Fast Thermalization from the Eigenstate Thermalization Hypothesis

The Eigenstate Thermalization Hypothesis (ETH) has played a major role in explaining thermodynamic phenomena in quantum systems. However, so far ETH has not been linked to the central question about the timescale of thermalization. In this paper, we rigorously show that ETH does imply that the system thermalizes quickly to the global Gibbs state, whenever it is in contact with an external heat bath. 

    We consider two models of thermalization. In the first, the system is weakly coupled to a bath of (quasi)-free Fermions. We provide a quantitative derivation, with finite error bounds, of the effective Lindbladian governing the joint evolution. We show it is given by a finitely-resolved version of Davies' generator, whose fixed points include the (approximate) Gibbs state. The second is Quantum Metropolis Sampling, a quantum algorithm for preparing Gibbs states on a quantum computer. In both cases, no guarantee for fast convergence was previously known for non-commuting Hamiltonians.

    The key feature of ETH we explore is that below a certain sufficiently small energy scale, the Hamiltonian can be modeled by random matrix theory. We show this gives quantum expander at nearby eigenstates of the Hamiltonian. This then implies fast convergence to the global Gibbs state, by mapping the problem to a one-dimensional classical random walk on the spectrum of the Hamiltonian. Our results explain finite-time thermalization in chaotic quantum systems and suggest a less stringent formulation of ETH in terms of quantum expanders.