Quantum Graphical Models and Belief Propagation

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.

58 pages, 9 figures

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• Determining the structure of real-space entanglement spectrum from approximate conditional independence
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• Double-Edge Factor Graphs: Definition, Properties, and Examples
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• Cavity approach to variational quantum mechanics
• Loop Series Expansions for Tensor Networks
• Algorithms for the Markov Entropy Decomposition
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• Low-temperature excitations within the Bethe approximation
• Quantum codes from classical graphical models
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Investigation of commuting Hamiltonian in quantum Markov network
• Bethe free-energy approximations for disordered quantum systems
• Error Correction for Reliable Quantum Computing
• Quantum Cluster Variational Method and Message Passing Algorithms Revisited
• Explaining quantum correlations through evolution of causal models
• Strong decay of correlations for Gibbs states in any dimension
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