  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.
