Sign problem in the Bethe approximation

TL;DR

This paper introduces a message-passing algorithm that leverages the Bethe approximation to estimate the energy of interacting fermionic systems, addressing the sign problem and improving upon mean-field results.

Contribution

It connects quantum expectations to classical averages with global interactions and applies the Bethe approximation to improve energy estimates for fermionic models.

Findings

Significant qualitative improvements over mean-field approximation.

Effective estimation of the fermion sign using global interactions.

Successful application to the Hubbard model on random regular graphs.

Abstract

We propose a message-passing algorithm to compute the Hamiltonian expectation with respect to an appropriate class of trial wave functions for an interacting system of fermions. To this end, we connect the quantum expectations to average quantities in a classical system with both local and global interactions, which are related to the variational parameters and use the Bethe approximation to estimate the average energy within the replica-symmetric approximation. The global interactions, which are needed to obtain a good estimation of the average fermion sign, make the average energy a nonlocal function of the variational parameters. We use some heuristic minimization algorithms to find approximate ground states of the Hubbard model on random regular graphs and observe significant qualitative improvements with respect to the mean-field approximation.