Eigenstate Thermalization Implies Efficient Quantum Metropolis Sampling
September 16th, 2019 CHARLES XU Caltech, USA

The problem of preparing or sampling from Gibbs states of many-body quantum systems is of great theoretical and practical interest, and a number of approximate sampling protocols have been proposed. These include the quantum Metropolis algorithm of Temme et al. which executes a random walk on energy eigenstates that converges to the Gibbs distribution, in a time scaling inversely with the gap of the transition matrix. In general the dependence of this runtime on system size N is unclear, but we are interested in conditions that allow for efficient Gibbs sampling. In this work we show that the Eigenstate Thermalization Hypothesis (ETH) suffices for this. That is, if the Hamiltonian and the operators implementing each Metropolis step obey ETH, the algorithm converges to the Gibbs state in poly(N) time. This can be argued heuristically by “coarse-graining” the spectrum, but our main result is a rigorous calculation of the random walk’s conductance, which gives a poly(N) upper bound on t he mixing time. We conclude that ETH implies fast thermalization of many-body quantum systems not just locally under unitary evolution, but also globally under the Metropolis algorithm’s model of system-bath dynamics.

Seminar, September 16, 2019, 15:00. ICFO’s Seminar Room

Hosted by Prof. Toni Acín