Quantum Cluster Variational Method and Message Passing Algorithms Revisited

TL;DR

This paper develops a framework combining quantum disordered systems with Kikuchi's Cluster Variational Method, translating it into message passing equations and population dynamics, and applies it to quantum Ising models.

Contribution

It introduces a novel extension of the CVM to quantum disordered systems using message passing and population dynamics, including a formalism for the average case scenario with a Replica-Symmetric ansatz.

Findings

Numerical solutions for quantum Ising models in 2D lattices.

Extension of message passing to quantum disordered systems.

Analysis of different approximations within the CVM framework.

Abstract

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