Abstract:
We apply the concepts of information geometry to study the mean-field
approximation for a general class of quantum statistical models namely the
third-order quantum Boltzmann machines (QBMs). The states we consider
are assumed to have at most third-order interactions with deterministic
coupling coefficients. The totality of such states 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 thirdorder
classical and quantum Boltzmann machines and demonstrate how
some information geometrical concepts, particularly, exponential and
mixture projections are useful in this case. It is obvious that our results for
third-order classical Boltzmann machines (GBMs) and QBMs emphasize
the validity and the importance of information geometrical point of view for
higher dimensional classical and quantum statistical models.
