Belief Propagation and Tensor Network Expansions for Many-Body Quantum Systems: Rigorous Results and Fundamental Limits

TL;DR

This paper rigorously analyzes belief propagation's effectiveness in approximating local observables in many-body quantum systems represented by PEPS, establishing conditions for success and failure.

Contribution

It provides the first rigorous criteria linking loop-decay conditions to the exponential decay of correlations and the success of BP in quantum tensor networks.

Findings

BP with cluster corrections approximates local observables with exponentially small error in gapped phases.

Loop-decay condition implies exponential decay of correlations, limiting BP's success at critical points.

Numerical simulations confirm analytical predictions, showing accuracy in gapped phases and failure near criticality.

Abstract

Belief propagation (BP) provides a scalable heuristic for contracting tensor networks on loopy graphs, but its success in quantum many-body settings has largely rested on empirical evidence. Developing upon a recently introduced cluster-expansion framework for tensor networks, we rigorously study the applicability of BP to many-body quantum systems. For a state represented as a PEPS satisfying a ``loop-decay" condition, we prove that BP supplemented by cluster corrections approximates local observables with exponentially small relative error, and we give explicit formulas expressing local expectation values as BP predictions dressed by connected clusters intersecting the observable region. This representation establishes a direct link between cluster corrections and physical correlation functions. As a result, we show that ``loop-decay" \emph{necessarily implies} exponential decay of connected correlations, yielding sharp, rigorous criteria for when BP can and cannot succeed, and ruling out its validity at critical points. Numerical simulations of the two- and three-dimensional transverse field Ising model at zero and finite temperature confirm our analytical predictions, demonstrating quantitative accuracy deep in gapped phases and systematic failure near criticality.