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Machine learning algorithms often take inspiration from established resultsand knowledge from statistical physics. A prototypical example is the Boltzmannmachine algorithm for supervised learning, which utilizes knowledge ofclassical 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 byminimizing a cost function inefficient. Recent advances in the study ofnon-equilibrium quantum integrable systems, which never thermalize, have leadto the exploration of a wider class of statistical ensembles. These systems maybe described by the so-called generalized Gibbs ensemble (GGE), whichincorporates a number of "effective temperatures". We propose that these GGE'scan be successfully applied as the basis of a Boltzmann-machine-like learningalgorithm, which operates by learning the optimal values of effectivetemperatures. We show that the GGE algorithm is an optimal quantum Boltzmannmachine: it is the only quantum machine that circumvents the quantumtraining-process problem. We apply a simplified version of the GGE algorithm,where quantum effects are suppressed, to the classification of handwrittendigits in the MNIST database. While lower error rates can be found with otherstate-of-the-art algorithms, we find that our algorithm reaches relatively lowerror rates while learning a much smaller number of parameters than would beneeded in a traditional Boltzmann machine, thereby reducing computational cost.

Machine learning algorithms based on generalized Gibbs ensembles