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Belief propagation -- a powerful heuristic method to solve inference problemsinvolving a large number of random variables -- was recently generalized toquantum theory. Like its classical counterpart, this algorithm is exact ontrees when the appropriate independence conditions are met and is expected toprovide reliable approximations when operated on loopy graphs. In this paper,we benchmark the performances of loopy quantum belief propagation (QBP) in thecontext of finite-tempereture quantum many-body physics. Our results indicatethat QBP provides reliable estimates of the high-temperature correlationfunction when the typical loop size in the graph is large. As such, it issuitable e.g. for the study of quantum spin glasses on Bethe lattices and thedecoding of sparse quantum error correction codes.

Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs