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

Quantum Physics [Submitted on 3 Apr 2026] Belief Propagation and Tensor Network Expansions for Many-Body Quantum Systems: Rigorous Results and Fundamental Limits Siddhant Midha, Grace M. Sommers, Joseph Tindall, Dmitry A. Abanin 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. Comments: 13 pages main text + supplementary, comments welcome Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech) Cite as: arXiv:2604.03228 [quant-ph]   (or arXiv:2604.03228v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.03228 Focus to learn more Submission history From: Siddhant Midha [view email] [v1] Fri, 3 Apr 2026 17:59:09 UTC (891 KB) Access Paper: view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cond-mat cond-mat.stat-mech References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)