Matrix Product Belief Propagation

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.

Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); Computational Physics (physics.comp-ph)

arXiv:2609.05598 [quant-ph]

(or arXiv:2609.05598v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.05598 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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