  Effectively compressing and optimizing tensor networks requires reliable
methods for fixing the latent degrees of freedom of the tensors, known as the
gauge. Here we introduce a new algorithm for gauging tensor networks using
belief propagation, a method that was originally formulated for performing
statistical inference on graphical models and has recently found applications
in tensor network algorithms. We show that this method is closely related to
known tensor network gauging methods. It has the practical advantage, however,
that existing belief propagation implementations can be repurposed for tensor
network gauging, and that belief propagation is a very simple algorithm based
on just tensor contractions so it can be easier to implement, optimize, and
generalize. We present numerical evidence and scaling arguments that this
algorithm is faster than existing gauging algorithms, demonstrating its usage
on structured, unstructured, and infinite tensor networks. Additionally, we
apply this method to improve the accuracy of the widely used simple update gate
evolution algorithm.
