Algorithmic Locality via Provable Convergence in Quantum Tensor Networks

TL;DR

This paper establishes a rigorous theoretical foundation for belief propagation in tensor networks, demonstrating efficient computation and a locality property that allows local updates to influence global states minimally.

Contribution

It introduces the first end-to-end theory of tensor network belief propagation for strongly injective projected entangled pair states, proving efficiency and locality of the algorithm.

Findings

BP fixed points can be found efficiently when injectivity exceeds a threshold

Local perturbations have rapidly decaying influence, enabling local updates

Local expectation values can be approximated from local data with controlled error

Abstract

Belief propagation has recently emerged as a powerful framework for evaluating tensor networks in higher dimensions, combining computational efficiency with provable analytical guarantees. In this work, we develop the first end-to-end theory of tensor network belief propagation for a class of projected entangled pair states satisfying \emph{strong injectivity}. We show that when the injectivity parameter exceeds a constant threshold, BP fixed points can be found efficiently, and a cluster-corrected BP algorithm computes physical quantities to $1/\mathrm{poly}(N)$ error in $\mathrm{poly}(N)$ time for an $N$ qubit system. We identify a striking phenomenon we term \emph{algorithmic locality}: local perturbations of the tensor network affect the BP fixed point with an influence decaying rapidly with distance. As a result, updates to the fixed point after a local perturbation can be carried out using only local recomputation. Moreover, through the cluster expansion, this locality extends to observables, implying that local expectation values can be approximated from local data with controlled accuracy. Our results provide the first rigorous guarantee for the effectiveness of tensor-network belief propagation on a wide class of many-body states, bridging a gap between widely used numerical practice and provable algorithmic performance.