Belief propagation is a message-passing algorithm, originally developed for approximate inference in graphical models, that estimates local marginal quantities by iteratively passing “messages” between neighboring nodes of a graph rather than exactly accounting for every variable at once. On a graph with no cycles it recovers the exact answer; on a graph with cycles (“loopy” belief propagation) it is only approximate, but in practice it often converges to a useful answer at a fraction of the cost of an exact computation.
Applied to tensor networks, belief propagation approximates the contraction of the network — the step that would otherwise cost exponentially more as the network’s loops and entanglement grow — by treating each tensor as a node and passing normalized messages along the network’s bonds until they converge, trading a controlled, problem-dependent amount of accuracy for a large reduction in computational cost. This makes it possible to simulate two- and three-dimensional quantum systems that an exact tensor-network contraction could not reach.
Related papers
- Tindall, Mello, Fishman, Stoudenmire, and Sels, “Dynamics of disordered quantum systems with two- and three-dimensional tensor networks”, Science 392, 868 (2026) — uses belief propagation as the core approximate-contraction method behind a large-scale classical simulation of disordered spin-glass dynamics. Covered in 2026-w32.
- Bermejo, Villalonga, Ware, Vidal, and Szasz, “Tensor Networks with Belief Propagation Cannot Feasibly Simulate Google’s Quantum Echoes Experiment” (2026) — shows the method’s accuracy is not universal: for circuits whose entanglement grows too fast to compress, belief propagation provably fails. Covered in 2026-w32.