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.
Highlights
There is a natural connection between classical probabilistic systems of many random variables and quantum many-body systems
We have defined the belief propagation method for PEPS contraction, which can be viewed as the ordinary Belief propagation (BP) method applied for double-edge factor graphs (DEFGs) [28] that are derived from the PEPS tensor network
Just as in ordinary graphical models, the fixed points of the BP iterations correspond to stationary points of the Bethe free energy, which is defined for the underlying PEPS tensor networks (TNs)
Summary
There is a natural connection between classical probabilistic systems of many random variables and quantum many-body systems. In this paper we show how an important class of inference and marginalization algorithms for graphical models, called belief propagation (BP), can be adapted and used for approximate tensor network contraction. Since the underlying approximation in the BP algorithm is the Bethe-Peierls approximation, our results imply that this type of approximation is at the center of the simple-update method It motivates the study of various improvements of the BP algorithms as potential tensor network contraction algorithms
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