Block belief propagation algorithm for two-dimensional tensor networks

Belief propagation is a well-studied algorithm for approximating local marginals of multivariate probability distribution over complex networks, while tensor network states are powerful tools for quantum and classical many-body problems. Building on a recent connection between the belief propagation algorithm and the problem of tensor network contraction, we propose a block belief propagation algorithm for contracting two-dimensional (2D) tensor networks and approximating the ground state of 2D systems. The advantages of our method are threefold: (1) the same algorithm works for both finite and infinite systems; (2) it allows natural and efficient parallelization; and (3) given its flexibility, it would allow us to deal with different unit cells. As applications, we use our algorithm to study the 2D Heisenberg and transverse Ising models, and show that the accuracy of the method is on par with state-of-the-art results.