  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 tensor networks
and approximating the ground state of $2D$ systems. The advantages of our
method are three-fold: 1) the same algorithm works for both finite and infinite
systems; 2) it allows natural and efficient parallelization; 3) given its
flexibility it would allow 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.
