Block belief propagation algorithm for two-dimensional tensor networks

TL;DR

This paper introduces a block belief propagation algorithm for contracting 2D tensor networks, enabling efficient approximation of ground states in quantum many-body systems with comparable accuracy to current leading methods.

Contribution

It presents a novel, flexible, and parallelizable belief propagation-based algorithm for 2D tensor network contraction applicable to finite and infinite systems.

Findings

Achieves accuracy comparable to state-of-the-art methods for 2D Heisenberg and Ising models.

Works efficiently for both finite and infinite systems.

Allows natural parallelization and handling of different unit cells.

Abstract

Belief propagation is a well-studied algorithm for approximating local marginals of multivariate probability distribution over complex networks, while tensor network states are powerful tools for quantum and classical many-body problems. Building on a recent connection between the belief propagation algorithm and the problem of tensor network contraction, we propose a block belief propagation algorithm for contracting two-dimensional tensor networks and approximating the ground state of $2D$ systems. The advantages of our method are three-fold: 1) the same algorithm works for both finite and infinite systems; 2) it allows natural and efficient parallelization; 3) given its flexibility it would allow to deal with different unit cells. As applications, we use our algorithm to study the $2D$ Heisenberg and transverse Ising models, and show that the accuracy of the method is on par with state-of-the-art results.