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 projected-entangled-pair-state 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 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 belief propagation as tensor networks algorithms.

中文翻译：

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张量网络收缩和置信度传播算法

信念传播是一种经过充分研究的消息传递算法，可在图形模型上运行，并可用于局部边界的近似推断和近似。所得的近似值等效于统计力学的Bethe-Peierls近似值。在这里，我们展示了该算法如何适应投影纠缠对状态张量网络的世界，并用作近似压缩方案。我们进一步表明，所得近似值等效于简单更新算法中使用的“均值字段”近似值，从而表明后者本质上是Bethe-Peierls近似值。