Information Geometry of Mean Field Approximation for Quantum Boltzmann Machines

摘要

Mean field theory(MFT), originated in statistical physics, has been widely used both in classical and quantum settings. In particular, mean field approximation(MFA) which is based on the MFT has been extensively used for the classical Boltzmann machine(CBM) and also several authors have discussed its properties in view of information geometry(IG). In this paper, we apply MFA to the quantum Boltzmann machine(QBM) and discuss its properties using the information geometrical concepts. The quantum relative entropy as a quantum divergence function is used for approximation, where e-(exponential) and m-(mixture) projections play an important role. We derive the naive mean field equations for QBMs from the viewpoint of IG. Finally, we outline the formulation which leads to the higher-order MFAs.