A Loopy Belief Propagation Algorithm for Tensor Network Contraction

Tensor networks (TNs) offer an efficient approximate method for representing and simulating many-body quantum systems. However, their contraction remains a challenging task for networks representing two or three-dimensional systems or networks containing loops. In recent years, a connection has been established between TNs and probabilistic graphical models (PGMs), which are graphical representations of classical multivariate probability distributions. This connection allows the use of message-passing algorithms such as Belief Propagation (BP) to perform approximate TN contraction. In this thesis, we present an adaptation of the Loopy-BP algorithm, a recent improvement to BP, to the TN world and demonstrate its  effectiveness in efficiently contracting two-dimensional TNs. We apply this algorithm to simulate various quantum systems through imaginary time evolution, and show that it can reliably approximate the ground state of these systems.