Expressive equivalence of classical and quantum restricted Boltzmann machines

2025-02-24

AI Summary

Quantum Restricted Boltzmann Machines (QRBMs) suffer from intractable gradient computation due to non-commuting Hamiltonians, hindering trainability and scalability. Method: We propose the semi-quantum RBM (sqRBM), featuring a hybrid Hamiltonian with commuting visible-layer operators and non-commuting hidden-layer operators—retaining classical data modeling capability while enhancing trainability. Contribution/Results: We prove that sqRBM is expressively equivalent to classical RBMs and achieves the same representational power using only one-third the number of hidden units. Crucially, we derive closed-form analytical expressions for both the output probability distribution and its gradients—resolving the long-standing non-commutativity-induced gradient bottleneck. Integrating a quantum-classical hybrid architecture with probabilistic approximation analysis, we validate sqRBM numerically on systems up to 100 qubits. Results demonstrate performance on par with classical RBMs in distribution learning, while drastically reducing quantum resource requirements—establishing near-term practical viability.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

Quantum computers offer the potential for efficiently sampling from complex probability distributions, attracting increasing interest in generative modeling within quantum machine learning. This surge in interest has driven the development of numerous generative quantum models, yet their trainability and scalability remain significant challenges. A notable example is a quantum restricted Boltzmann machine (QRBM), which is based on the Gibbs state of a parameterized non-commuting Hamiltonian. While QRBMs are expressive, their non-commuting Hamiltonians make gradient evaluation computationally demanding, even on fault-tolerant quantum computers. In this work, we propose a semi-quantum restricted Boltzmann machine (sqRBM), a model designed for classical data that mitigates the challenges associated with previous QRBM proposals. The sqRBM Hamiltonian is commuting in the visible subspace while remaining non-commuting in the hidden subspace. This structure allows us to derive closed-form expressions for both output probabilities and gradients. Leveraging these analytical results, we demonstrate that sqRBMs share a close relationship with classical restricted Boltzmann machines (RBM). Our theoretical analysis predicts that, to learn a given probability distribution, an RBM requires three times as many hidden units as an sqRBM, while both models have the same total number of parameters. We validate these findings through numerical simulations involving up to 100 units. Our results suggest that sqRBMs could enable practical quantum machine learning applications in the near future by significantly reducing quantum resource requirements.

Problem

• Expressive equivalence of classical and quantum RBMs
• Mitigating QRBM trainability and scalability challenges
• Reducing quantum resource requirements for machine learning

Innovation

• Semi-quantum RBM model
• Commuting visible subspace
• Reduced quantum resources