  Boltzmann machine is a powerful tool for modeling probability distributions
that govern the training data. A thermal equilibrium state is typically used
for Boltzmann machine learning to obtain a suitable probability distribution.
The Boltzmann machine learning consists of calculating the gradient of the loss
function given in terms of the thermal average, which is the most time
consuming procedure. Here, we propose a method to implement the Boltzmann
machine learning by using Noisy Intermediate-Scale Quantum (NISQ) devices. We
prepare an initial pure state that contains all possible computational basis
states with the same amplitude, and apply a variational imaginary time
simulation. Readout of the state after the evolution in the computational basis
approximates the probability distribution of the thermal equilibrium state that
is used for the Boltzmann machine learning. We actually perform the numerical
simulations of our scheme and confirm that the Boltzmann machine learning works
well by our scheme.
