Quantum Gibbs Sampling: the commuting case

Physical systems in nature very often are in thermal equilibrium. Statistical mechanics provides a microscopic theory justifying the relevance of thermal states of matter. However, fully understanding the ubiquity of this class of states from the laws of quantum theory remains an important topic in theoretical physics. The problem can be broken up into two sets of questions: (i) under what conditions does a system thermalize in the long time limit, and (ii) assuming a system does eventually thermalize, how much time does one have to wait before this is so? Our work is concerned with the latter question in the setting of quantum lattice spin systems. The problem of the speed of thermalization is also of practical relevance in the context of quantum simulators, where one wants to analyze the properties of a real physical system by simulating a controllable idealization of it on a classical or quantum computer. Given that many of the systems which one would want to simulate are thermal, it is an important task to develop simulation and sampling algorithms that can prepare large classes of thermal states of local Hamiltonians. A large body of work has already been done on the classical problem, starting with the development and analysis of Gibbs sampling algorithms of lattice systems called Glauber dynamics, which include the Metropolis and Heat-bath algorithms as spacial cases. A peculiar feature of many of these algorithms is that they often provide reliable results in practice, but a systematic certification of their accuracy and efficiency is often elusive. Although a very hard problem in general, estimation of the convergence time of classical Gibbs samplers has seen a number of breakthroughs in the past few decades. The centerpiece of this theory is a structural theorem which says that the Gibbs state of a local Hamiltonian on a lattice has exponentially decaying correlations if, and only if, the Glauber dynamics are rapidly mixing. In this submission, we extend this main structural theorem to the quantum setting. In this submission we will restrict ourselves to commuting Hamiltonians. It is worth noting that the case of commuting Hamiltonians does not effectively reduce to classical systems, as these allow for intrinsic quantum phenomena, such as topological quantum order. In particular, this setting encompasses all stabilizer Hamiltonians, which have been a useful playground for exploring unique quantum features of many-body systems. The physical relevance of our results is twofold. First, we consider a class of Gibbs samplers (called Davies generators [2]) which can be derived from the weak coupling of a finite quantum system to a large thermal bath. Hence our analysis pertains to the time it takes to reach thermal equilibrium in naturally occurring systems. Secondly, all Gibbs samplers which we consider are local and bounded maps, and therefore can be prepared by dissipative engineering or digital simulation on quantum computers, or quantum simulators [? ].