  Quantum Boltzmann machines (QBMs) are machine-learning models for both
classical and quantum data. We give an operational definition of QBM learning
in terms of the difference in expectation values between the model and target,
taking into account the polynomial size of the data set. By using the relative
entropy as a loss function this problem can be solved without encountering
barren plateaus. We prove that a solution can be obtained with stochastic
gradient descent using at most a polynomial number of Gibbs states. We also
prove that pre-training on a subset of the QBM parameters can only lower the
sample complexity bounds. In particular, we give pre-training strategies based
on mean-field, Gaussian Fermionic, and geometrically local Hamiltonians. We
verify these models and our theoretical findings numerically on a quantum and a
classical data set. Our results establish that QBMs are promising machine
learning models.
