  quantum system interacting with other quantum systems experiences these other
systems asan effective environment. The environment is the result of
integrating out all the other degrees of freedom in the network, and can be
represented by a Feynman-Vernon influence functional (IF)acting on system of
interest. A network is characterized by the constitutive systems, how they
interact, and the topology of those interactions. Here we show that for
networks having the topology of locally tree-like graphs, the Feynman-Vernon
influence functional can be determined in a new version of the cavity or Belief
Propagation (BP) method. In the BP update stage, cavity IFs are mapped to
cavity IFs, while in the BP output stage cavity IFs are combined to output IFs.
We compute the fixed point of of this version of BP for harmonic oscillator
systems interacting uniformly. We discuss Replica Symmetry and the effects of
disorder in this context.
