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.
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.
Discrete diffusion models represent a significant advance in generative modeling, demonstrating remarkable success in synthesizing complex, high-quality discrete data. However, to avoid exponential computational costs, they typically rely on calculating per-dimension transition probabilities when learning high-dimensional distributions. In this study, we rigorously prove that this approach leads to a 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, offering a theoretical pathway to faithfully capture the true joint distribution. 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 that utilizes temporal information for parameter sharing and incorporates 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.
Deep generative models are key-enabling technology to computer vision, text generation, and large language models. Denoising diffusion probabilistic models (DDPMs) have recently gained much attention due to their ability to generate diverse and high-quality samples in many computer vision tasks, as well as to incorporate flexible model architectures and a relatively simple training scheme. Quantum generative models, empowered by entanglement and superposition, have brought new insight to learning classical and quantum data. Inspired by the classical counterpart, we propose the quantum denoising diffusion probabilistic model (QuDDPM) to enable efficiently trainable generative learning of quantum data. QuDDPM adopts sufficient layers of circuits to guarantee expressivity, while it introduces multiple intermediate training tasks as interpolation between the target distribution and noise to avoid barren plateau and guarantee efficient training. We provide bounds on the learning error and demonstrate QuDDPM's capability in learning correlated quantum noise model, quantum many-body phases, and topological structure of quantum data. The results provide a paradigm for versatile and efficient quantum generative learning.
Quantum generative models exploit quantum superposition and entanglement to enhance learning efficiency for both classical and quantum data. Recently, inspired by classical diffusion frameworks, the quantum denoising diffusion probabilistic model has emerged as a powerful tool for learning correlated noise models, many-body phases, and topological data structures. However, we demonstrate that this framework is currently restricted to small-scale systems. As the system size increases, a severe barren plateau problem emerges, fundamentally limiting the model's scalability. We provide rigorous theoretical proofs and experimental validation to identify the origin of this barren plateau, distinct from previously known causes. To restore trainability, we introduce an enhanced architecture that effectively mitigates the barren plateau phenomenon and guarantees the model's trainability in the tested settings. Building on this architecture, we further propose a conditional quantum denoising diffusion probabilistic model, capable of generating ground states based on Hamiltonian parameters, expanding the utility of quantum generative models for complex quantum state preparation to a certain extent. Our approach not only holds the potential to address the scalability and trainability bottlenecks of quantum diffusion models, but also provides a robust tool for exploring complex quantum matter and state preparation in the NISQ era.
Quantum diffusion models provide a physics-consistent route to generative learning by formulating noising and denoising directly on quantum states. However, applying such models to classical high-dimensional data is constrained by the qubit cost of state encoding and the computational burden of simulating large density operators. We propose a scalable hybrid generative pipeline that combines a classical autoencoder for dimensionality reduction with a mixed-state quantum denoising diffusion probabilistic model (MSQuDDPM) operating in the learned latent space. The autoencoder compresses data into compact latent codes that can be embedded into a small-qubit Hilbert space, after which the quantum diffusion model learns a generative distribution over latent density operators and decodes samples back to the original domain. Algorithmically, we simplify the reverse dynamics by predicting an estimate of the clean state ρ0 at timestep t and computing the one-step reverse update via an analytic backward propagation rule, rather than learning an explicit predictor for ρt−1. We demonstrate the proposed approach on MNIST image generation and discuss how mixed-state quantum diffusion can serve as a practical backbone for hybrid quantum--classical generative modeling under realistic qubit budgets.
Generative models for quantum data pose significant challenges but hold immense potential in fields such as chemoinformatics and quantum physics. Quantum denoising diffusion probabilistic models (QuDDPMs) enable efficient learning of quantum data distributions by progressively scrambling and denoising quantum states. However, existing implementations typically rely on circuit-based random unitary dynamics, which can be costly to implement and sensitive to control imperfections, particularly on analog quantum hardware. We propose the chaotic quantum diffusion model, a framework that generates projected ensembles via chaotic Hamiltonian time evolution, providing a flexible and hardware-compatible diffusion mechanism. Requiring only global, time-independent control, our approach substantially reduces implementation overhead across diverse analog quantum platforms while achieving accuracy comparable to QuDDPMs. This method improves trainability and robustness, broadening the applicability of quantum generative modeling.
Quantum computers could solve problems beyond the reach of classical devices, but this potential depends on quantum error correction (QEC) to protect fragile quantum states from noise. A central challenge in QEC is decoding: inferring likely physical errors from syndrome patterns generated by repeated stabilizer measurements. Existing decoders, including graph-based and neural approaches, typically return a single correction hypothesis and therefore discard the richer posterior structure of the error distribution conditioned on the observed syndrome. Here we recast QEC decoding as posterior inference using discrete denoising diffusion, exploiting the analogy between stochastic error accumulation and the forward diffusion process. We introduce DiffQEC, a generative decoder that combines a syndrome processor for multi-round spatial-temporal syndrome histories with syndrome feature modulation to condition denoising on the observed syndrome throughout inference. On experimental data from Google's superconducting quantum processor, DiffQEC reduces logical error rates by up to 10.2% relative to minimum-weight perfect matching and by about 5% relative to tensor-network decoding. These improvements persist for larger code distances up to 17 under depolarizing noise and for logical circuits of increasing depth. Beyond accuracy, the learned posterior provides confidence estimates for post-selection and reveals physically meaningful error structure, establishing posterior generative decoding as a practical framework for QEC.