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

Discrete diffusion models typically rely on dimension-wise factorization to avoid computational intractability. However, we rigorously prove this approach leads to worst-case errors scaling linearly with data dimension, fundamentally failing to capture inter-dimensional correlations. 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, we establish a theoretical foundation for quantum-enhanced diffusion models. We design a quantum circuit that utilizes temporal information for parameter sharing and incorporates learnable classical-data-controlled rotations for encoding. Crucially, our approach enables single-step sampling from pure noise to eliminate iterative bottlenecks, while also supporting retraining-free conditional inference, a flexibility often absent in existing quantum generative models such as quantum circuit Born machines. Simulations demonstrate that QD3PM significantly outperforms the parameter-matched classical baseline in modeling inter-dimensional correlations and exhibits superior robustness against quantum noise compared to quantum generative adversarial networks and quantum variational autoencoders. Hence, our work establishes a new theoretical paradigm by leveraging quantum advantages in joint distribution learning.