Fast Thermalization from the Eigenstate Thermalization Hypothesis

Chi-Fang Chen, Fernando G. S. L. Brandão

Published 2021-12-14, updated 2022-03-07Version 2

The Eigenstate Thermalization Hypothesis (ETH) has played a major role in explaining thermodynamic phenomena in closed quantum systems. However, no connection has been known between ETH and the timescale of thermalization for open system dynamics. This paper rigorously shows that ETH indeed implies fast thermalization to the global Gibbs state. We show fast convergence for two models of thermalization. In the first, the system is weakly coupled to a bath of quasi-free Fermions that we routinely refresh. We derive a finite-time version of Davies' generator, with explicit error bounds and resource estimates, that describes the joint evolution. 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, partly due to technical issues with a finite energy resolution. The critical feature of ETH we exploit is that operators in the energy basis can be modeled by independent random matrices in a near-diagonal band. 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 energy eigenstates. Our results explain finite-time thermalization in chaotic open quantum systems and suggest an alternative formulation of ETH in terms of quantum expanders, which we investigate numerically for small systems.