For PEPS states with loop-decay, BP with cluster corrections approximates local observables exponentially accurately, and loop-decay necessarily implies exponential decay of connected correlations, ruling out BP at critical points.
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A stochastic MCMC sampling method with umbrella sampling provides unbiased loop corrections to belief propagation for exact factorization-based tensor network contraction on loopy graphs with symmetric potentials.
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Stochastic Loop Corrections to Belief Propagation for Tensor Network Contraction
A stochastic MCMC sampling method with umbrella sampling provides unbiased loop corrections to belief propagation for exact factorization-based tensor network contraction on loopy graphs with symmetric potentials.