Existence and efficient findability of belief-propagation fixed points for tensor networks

Determine precise classes of tensor networks, such as projected entangled pair states (PEPS) on arbitrary graphs, for which belief-propagation message-passing equations admit fixed points and establish algorithms with rigorous convergence guarantees to efficiently find such fixed points; in particular, ascertain conditions under which fixed points exist and message-passing converges for tensor-network contraction.

Background

The cluster and cluster–cumulant expansions developed in the paper provide rigorous error guarantees only when expanded around a suitable belief-propagation (BP) fixed point that exhibits loop decay. In practice, however, tensor networks can possess multiple BP fixed points, and standard message-passing may converge to an inappropriate stable solution (e.g., the symmetry-broken solution in a confusion regime) or fail to converge.

The authors emphasize that expanding around a poor fixed point invalidates the guarantees, highlighting a foundational algorithmic gap: when do fixed points exist for a given tensor network, and when can they be efficiently and reliably found by message-passing with provable convergence? This gap motivates a formal characterization of existence and convergence conditions for BP on tensor networks.