Mean-field equations for higher-order quantum statistical models : an information geometric approach

This work is a simple extension of \cite{NNjpa}. We apply the concepts of information geometry to study the mean-field approximation for a general class of quantum statistical models namely the higher-order quantum Boltzmann machines (QBMs). The states we consider are assumed to have at most third-order interactions with deterministic coupling coefficients. Such states, taken together, can be shown to form a quantum exponential family and thus can be viewed as a smooth manifold. In our work, we explicitly obtain naive mean-field equations for the third-order classical and quantum Boltzmann machines and demonstrate how some information geometrical concepts, particularly, exponential and mixture projections used to study the naive mean-field approximation in \cite{NNjpa} can be extended to a more general case. Though our results do not differ much from those in \cite{NNjpa}, we emphasize the validity and the importance of information geometrical point of view for higher dimensional classical and quantum statistical models.