Beyond BP: Cluster Corrected Tensor Network Contraction with Exponential Convergence

Tensor network contraction is a fundamental computational challenge with applications ranging from quantum simulation to error correction. While belief propagation (BP) provides a powerful approximation algorithm for this task, its accuracy limitations remain poorly understood, and systematic improvements have been elusive. In this talk, we present a rigorous theoretical framework for BP in tensor networks, drawing on insights from statistical mechanics to construct a cluster expansion that systematically improves the BP approximation. We prove that this cluster expansion converges exponentially fast under broadly applicable conditions, yielding (1) a rigorous error bound for BP and (2) an algorithm for systematically improving BP with exponential convergence. We demonstrate the effectiveness of our approach by computing thermodynamic properties of the two-dimensional Ising model and discuss potential applications in decoding quantum error-correcting codes and simulating quantum many-body systems.