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Fast Thermalization from the Eigenstate Thermalization Hypothesis

Chen, Chi-Fang and Brandão, Fernando G. S. L. (2021) Fast Thermalization from the Eigenstate Thermalization Hypothesis. . (Unpublished)

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The Eigenstate Thermalization Hypothesis (ETH) has played a major role in explaining thermodynamic phenomena in quantum systems. However, so far, no connection has been known between ETH and the timescale of thermalization. In this paper, we rigorously show 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 control. We derive a finitely-resolved version of Davies' generator, with explicit error bounds and resource estimates, that describes the joint evolution at finite times. 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 the Hamiltonian can be modeled by random matrix theory below a sufficiently small energy scale. 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 open quantum systems and suggest an alternative formulation of ETH in terms of quantum expanders, which we confirm numerically for small systems.

Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription Paper
Chen, Chi-Fang0000-0001-5589-7896
Brandão, Fernando G. S. L.0000-0003-3866-9378
Additional Information:We thank Charles Xu for early discussions on the topic of this paper. We thank Robert (Hsin-Yuan) Huang for suggesting to check quantum expander properties numerically. We thank Cambyse Rouz  for discussions on approximate tensorization. CFC is supported by Caltech RA fellowship and the Eddleman Fellowship.
Group:Institute for Quantum Information and Matter, AWS Center for Quantum Computing
Funding AgencyGrant Number
Eddleman FellowshipUNSPECIFIED
Record Number:CaltechAUTHORS:20220202-191908990
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:113225
Deposited By: George Porter
Deposited On:02 Feb 2022 22:12
Last Modified:02 Feb 2022 22:12

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