternative highly coherent quantum generalization of detailed balanced dynamics that resembles another physically derived master equation, and propose a smooth interpolation between this and earlier constructions. We study generic properties of all constructions, including the uniqueness of the fixed-point and the locality of the resulting operators. We hope our results provide a systematic approach to the possible quantum generalizations of classical Glauber and Metropolis dynamics.

翻译：经典马尔可夫链蒙特卡罗方法对于模拟统计物理系统至关重要，并已证明同样适用于其他具有复杂自由度的系统。受统计物理学起源的启发，Chen、Kastoryano与Gilyén [CKG23] 提出了一种对应于格劳伯动力学的连续时间量子热力学类比模型，该模型具有以下特性：(i) 严格满足细致平衡条件，(ii) 可高效实现，(iii) 对几何局域系统具有准局域性。在物理上，他们的构造给出了从弱系统-环境相互作用导出的戴维斯生成元的一种平滑变体。本工作中，我们提出了一种可高效实现的离散时间量子对应方法，用于Metropolis采样，同样具备上述理想特性(i)-(iii)。此外，我们提出了另一种高度相干的细致平衡动力学量子推广，其形式类似于另一类从物理推导得到的主方程，并构建了该模型与早期构造之间的平滑插值。我们研究了所有构造的通用性质，包括不动点的唯一性以及所得算符的局域性。我们希望本研究能为经典格劳伯动力学与Metropolis动力学的可能量子推广提供系统化的研究路径。