[en] We study the mean-field approximation for a general class of quantum Ising spin states from an information geometrical point of view. The states we consider are assumed to have at most second-order interactions with arbitrary but deterministic coupling coefficients. We call such a state a quantum Boltzmann machine (QBM) for the reason that it can be regarded as a quantum extension of the equilibrium distribution of a (classical) Boltzmann machine (CBM), which is a well-known stochastic neural network model. 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. We elaborate on the significance and usefulness of information geometrical concepts, in particular the e- and m-projections, in studying the naive mean-field approximation for QBMs. We also discuss the higher-order corrections to the naive mean-field approximation based on the idea of the Plefka expansion in statistical physics. We elucidate the geometrical essence of the corrections and provide the expansion coefficients with expressions in terms of information geometrical quantities. Here, one may note this work as the information geometrical interpretation of (Plefka T 2006 Phys. Rev. E 73 016129) and as the quantum extension of (Tanaka T 2000 Neural Comput. 12 1951-68)