Generative training of quantum Boltzmann machines with hidden units

TL;DR

This paper introduces two novel quantum training methods for quantum Boltzmann machines using quantum relative entropy, overcoming previous mathematical challenges in gradient evaluation for fully quantum generative models.

Contribution

It presents the first fully quantum generative training methods for quantum Boltzmann machines with hidden units, utilizing variational bounds and advanced approximation techniques.

Findings

Proposes a variational upper bound approach for restricted quantum Boltzmann machines.

Develops high-order divided difference methods for gradient approximation.

Demonstrates efficiency under assumptions of Gibbs state preparation and sparse Hamiltonians.

Abstract

In this article we provide a method for fully quantum generative training of quantum Boltzmann machines with both visible and hidden units while using quantum relative entropy as an objective. This is significant because prior methods were not able to do so due to mathematical challenges posed by the gradient evaluation. We present two novel methods for solving this problem. The first proposal addresses it, for a class of restricted quantum Boltzmann machines with mutually commuting Hamiltonians on the hidden units, by using a variational upper bound on the quantum relative entropy. The second one uses high-order divided difference methods and linear-combinations of unitaries to approximate the exact gradient of the relative entropy for a generic quantum Boltzmann machine. Both methods are efficient under the assumption that Gibbs state preparation is efficient and that the Hamiltonian are given by a sparse row-computable matrix.