The generation and preservation of complex quantum states against environmental noise are paramount challenges in advancing continuous-variable (CV) quantum information processing. This paper introduces a novel framework based on continuous-variable quantum diffusion principles, synergizing them with CV quantum neural networks (CVQNNs) to address these dual challenges. For the task of state generation, our Continuous-Variable Quantum Diffusion Generative model (CVQD-G) employs a physically driven forward diffusion process using a thermal loss channel, which is then inverted by a learnable, parameter-efficient backward denoising process based on a CVQNN with time-embedding. This framework's capability is further extended for state recovery by the Continuous-Variable Quantum Diffusion Restoration model (CVQD-R), a specialized variant designed to restore quantum states, particularly coherent states with unknown parameters, from thermal degradation. Extensive numerical simulations validate these dual capabilities, demonstrating the high-fidelity generation of diverse Gaussian (coherent, squeezed) and non-Gaussian (Fock, cat) states, typically with fidelities exceeding 99%, and confirming the model's ability to robustly restore corrupted states. Furthermore, a comprehensive complexity analysis reveals favorable training and inference costs, highlighting the framework's efficiency, scalability, and its potential as a robust tool for quantum state engineering and noise mitigation in realistic CV quantum systems.
The generation and preservation of complex quantum states against environmental noise are paramount challenges in advancing continuous-variable (CV) quantum information processing. This paper introduces a novel framework based on continuous-variable quantum diffusion principles, synergizing them with CV quantum neural networks (CVQNNs) to address these dual challenges. For the task of state generation, our Continuous-Variable Quantum Diffusion Generative model (CVQD-G) employs a physically driven forward diffusion process using a thermal loss channel, which is then inverted by a learnable, parameter-efficient backward denoising process based on a CVQNN with time-embedding. This framework's capability is further extended for state recovery by the Continuous-Variable Quantum Diffusion Restoration model (CVQD-R), a specialized variant designed to restore quantum states, particularly coherent states with unknown parameters, from thermal degradation. Extensive numerical simulations validate these dual capabilities, demonstrating the high-fidelity generation of diverse Gaussian (coherent, squeezed) and non-Gaussian (Fock, cat) states, typically with fidelities exceeding 99%, and confirming the model's ability to robustly restore corrupted states. Furthermore, a comprehensive complexity analysis reveals favorable training and inference costs, highlighting the framework's efficiency, scalability, and its potential as a robust tool for quantum state engineering and noise mitigation in realistic CV quantum systems.
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.
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.
Continuous-variable (CV) quantum systems provide a versatile platform for quantum information processing, in which quantum states can be represented in the quadrature phase space. In realistic implementations, environmental noise, primarily photon loss and dephasing, progressively degrades these states. Machine-learning-based quantum error mitigation (QEM) has recently emerged as a promising approach to suppress such noise; however, existing methods are typically limited to the training horizon and require training data that cover the entire evolution, which is experimentally demanding. Here we introduce a framework for extrapolative quantum error mitigation based on a time-conditioned Swin Transformer. By explicitly embedding the evolution time via adaptive layer normalization, the model learns a correction map that accounts for the continuous accumulation of noise while capturing nonlocal phase-space correlations. Numerical simulations under both Markovian and non-Markovian noise demonstrate accurate state recovery in the long-time regime, where existing approaches deteriorate. Our results establish extrapolative QEM as a practical route to mitigating noise in CV quantum systems without exhaustive training data.
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.
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.
Continuous-variables (CV) quantum optics is a natural formalism for neural networks (NNs) due to its ability to reproduce the information processing of such trainable interconnected systems. In quantum optics, Gaussian operators induce affine mappings on the quadratures of optical modes while non-Gaussian resources (the challenging piece for physical implementation) originate the nonlinear effects, unlocking quantum analogs of an artificial neuron. This work presents a novel experimentally-feasible framework for continuous-variable quantum optical neural networks (QONNs) developed with available photonic components: coherent states as input encoding, a general Gaussian transformation followed by multi-mode photon subtractions as the processing layer, and homodyne detection as outputs readout. The closed-form expressions of such architecture are derived demonstrating the family of adaptive activations and the quantum-optical neurons that emerge from the amount of photon-subtracted modes, proving that the proposed design satisfies the Universal Approximation Theorem within a single layer. To classically simulate the QONN training, the high-performance QuaNNTO library has been developed based on Wick-Isserlis expansion and Bogoliubov transformations, allowing multi-layer exact expectation values of non-Gaussian states without truncating the infinite-dimensional Hilbert space. Experiments on supervised learning and state-preparation tasks show balanced-resource efficiency with strong expressivity and generalization capabilities, illustrating the potential of the architecture for scalable photonic quantum machine learning and for quantum applications such as complex non-Gaussian gate synthesis.