Tensor Networks contraction and the Belief Propagation algorithm

Abstract

Belief Propagation is a well-studied message-passing algorithm that runs over graphical models and can be used for approximate inference and approximation of local marginals. The resulting approximations are equivalent to the Bethe-Peierls approximation of statistical mechanics. Here we show how this algorithm can be adapted to the world of PEPS tensor networks and used as an approximate contraction scheme. We further show that the resultant approximation is equivalent to the ``mean field'' approximation that is used in the Simple-Update algorithm, thereby showing that the latter is a essentially the Bethe-Peierls approximation. This shows that one of the simplest approximate contraction algorithms for tensor networks is equivalent to one of the simplest schemes for approximating marginals in graphical models in general, and paves the way for using improvements of BP as tensor networks algorithms.

RevTeX 4.1, 14 pages, 13 figures. Comments are welcome. Version2: very minor modifications

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• Dynamics of disordered quantum systems with two- and three-dimensional tensor networks
• Loop Series Expansions for Tensor Networks
• Improved real-space parallelizable matrix-product state compression and its application to unitary quantum dynamics simulation
• Solvable Quantum Circuits in Tree+1 Dimensions
• Chemical master equation parameter exploration using DMRG
• Approximate Contraction of Arbitrary Tensor Networks with a Flexible and Efficient Density Matrix Algorithm
• Tensor networks for $p$-spin models
• Nonperturbative decay of bipartite discrete time crystals
• Simulating dynamics of the two-dimensional transverse-field Ising model: a comparative study of large-scale classical numerics
• Evaluating the Limits of QAOA Parameter Transfer at High-Rounds on Sparse Ising Models With Geometrically Local Cubic Terms
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• Tensor Network Loop Cluster Expansions for Quantum Many-Body Problems
• Nonequilibrium steady-state dynamics of Markov processes on graphs