Quantum Graphical Models and Belief Propagation Export

@misc{citeulike:1558066, 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.}, archivePrefix = {arXiv}, author = {Leifer, Matthew and Poulin, David}, citeulike-article-id = {1558066}, citeulike-linkout-0 = {http://arxiv.org/abs/0708.1337}, citeulike-linkout-1 = {http://arxiv.org/pdf/0708.1337}, day = {9}, eprint = {0708.1337}, keywords = {belief-propagation, chain, graphical-models, information, information-theory, markov, physics, propagation, quantum, quantum-information, trees}, month = {Aug}, posted-at = {2007-09-07 18:38:26}, priority = {2}, title = {Quantum Graphical Models and Belief Propagation}, url = {http://arxiv.org/abs/0708.1337}, year = {2007} }

TY - ELEC ID - citeulike:1558066 L3 - citeulike-article-id:1558066 N2 - 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. TI - Quantum Graphical Models and Belief Propagation KW - belief-propagation KW - chain KW - graphical-models KW - information KW - information-theory KW - markov KW - physics KW - propagation KW - quantum KW - quantum-information KW - trees AU - Leifer, Matthew AU - Poulin, David PY - 2007/Aug/09/ UR - http://arxiv.org/abs/0708.1337 ER -