Variational Microcanonical Estimator

TL;DR

This paper introduces a variational quantum algorithm to estimate microcanonical expectation values in quantum many-body systems, leveraging weakly entangled superpositions and ensemble averaging to improve accuracy and address eigenstate thermalization hypothesis challenges.

Contribution

It presents a novel variational approach for microcanonical estimation that generates superpositions of eigenstates and analyzes error sources related to eigenstate thermalization.

Findings

Algorithm converges for shallow circuits proportional to system size.

Accurate thermal estimates achieved at intermediate energy densities.

Error analysis links to eigenstate thermalization hypothesis and matrix element behavior.

Abstract

We propose a variational quantum algorithm for estimating microcanonical expectation values in models obeying the eigenstate thermalization hypothesis. Using a relaxed criterion for convergence of the variational optimization loop, the algorithm generates weakly entangled superpositions of eigenstates at a given target energy density. An ensemble of these variational states is then used to estimate microcanonical averages of local operators, with an error whose dominant contribution decreases initially as a power law in the size of the ensemble and is ultimately limited by a small bias. We apply the algorithm to the one-dimensional mixed-field Ising model, where it converges for ansatz circuits of depth roughly linear in system size. The most accurate thermal estimates are produced for intermediate energy densities. In our error analysis, we find connections with recent works investigating the underpinnings of the eigenstate thermalization hypothesis. In particular, the failure of energy-basis matrix elements of local operators to behave as \textit{independent} random variables is a potential source of error that the algorithm can overcome by averaging over an ensemble of variational states.