Contracting Tensor Networks with Generalized Belief Propagation

TL;DR

This paper introduces a generalized belief propagation method for tensor network contraction, enabling more accurate and efficient approximations across various complex models and network structures.

Contribution

It extends belief propagation to a hierarchy of overlapping regions, improving tensor network contraction accuracy and efficiency with practical implementations.

Findings

Successfully applied GBP to 2D and 3D tensor networks

Achieved accurate partition function calculations for Ising models

Computed ground state degeneracies and observables in quantum states

Abstract

Recent years have seen a growing interest in the use of belief propagation - an algorithm originally introduced for performing statistical inference on graphical models - for approximate, but highly efficient, tensor network contraction. Here, we detail how to apply generalized belief propagation (GBP) - where messages are passed within a hierarchy of overlapping regions of the tensor network - to approximately contract tensor networks and obtain accurate results. The original belief propagation algorithm is a corner case of this approach, corresponding to a particularly simple choice of regions of the tensor network. We implement the GBP algorithm for a number of different region choices on a range of two- and three-dimensional, infinite and finite tensor networks, solving the corresponding fixed point equations both numerically and, in certain tractable cases, analytically. Our examples include calculating the partition function of the fully frustrated Ising model, computing the ground state degeneracy of three-dimensional ice models, measuring observables on the deformed AKLT quantum state and evaluating the norm of randomly generated tensor network states.