  Boltzmann machines are the basis of several deep learning methods that have
been successfully applied to both supervised and unsupervised machine learning
tasks. These models assume that a dataset is generated according to a Boltzmann
distribution, and the goal of the training procedure is to learn the set of
parameters that most closely match the input data distribution. Training such
models is difficult due to the intractability of traditional sampling
techniques, and proposals using quantum annealers for sampling hope to mitigate
the sampling cost. However, real physical devices will inevitably be coupled to
the environment, and the strength of this coupling affects the effective
temperature of the distributions from which a quantum annealer samples. To
counteract this problem, error correction schemes that can effectively reduce
the temperature are needed if there is to be some benefit in using quantum
annealing for problems at a larger scale, where we might expect the effective
temperature of the device to not be sufficiently low. To this end, we have
applied nested quantum annealing correction (NQAC) to do unsupervised learning
with a small bars and stripes dataset, and to do supervised learning with a
coarse-grained MNIST dataset. For both datasets we demonstrate improved
training and a concomitant effective temperature reduction at higher noise
levels relative to the unencoded case. We also find better performance overall
with longer anneal times and offer an interpretation of the results based on a
comparison to simulated quantum annealing and spin vector Monte Carlo. A
counterintuitive aspect of our results is that the output distribution
generally becomes less Gibbs-like with increasing nesting level and increasing
anneal times, which shows that improved training performance can be achieved
without equilibration to the target Gibbs distribution.
