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  • Open Access

Beyond Belief Propagation: Cluster-Corrected Tensor Network Contraction with Exponential Convergence

Siddhant Midha1,*,‡ and Yifan F. Zhang2,*,†

  • *These authors contributed equally to this work.
  • †Contact author: sm7456@princeton.edu
  • ‡Contact author: yz4281@princeton.edu
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PRX Quantum 7, 033010 – Published 9 July, 2026

DOI: https://doi.org/10.1103/sdlx-d3tn

Abstract

Tensor network contraction on arbitrary graphs 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 are poorly understood and systematic improvements remain elusive. Here, we develop a rigorous theoretical framework for BP in tensor networks, leveraging insights from statistical mechanics to devise a cluster expansion that systematically improves the BP approximation. We prove that the cluster expansion converges exponentially fast if an object called the loop contribution decays sufficiently fast with the loop size, giving a rigorous error bound on BP. We also provide a simple and efficient algorithm to compute the cluster expansion to arbitrary order. We demonstrate the efficacy of our method on the two-dimensional Ising model, where we find that our method significantly improves upon BP and existing corrective algorithms such as loop series expansion. Our work opens the door to a systematic theory of BP for tensor networks and its applications in decoding quantum error-correcting codes and simulating quantum systems.

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Popular Summary

Simulating quantum systems on classical computers is famously hard: the amount of information needed grows explosively with the number of particles. Tensor networks tame this explosion by storing only the most essential information, but extracting predictions from them can still be extremely expensive, especially for two- and three-dimensional systems. A fast approximate shortcut called belief propagation (BP) has recently gained popularity, yet it has remained unclear when it works, and how it can be improved. This work introduces a systematic way to correct belief propagation by accounting for small, localized packets of information, called clusters, that account for "excitations" that BP misses. Each added cluster trades a bit more computing time for greater accuracy, letting users dial in the precision they need. This approach could make quantum simulations more trustworthy and efficient, with applications ranging from designing new algorithms to correcting errors in future quantum computers.

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PRX Quantum 7, 033010– Published 9 July, 2026
  • Received 24 October 2025
  • Accepted 28 May 2026
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