  We continue our numerical study of quantum belief propagation initiated in
[Phys. Rev. A, 77 (2008), p. 052318]. We demonstrate how the method can be
expressed in terms of an effective thermal potential that materializes when the
system presents quantum correlations, but is insensitive to classical
correlations. The thermal potential provides an efficient means to assess the
precision of belief propagation on graphs with no loops. We illustrate these
concepts using the one-dimensional quantum Ising model and compare our results
with exact solutions. We also use the method to study the transverse field
quantum Ising spin glass for which we obtain a phase diagram that is largely in
agreement with the one obtained in [arXiv:0706.4391] using a different
approach. Finally, we introduce the coarse grained belief propagation (CGBP)
algorithm to improve belief propagation at low temperatures. This method
combines the reliability of belief propagation at high temperatures with the
ability of entanglement renormalization to efficiently describe low energy
subspaces of quantum systems with local interactions. With CGBP, thermodynamic
properties of quantum systems can be calculated with a high degree of accuracy
at all temperatures.
