Realizing Quantum Boltzmann Machines Through Eigenstate Thermalization

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.