Message-Passing Enhanced Approximate Contraction of Tensor Networks

Tensor networks play important roles in numerous problems, including quantum physics, statistical physics, and combinatorial optimization. Since exact contraction of tensor networks is a #P-hard problem, developing efficient and accurate approximation methods is of considerable interest. One important approach is bond compression; however, achieving accurate bond truncation requires environmental information that is difficult to approximate well. Another route is to transform the problem into a statistical physics problem and represent it through message passing algorithms using Bethe free energy and related loop corrections, but this approach faces challenges of difficult convergence and strong dependence on graph topology. Therefore, we combine these two approximation ideas by leveraging tensor networks' ability to capture global structure and expand the range of locally exact computations to enhance the stability and accuracy of message passing algorithms, while using message passing to provide more accurate environmental information for tensor network bond compression gauging. This yields a new tensor network approximation contraction method with higher accuracy and broader applicability.