Realizing Quantum Boltzmann Machines Through Eigenstate Thermalization

  • Yudong Cao

This paper proposes a scheme for directly enhancing the classical Boltzmann machine by approximately sampling from the Boltzmann distribution of a quantum Hamiltonian, a task which is arguably beyond the capability of the classical computer. This technique is well suited for implementation on a diverse range of near-term devices including gate-model devices and analog physical simulators such as neutral atoms. It also allows for seamless integration with existing neural network architectures, allowing for a broad range of potential applications. 



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 observ- ables 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 Eigen- state Thermalization Hypothesis, and thus through Hamiltonian simulation an efficient method for training quantum Boltzmann machines on near-term quantum devices. Furthermore, under realis- tic 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.

Yudong Cao
Zapata Author

Yudong Cao , Ph.D.

CTO & Founder