Beyond Belief Propagation: Cluster-Corrected Tensor Network Contraction with Exponential Convergence
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915578715832320 |
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| author | Midha, Siddhant Zhang, Yifan F. |
| author_facet | Midha, Siddhant Zhang, Yifan F. |
| contents | 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 \emph{cluster expansion} that systematically improves the BP approximation. We prove that the cluster expansion converges exponentially fast if an object called the \emph{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 classical and quantum error-correcting codes and simulating quantum systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beyond Belief Propagation: Cluster-Corrected Tensor Network Contraction with Exponential Convergence Midha, Siddhant Zhang, Yifan F. Quantum Physics Statistical Mechanics 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 \emph{cluster expansion} that systematically improves the BP approximation. We prove that the cluster expansion converges exponentially fast if an object called the \emph{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 classical and quantum error-correcting codes and simulating quantum systems. |
| title | Beyond Belief Propagation: Cluster-Corrected Tensor Network Contraction with Exponential Convergence |
| topic | Quantum Physics Statistical Mechanics |
| url | https://arxiv.org/abs/2510.02290 |