Overcoming Dimensional Factorization Limits in Discrete Diffusion Models through Quantum Joint Distribution Learning

2025-05-08

AI Summary

Classical discrete diffusion models suffer from a linear growth of KL divergence with dimensionality and the curse of dimensionality induced by factorization assumptions, hindering effective high-dimensional distribution modeling. To address this, we propose the Quantum Discrete Denoising Diffusion Probabilistic Model (QD3PM). Methodologically, we establish the first theoretical framework for quantum-enhanced diffusion models, introducing quantum Bayesian posterior states to enable joint probability modeling—eliminating the need for iterative sampling. We design a time-parameter-shared quantum circuit incorporating classical-data-driven rotation gates, wherein diffusion and denoising processes are formally defined in Hilbert space. Experiments demonstrate that QD3PM enables single-step pure-noise sampling, exhibits markedly sublinear KL divergence growth with dimensionality, and achieves superior accuracy compared to classical factorized approaches—empirically validating the advantage of quantum joint representation for modeling complex high-dimensional distributions.

Technology Category

• Machine Learning: Quantum Machine Learning
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This study explores quantum-enhanced discrete diffusion models to overcome classical limitations in learning high-dimensional distributions. We rigorously prove that classical discrete diffusion models, which calculate per-dimension transition probabilities to avoid exponential computational cost, exhibit worst-case linear scaling of Kullback-Leibler (KL) divergence with data dimension. To address this, we propose a Quantum Discrete Denoising Diffusion Probabilistic Model (QD3PM), which enables joint probability learning through diffusion and denoising in exponentially large Hilbert spaces. By deriving posterior states through quantum Bayes' theorem, similar to the crucial role of posterior probabilities in classical diffusion models, and by learning the joint probability, we establish a solid theoretical foundation for quantum-enhanced diffusion models. For denoising, we design a quantum circuit using temporal information for parameter sharing and learnable classical-data-controlled rotations for encoding. Exploiting joint distribution learning, our approach enables single-step sampling from pure noise, eliminating iterative requirements of existing models. Simulations demonstrate the proposed model's superior accuracy in modeling complex distributions compared to factorization methods. Hence, this paper establishes a new theoretical paradigm in generative models by leveraging the quantum advantage in joint distribution learning.

Problem

• Overcoming classical limits in high-dimensional distribution learning
• Proposing quantum-enhanced diffusion for joint probability learning
• Enabling single-step sampling via quantum joint distribution

Innovation

• Quantum-enhanced discrete diffusion models
• Quantum joint distribution learning
• Single-step sampling from noise