Efficient and accurate tensor network simulations of quantum systems using belief propagation and cluster expansions

Tensor networks provide a powerful ansatz for quantum many-body systems, but typical methods can become prohibitively expensive, especially in two or more dimensions. Cheaper algorithms such as belief propagation and simple update allow for comparitively large amounts of entanglement, at the cost of only directly addressing tree like correlations. We explore and demonstrate methods that allow one to affordably include loopy correlations back into such calculations, significantly increasing their accuracy.