  Machine learning algorithms often take inspiration from established results
and knowledge from statistical physics. A prototypical example is the Boltzmann
machine algorithm for supervised learning, which utilizes knowledge of
classical thermal partition functions and the Boltzmann distribution. Recently,
a quantum version of the Boltzmann machine was introduced by Amin, et. al.,
however, non-commutativity of quantum operators renders the training process by
minimizing a cost function inefficient. Recent advances in the study of
non-equilibrium quantum integrable systems, which never thermalize, have lead
to the exploration of a wider class of statistical ensembles. These systems may
be described by the so-called generalized Gibbs ensemble (GGE), which
incorporates a number of "effective temperatures". We propose that these GGE's
can be successfully applied as the basis of a Boltzmann-machine-like learning
algorithm, which operates by learning the optimal values of effective
temperatures. We show that the GGE algorithm is an optimal quantum Boltzmann
machine: it is the only quantum machine that circumvents the quantum
training-process problem. We apply a simplified version of the GGE algorithm,
where quantum effects are suppressed, to the classification of handwritten
digits in the MNIST database. While lower error rates can be found with other
state-of-the-art algorithms, we find that our algorithm reaches relatively low
error rates while learning a much smaller number of parameters than would be
needed in a traditional Boltzmann machine, thereby reducing computational cost.
