Using the Belief Propagation algorithm for finding a Tensor-Network approximation of many-body ground states

Tensor Networks (TN) are a mathematical framework that enables an efficient description of many-body quantum states using a network of interconnected tensors. In the last decade, there has been a significant progress in the usage of TN in a large number of fields such as condensed matter, cosmology, quantum information, machine learning etc. One of the main challenges in the TN world is the contraction of TN in high spatial dimensions (>1D), which can take an exponential time. This presents critical issues in TN optimizations, for example when trying to find a TN approximation to a ground state of a many-body Hamiltonian.  Over the years, many TN algorithms have been developed for dealing with this problem, which all have their strengths and their weaknesses. In this talk I will present a new approach for dealing with TN contraction. It is based on a well-known method from the field of statistical mechanics and computer science, which is called Belief Propagation (BP). Originally, this method was developed for approximating the marginals of many-body classical Gibbs distributions, or, more generally, Probabilistic Graphical Models (PGM). As both problems involve a summation over an exponential number of configurations, it turns that the BP algorithm can be imported to the quantum setup. To that aim, I will introduce a new type of PGM that supports the quantum behavior of a TN. Then, based on this connection, I will present the Belief Propagation Update (BPU) algorithm for the contraction of TN and benchmark it against other well-known TN  algorithms on a 4x4 and 10x10 Antiferromagnetic Heisenberg model.