Generative modeling using evolved quantum Boltzmann machines

TL;DR

This paper introduces a practical method for training quantum Boltzmann machines to perform Born-rule generative modeling, enabling quantum models to efficiently learn complex probability distributions.

Contribution

It proposes a novel training approach using variational representations and gradient estimators, applicable to evolved quantum Boltzmann machines with convergence guarantees.

Findings

Developed a practical training algorithm for quantum Boltzmann machines.

Extended the method to other distinguishability measures.

Presented four hybrid quantum-classical optimization algorithms.

Abstract

Born-rule generative modeling, a central task in quantum machine learning, seeks to learn probability distributions that can be efficiently sampled by measuring complex quantum states. One hope is for quantum models to efficiently capture probability distributions that are difficult to learn and simulate by classical means alone. Quantum Boltzmann machines were proposed about one decade ago for this purpose, yet efficient training methods have remained elusive. In this paper, I overcome this obstacle by proposing a practical solution that trains quantum Boltzmann machines for Born-rule generative modeling. Two key ingredients in the proposal are the Donsker-Varadhan variational representation of the classical relative entropy and the quantum Boltzmann gradient estimator of [Patel et al., arXiv:2410.12935]. I present the main result for a more general ansatz known as an evolved quantum Boltzmann machine [Minervini et al., arXiv:2501.03367], which combines parameterized real- and imaginary-time evolution. I also show how to extend the findings to other distinguishability measures beyond relative entropy. Finally, I present four different hybrid quantum-classical algorithms for the minimax optimization underlying training, and I discuss their theoretical convergence guarantees.