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- W2276711936 abstract "In this thesis, we consider quantum extension of the well-known stochastic neural network model called (classical) Boltzmann machine (CBM) from an information geometrical point of view. The new model is called quantum Boltzmann machine (QBM). We investigate some properties and geometrical aspects of QBM analogous to those of CBM. Furthermore, we study the mean-field approximation for such a model from the information geometrical point of view. This is in some sense motivated by the application of mean-field approximation for probabilistic inference in the graphical models in the classical probability theory. Although the problem tackled in the present thesis is somewhat deviated from the major field of quantum information theory, we have elucidated the relationships among several well-known fields such as statistical physics, differential geometry, information theory and statistics using the concepts of the new emerging subject of quantum information geometry. We first define QBMs which can be considered as a general class of quantum Ising spin models. The states we consider are assumed to have at most secondorder interactions with arbitrary but deterministic coupling coefficients. We call such a state a QBM for the reason that it can be regarded as a quantum extension of the equilibrium distribution of CBM. The totality of QBMs is then shown to form a quantum exponential family and thus can be considered as a smooth manifold having similar geometrical structures to those of CBMs. The information geometrical structure of the manifold of QBMs is discussed and the problem of approximating a given quantum state (density operator) by a QBM is also treated. We also define a restricted class of QBMs called the strongly separable QBMs (SSQBMs). We consider the dynamics of SSQBMs and propose a new state renewal rule based on that of CBM. The geometrical structure of the totality of SSQBMs is shown to be equivalent to that of the totality of CBMs. Approximation process for SSQBMs is also studied. Finally, we briefly discuss the parameter estimation of a SSQBM. Next, we study the mean-field approximation for QBMs from an information geometrical point of view. We elaborate on the significance and usefulness of information" @default.
- W2276711936 created "2016-06-24" @default.
- W2276711936 creator A5002974980 @default.
- W2276711936 date "2008-01-01" @default.
- W2276711936 modified "2023-09-27" @default.
- W2276711936 title "Information geometrical study of quantum Boltzmann machines" @default.
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