Matrix Product Belief Propagation

• URL: http://arxiv.org/abs/2609.05598v1
• Date: Fri, 04 Sep 2026 18:00:00 GMT
• Title: Matrix Product Belief Propagation
• Abstract summary: "matrix product belief propagation" (MP-BP) is a controlled method for the approximate contraction of two-dimensional tensor networks and graphical models.<n>MP-BP can be understood as a practical implementation of Baxter's corner-transfer-matrix method.<n>We numerically demonstrate the qualitatively improved convergence for a variety of classical and quantum 2D tensor network problems.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce "matrix product belief propagation" (MP-BP) as a controlled method for the approximate contraction of two-dimensional tensor networks and graphical models, which generically leads to a quadratic reduction of errors relative to existing methods: at similar computational effort, the number of accurate digits is asymptotically doubled. For infinite systems, MP-BP can be understood as a practical implementation of Baxter's corner-transfer-matrix method; for reflection-symmetric networks, it further coincides with the boundary matrix-product-state and corner-transfer-matrix renormalization group algorithms as conventionally implemented. In the limit of unit matrix-product rank, MP-BP is equivalent to belief propagation, and more generally, can be understood as a generalization of BP in which the "messages" live on surfaces and are assumed to have a matrix-product factorization. We further extend the quadratic improvement to the computation of expectation values, and numerically demonstrate the qualitatively improved convergence for a variety of classical and quantum 2D tensor network problems.

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