Stochastic Loop Corrections to Belief Propagation for Tensor Network Contraction

TL;DR

This paper introduces a stochastic method to correct belief propagation errors in tensor network contraction, improving accuracy in complex models like the 2D Ising model.

Contribution

It presents a novel hybrid approach combining stochastic sampling and belief propagation to accurately account for loop corrections in tensor network calculations.

Findings

Achieves unbiased estimates with controllable error across parameters

Successfully applied to the 2D ferromagnetic Ising model

Generalizes to any pairwise Markov random field with symmetric potentials

Abstract

Tensor network contraction is a fundamental computational challenge underlying quantum many-body physics, statistical mechanics, and machine learning. Belief propagation (BP) provides an efficient approximate solution, but introduces systematic errors on graphs with loops. Here, we introduce a hybrid method that achieves accurate results by stochastically sampling loop corrections to BP and showcase our method by applying it to the two-dimensional ferromagnetic Ising model. For any pairwise Markov random field with symmetric edge potentials, our approach exploits an exact factorization of the partition function into the BP contribution and a loop correction factor summing over all valid loop configurations, weighted by edge weights derived directly from the potentials. We sample this sum using Markov chain Monte Carlo with moves that preserve the loop constraint, combined with umbrella sampling to ensure efficient exploration across all correlation strengths. Our stochastic approach provides unbiased estimates with controllable statistical error in any parameter regime.