  Quantum Boltzmann machines are natural quantum generalizations of Boltzmann
machines that are expected to be more expressive than their classical
counterparts, as evidenced both numerically for small systems and
asymptotically under various complexity theoretic assumptions. However,
training quantum Boltzmann machines using gradient-based methods requires
sampling observables in quantum thermal distributions, a problem that is
NP-hard. In this work, we find that the locality of the gradient observables
gives rise to an efficient sampling method based on the Eigenstate
Thermalization Hypothesis, and thus through Hamiltonian simulation an efficient
method for training quantum Boltzmann machines on near-term quantum devices.
Furthermore, under realistic assumptions on the moments of the data
distribution to be modeled, the distribution sampled using our algorithm is
approximately the same as that of an ideal quantum Boltzmann machine. We
demonstrate numerically that under the proposed training scheme, quantum
Boltzmann machines capture multimodal Bernoulli distributions better than
classical restricted Boltzmann machines with the same connectivity structure.
We also provide numerical results on the robustness of our training scheme with
respect to noise.
