Information geometrical study of quantum Boltzmann machines
Nihal Yapage · Institutional Repositories DataBase (IRDB) · 2008
In this thesis, we consider quantum extension of the well-known stochastic neuralnetwork model called (classical) Boltzmann machine (CBM) from an informationgeometrical point of view. The new model is called quantum Boltzmann machine(QBM).We investigate some properties and geometrical aspects of QBM analogous tothose of CBM. Furthermore, we study the mean-field approximation for such a modelfrom the information geometrical point of view. This is in some sense motivated bythe application of mean-field approximation for probabilistic inference in the graphicalmodels in the classical probability theory. Although the problem tackled in the presentthesis is somewhat deviated from the major field of quantum information theory, wehave elucidated the relationships among several well-known fields such as statisticalphysics, differential geometry, information theory and statistics using the concepts ofthe new emerging subject of quantum information geometry.We first define QBMs which can be considered as a general class of quantumIsing spin models. The states we consider are assumed to have at most secondorderinteractions with arbitrary but deterministic coupling coefficients. We call sucha state a QBM for the reason that it can be regarded as a quantum extension ofthe equilibrium distribution of CBM. The totality of QBMs is then shown to forma quantum exponential family and thus can be considered as a smooth manifoldhaving similar geometrical structures to those of CBMs. The information geometricalstructure of the manifold of QBMs is discussed and the problem of approximating agiven 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 renewalrule based on that of CBM. The geometrical structure of the totality of SSQBMs isshown to be equivalent to that of the totality of CBMs. Approximation process forSSQBMs is also studied. Finally, we briefly discuss the parameter estimation of aSSQBM.Next, we study the mean-field approximation for QBMs from an information geometricalpoint of view. We elaborate on the significance and usefulness of informationgeometrical concepts, in particular the e-(exponential) and m-(mixture) projections,in studying the naive mean-field approximation for QBMs and derive the naive meanfieldequation explicitly. We also discuss the higher-order corrections to the naivemean-field approximation based on the idea of Plefka expansion in statistical physics.We elucidate the geometrical essence of the corrections and provide the expansioncoefficients with expressions in terms of information geometrical quantities. Here,one may note this work as the information geometrical interpretation of [Ple06] andas the quantum extension of [Tan00].