Quantum generalizations of Glauber and Metropolis dynamics

Abstract: 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.