Quantum Graphical Models and Belief Propagation

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.Comment: 58 pages, 9 figure

Bayesian Symbol Detection in Wireless Relay Networks via Likelihood-Free Inference

This paper presents a general stochastic model developed for a class of cooperative wireless relay networks, in which imperfect knowledge of the channel state information at the destination node is assumed. The framework incorporates multiple relay nodes operating under general known non-linear processing functions. When a non-linear relay function is considered, the likelihood function is generally intractable resulting in the maximum likelihood and the maximum a posteriori detectors not admitting closed form solutions. We illustrate our methodology to overcome this intractability under the example of a popular optimal non-linear relay function choice and demonstrate how our algorithms are capable of solving the previously intractable detection problem. Overcoming this intractability involves development of specialised Bayesian models. We develop three novel algorithms to perform detection for this Bayesian model, these include a Markov chain Monte Carlo Approximate Bayesian Computation (MCMC-ABC) approach; an Auxiliary Variable MCMC (MCMC-AV) approach; and a Suboptimal Exhaustive Search Zero Forcing (SES-ZF) approach. Finally, numerical examples comparing the symbol error rate (SER) performance versus signal to noise ratio (SNR) of the three detection algorithms are studied in simulated examples