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.
Related
- Matrix-Product Belief Propagation for continuous-state-space variables
- Dithered Belief Propagation for Real-Time Quantum Memory Decoding
- Logical Neural Belief Propagation for Linear-Complexity Decoding of Surface Codes
- Joint Trust-Distrust Structure Shapes Belief Propagation and Collective Dynamics
- DMRG using Belief Propagation
BibTeX
@misc{woolls2026matrix,
title = {{Matrix Product Belief Propagation}},
author = {Gabriel Woolls and Shahin Jahanbani and Adarsh Pashikanti and Matthew T. Fishman and Joseph Tindall and Michael P. Zaletel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05598},
eprint = {2609.05598},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05598},
}
RIS
TY - UNPB
TI - Matrix Product Belief Propagation
AU - Gabriel Woolls
AU - Shahin Jahanbani
AU - Adarsh Pashikanti
AU - Matthew T. Fishman
AU - Joseph Tindall
AU - Michael P. Zaletel
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05598
UR - https://arxiv.org/abs/2609.05598
AB - 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.
ER -