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We present a general framework to study quantum disordered systems in thecontext of the Kikuchi's Cluster Variational Method (CVM). The method relies inthe solution of message passing-like equations for single instances or in theiterative solution of complex population dynamic algorithms for an average casescenario. We first show how a standard application of the Kikuchi's ClusterVariational Method can be easily translated to message passing equations forspecific instances of the disordered system. We then present an "ad-hoc"extension of these equations to a population dynamic algorithm representing anaverage case scenario. At the Bethe level, these equations are equivalent tothe dynamic population equations that can be derived from a proper CavityAnsatz. However, at the plaquette approximation, the interpretation is moresubtle and we discuss it taking also into account previous results in classicaldisordered models. Moreover, we develop a formalism to properly deal with theaverage case scenario using a Replica-Symmetric ansatz within this CVM forquantum disordered systems. Finally, we present and discuss numerical solutionsof the different approximations for the Quantum Transverse Ising model and theQuantum Random Field Ising model in two dimensional lattices.

Quantum Cluster Variational Method and Message Passing Algorithms Revisited