Quantum Graphical Models and Belief Propagation

TL;DR

This paper extends classical belief propagation algorithms to quantum graphical models, introducing new quantum structures, properties, and a quantum belief propagation algorithm with applications in quantum error correction and many-body systems.

Contribution

It generalizes classical probabilistic graphical models to the quantum domain, defining quantum analogs and analyzing their properties and algorithms.

Findings

Quantum belief propagation converges on tree-structured networks.

Quantum Markov Networks exhibit properties similar to classical cases.

Applications include quantum error correction and simulation of quantum systems.

Abstract

Belief Propagation algorithms acting on Graphical Models of classical probability distributions, such as Markov Networks, Factor Graphs and Bayesian Networks, are amongst the most powerful known methods for deriving probabilistic inferences amongst large numbers of random variables. This paper presents a generalization of these concepts and methods to the quantum case, based on the idea that quantum theory can be thought of as a noncommutative, operator-valued, generalization of classical probability theory. Some novel characterizations of quantum conditional independence are derived, and definitions of Quantum n-Bifactor Networks, Markov Networks, Factor Graphs and Bayesian Networks are proposed. The structure of Quantum Markov Networks is investigated and some partial characterization results are obtained, along the lines of the Hammersely-Clifford theorem. A Quantum Belief Propagation algorithm is presented and is shown to converge on 1-Bifactor Networks and Markov Networks when the underlying graph is a tree. The use of Quantum Belief Propagation as a heuristic algorithm in cases where it is not known to converge is discussed. Applications to decoding quantum error correcting codes and to the simulation of many-body quantum systems are described.