Mixed quantum states are the native description of many physically important quantum systems, making their generation a fundamental task in quantum information processing. However, constructing a diffusion process that generates density operators while keeping every reverse step physically valid remains nontrivial. This work introduces Quantum Generative Diffusion Model (QGDM), a fully quantum-mechanical model whose forward and backward processes are grounded in quantum channel theory. Through a non-unitary forward process, any target quantum state can be transformed into a completely mixed state. A trainable backward process recovers the former from the latter. We introduce partial trace to make the backward process non-unitary, and share parameters across timesteps by incorporating temporal information as an input. We present QGDM's resource-efficient version to reduce auxiliary qubits while preserving generative capabilities. We theoretically analyze the denoising design, showing it avoids a low-loss shortcut that traps training and cause generation failure. Simulations confirm that QGDM outperforms quantum generative adversarial networks on random pure- and mixed-state generation, with better noise robustness than other quantum generative models and a task-specialized approach for practical Gibbs state generation. Hence, QGDM provides a channel-based diffusion framework for learning fixed mixed-state targets, extending quantum generative modeling toward realistic quantum information settings.
Mixed quantum states are the native description of many physically important quantum systems, making their generation a fundamental task in quantum information processing. However, constructing a diffusion process that generates density operators while keeping every reverse step physically valid remains nontrivial. This work introduces Quantum Generative Diffusion Model (QGDM), a fully quantum-mechanical model whose forward and backward processes are grounded in quantum channel theory. Through a non-unitary forward process, any target quantum state can be transformed into a completely mixed state. A trainable backward process recovers the former from the latter. We introduce partial trace to make the backward process non-unitary, and share parameters across timesteps by incorporating temporal information as an input. We present QGDM's resource-efficient version to reduce auxiliary qubits while preserving generative capabilities. We theoretically analyze the denoising design, showing it avoids a low-loss shortcut that traps training and cause generation failure. Simulations confirm that QGDM outperforms quantum generative adversarial networks on random pure- and mixed-state generation, with better noise robustness than other quantum generative models and a task-specialized approach for practical Gibbs state generation. Hence, QGDM provides a channel-based diffusion framework for learning fixed mixed-state targets, extending quantum generative modeling toward realistic quantum information settings.
Generative quantum machine learning has gained significant attention for its ability to produce quantum states with desired distributions. Among various quantum generative models, quantum denoising diffusion probabilistic models (QuDDPMs) [Zhang et al., Phys. Rev. Lett. 132, 100602 (2024)] provide a promising approach with stepwise learning that resolves the training issues. However, the requirement of high-fidelity scrambling unitaries in QuDDPMs poses a challenge to near-term implementation. We propose a mixed-state quantum denoising diffusion probabilistic model (MSQuDDPM) to eliminate the need for scrambling unitaries. Our approach focuses on adapting the quantum noise channels to the model architecture, which integrates depolarizing noise channels in the forward diffusion process and parameterized quantum circuits with projective measurements in the backward denoising steps. We also introduce several techniques to improve MSQuDDPM, including a cosine-exponent schedule of noise interpolation, the use of a single-qubit random ancilla, and superfidelity-based cost functions to enhance the convergence. We evaluate MSQuDDPM on quantum ensemble generation tasks, demonstrating its successful performance.
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.
We present a channel-constrained Markovian quantum diffusion (CCMQD) model that prepares quantum states by rigorously framing the generative process within the dynamics of open quantum systems. Our model interprets the forward diffusion process as natural decoherence using quantum master equations, whereas the reverse denoising is achieved by learning inverse quantum channels. Our core innovation is a comprehensive channel-constrained framework: we model the diffusion and denoising steps as quantum channels defined by Kraus operators, ensure their physical validity through optimization on the Stiefel manifold, and introduce tailored training strategies and loss functions that leverage this constrained structure for high-fidelity state reconstruction. Experimental validation on systems ranging from single qubits to entangled states 7 -qubits demonstrates high-fidelity state generation, achieving fidelities exceeding 0.998 under both random and depolarizing noise conditions. This work confirms that quantum diffusion can be characterized as a controlled Markov evolution, demonstrating that environmental interactions are not limited to being a source of decoherence but can also be utilized to achieve high-fidelity quantum state synthesis.
Generative Artificial Intelligence is one of the most relevant technologies of our century, enabling digital machines not only to analyze or classify data but also to generate new content such as images, text, and video. This technology relies on deep learning algorithms designed to capture complex relationships from finite data sets. In this context, Diffusion Models have recently emerged as state-of-the-art in deep generative modeling thanks to their ability to produce high-quality data. Their pipeline typically consists of a forward diffusion process, where data samples are progressively corrupted into pure (classical) noise, and a reverse denoising process, where deep neural networks learn to reverse the diffusion process and then generate new synthetic samples. In parallel, Quantum Machine Learning represents one of the most promising applications of quantum computing. However, currently available quantum processors, known as Noisy Intermediate-Scale Quantum (NISQ) devices, are composed of tens or hundreds of noisy qubits. Therefore, designing scalable and efficient practical quantum algorithms represents a significant challenge. This thesis proposes the first generalization of classical Denoising Diffusion Probabilistic Models to the quantum domain, introducing the novel generative paradigm of Quantum Diffusion Models. In particular, the three possible scenarios are formalized and implemented: (i) classical data corrupted with Gaussian noise and a reverse process implemented via a quantum neural network; (ii) quantum data corrupted with quantum noise and a denoising process through a classical deep neural network; and (iii) a fully quantum approach, where both forward and backward processes are implemented within a quantum framework. Subsequently, the impact of different stochastic dynamics in the forward process is analyzed using the formalism of Quantum Stochastic Walks. We observe that a specific hybrid quantum-classical dynamics produces more statistically robust models, capable of generating sets of images closely resembling real ones compared to purely classical dynamics. Furthermore, we propose a hybrid quantum-classical diffusion model based on Quantum Walks that exploits the intrinsic noise of real quantum processors as a useful resource for generating images. This model is implementable on current NISQ devices and scalable, as it allows the generation of images of any size, requiring q + 1 qubits for grayscale images with pixel intensities in the range [0, 2^q - 1]. Finally, an application in the context of medical imaging is presented, specifically the generation of 64 × 64 color images of blood cells from the BloodMNIST dataset and 200 × 200 brain MRI images from the BraTS2020 dataset.
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.
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.