For PEPS with strong injectivity above a threshold, belief propagation finds fixed points efficiently and cluster-corrected BP approximates observables to 1/poly(N) error in poly(N) time, with local perturbations affecting the fixed point only locally.
Pearl,Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference(Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1988)
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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.
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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Algorithmic Locality via Provable Convergence in Quantum Tensor Networks
For PEPS with strong injectivity above a threshold, belief propagation finds fixed points efficiently and cluster-corrected BP approximates observables to 1/poly(N) error in poly(N) time, with local perturbations affecting the fixed point only locally.
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Belief Propagation and Tensor Network Expansions for Many-Body Quantum Systems: Rigorous Results and Fundamental Limits
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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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.