Generative modeling using evolved quantum Boltzmann machines

2025-12-02

AI Summary

Training quantum Boltzmann machines (QBMs) to learn classically intractable probability distributions remains computationally prohibitive in quantum generative modeling. Method: We propose the first efficient training framework for QBMs based on the relative entropy, integrating the Donsker–Varadhan variational representation with a novel quantum gradient estimator. Our evolutionary training paradigm employs real- and imaginary-time evolution circuits for unified parameterization, a hybrid quantum-classical architecture, and min-max optimization. Contribution/Results: Theoretically, we design four algorithms with provable convergence guarantees. Experimentally, our framework efficiently models strongly correlated and non-Gaussian distributions that are challenging for classical methods. Crucially, this work achieves the first scalable QBM training under the relative entropy objective and generalizes to broader classes of $f$-divergences, establishing a new pathway toward practical quantum generative models.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• Training quantum Boltzmann machines efficiently
• Learning distributions via quantum generative modeling
• Developing hybrid quantum-classical training algorithms

Innovation

• Trains quantum Boltzmann machines using Donsker-Varadhan representation
• Employs quantum Boltzmann gradient estimator for efficient training
• Extends to evolved quantum Boltzmann machines with time evolution