Loop Series Expansions for Tensor Networks

TL;DR

This paper introduces a loop series expansion method to systematically improve belief propagation approximations for tensor network contractions, significantly enhancing accuracy with minimal additional computational cost.

Contribution

It presents a novel framework for expanding tensor networks into a hierarchy of component networks, improving contraction accuracy over standard BP.

Findings

Improves contraction accuracy of tensor networks by several orders of magnitude.

Applicable to iPEPS representing ground states of AKLT models and random tensors.

Provides a scalable approach for accurate tensor network evaluation beyond existing methods.

Abstract

Belief propagation (BP) can be a useful tool to approximately contract a tensor network, provided that the contributions from any closed loops in the network are sufficiently weak. In this manuscript we describe how a loop series expansion can be applied to systematically improve the accuracy of a BP approximation to a tensor network contraction, in principle converging arbitrarily close to the exact result. More generally, our result provides a framework for expanding a tensor network as a sum of component networks in a hierarchy of increasing complexity. We benchmark this proposal for the contraction of iPEPS, either representing the ground state of an AKLT model or with randomly defined tensors, where it is shown to improve in accuracy over standard BP by several orders of magnitude whilst incurring only a minor increase in computational cost. These results indicate that the proposed series expansions could be a useful tool to accurately evaluate tensor networks in cases that otherwise exceed the limits of established contraction routines.