Classical Markov Chain Monte Carlo methods have been essential for simulating statistical physical systems and have proven well applicable to other systems with many degrees of freedom. Motivated by the statistical physics origins, Chen, Kastoryano, and Gilyén [CKG23] proposed a continuous-time quantum thermodynamic analogue to Glauber dynamics that is (i) exactly detailed balanced, (ii) efficiently implementable, and (iii) quasi-local for geometrically local systems. Physically, their construction resembles the dissipative dynamics arising from weak system-bath interaction. In this work, we give an efficiently implementable discrete-time counterpart to any continuous-time quantum Gibbs sampler. Our construction preserves the desirable features (i)-(iii) while does not decrease the spectral gap. 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. Moreover, we show how to make earlier Metropolis-style Gibbs samplers (which estimate energies both before and after jumps) exactly detailed balanced. We study generic properties of all constructions, including the uniqueness of the fixed point, the (quasi-)locality of the resulting operators. Finally, we prove that the spectral gap of our new highly coherent Gibbs sampler is constant at high temperatures, thereby it mixes fast. We hope that our systematic approach to quantum Glauber and Metropolis dynamics will lead to widespread applications in various domains.

翻译：经典马尔可夫链蒙特卡洛方法在模拟统计物理系统方面至关重要，并已被证明广泛适用于其他具有多自由度的系统。受统计物理学起源的启发，Chen、Kastoryano与Gilyén [CKG23] 提出了一种连续时间量子热力学类比于Glauber动力学的构造，该构造具有以下特性：(i) 严格满足细致平衡条件，(ii) 可高效实现，(iii) 对于几何局域系统是准局域的。在物理上，他们的构造类似于弱系统-环境相互作用产生的耗散动力学。在本工作中，我们为任意连续时间量子吉布斯采样器提出了一种可高效实现的离散时间对应方案。我们的构造保留了上述理想特性(i)-(iii)，同时不会降低谱隙。此外，我们提出了另一种高度相干的细致平衡动力学的量子广义化方案，该方案类似于另一类从物理推导得到的主方程，并提出了此方案与早期构造之间的平滑插值。更进一步，我们展示了如何使早期的Metropolis风格吉布斯采样器（其在跃迁前后均估算能量）严格满足细致平衡条件。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（准）局域性。最后，我们证明了我们新的高度相干吉布斯采样器在高温下的谱隙为常数，因而具有快速混合特性。我们希望我们对量子Glauber与Metropolis动力学的系统化研究方法将推动其在各个领域的广泛应用。