Probabilistic information processing and statistical mechanics of information by optimal controlling of quantum fluctuation

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We investigated error-correcting codes described by infinite range Ising spin glasses in a quantum-mechanical transverse field. We evaluated the statistical performance for the Bayes-optimal solution by making use of the replica method and discussed the relationship between the Shannon's bound and the critical point at which the Ferromagnet-SpinGlass phase transition takes place. We also constructed a quantum-mechanical variant of the TAP equation to achieve the performance near the Shannon's bound. We next proposed a theoretical framework to describe the dynamical process of the system evolving according to the quantum Monte Carlo method and derived the differential equations with respect to the macroscopic quantities such as a magnetization from the microscopic master equation. We clarified the validity of the static approximation for some limited cases. We also obtained several results in the topic of spin wave theoretical analysis of quantum annealing with degeneracy, phase transition in the quantum anti-ferromagnets and the so-called digital halftoning by collaborating researchers in and out of our research project.