Abstract Denoising diffusion probabilistic models have become increasingly important in machine learning, but their quantum counterpart has not been studied yet. In this work, we propose a quantum version of a generative diffusion model. In this algorithm, artificial neural networks are replaced with parameterized quantum circuits, in order to manipulate quantum states directly. We present both a full quantum version and a latent classical-quantum version of the algorithm. In the latent model, the parameterized quantum circuits are trained in a low dimensional representation of data, obtained by employing a pre-trained classical autoencoder. For both models, we show a method for conditioning the output distribution, utilizing ancillary qubits. The models' performance have been evaluated using quantitative metrics complemented by qualitative assessments. For the latent model, we show the implementation of a simplified version on real quantum hardware. The execution on a NISQ device allows to evaluate the performance of the algorithm in the presence of noise.
Abstract Denoising diffusion probabilistic models have become increasingly important in machine learning, but their quantum counterpart has not been studied yet. In this work, we propose a quantum version of a generative diffusion model. In this algorithm, artificial neural networks are replaced with parameterized quantum circuits, in order to manipulate quantum states directly. We present both a full quantum version and a latent classical-quantum version of the algorithm. In the latent model, the parameterized quantum circuits are trained in a low dimensional representation of data, obtained by employing a pre-trained classical autoencoder. For both models, we show a method for conditioning the output distribution, utilizing ancillary qubits. The models' performance have been evaluated using quantitative metrics complemented by qualitative assessments. For the latent model, we show the implementation of a simplified version on real quantum hardware. The execution on a NISQ device allows to evaluate the performance of the algorithm in the presence of noise.
Classical diffusion models have achieved remarkable results in terms of sample quality and variability. The quantum counterpart of these models is still to be explored thoroughly and offers a promising research direction in the field of quantum generative models. In this work, we introduce two versions of a quantum diffusion model, using parameterized quantum circuits to sample from quantum state distributions, while performing training via classical simulations. The first model is a quantum-hardware-oriented design, based on denoising diffusion probabilistic models, for which we demonstrate an implementation on real quantum hardware. The second model is a hybrid approach, based on denoising diffusion implicit models, that uses a classical neural network to enhance the expressive power. We show that the second version produces samples of higher quality and variability, but requires additional classical-quantum encoding and decoding overhead, underscoring a trade-off between expressive power and hardware efficiency. We demonstrate our method on MNIST images encoded as quantum states, as sample quality is easier to visualize and objective metrics are more widely available. While our results do not surpass classical diffusion models in the generation of classical samples, this study suggests the viability of using quantum diffusion models for learning quantum state distributions.
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 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.
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.
Generative models have achieved remarkable success in data synthesis, though recent advances driven by increasing model scale have introduced challenges in computational cost and efficiency. Quantum machine learning offers a promising alternative, representing complex data distributions using compact, highly expressive models. Here, we propose QDiffusion-TS, the first quantum generative diffusion model for time series synthesis, and validate it on the IQM quantum processor. The framework extends a classical diffusion architecture by replacing feed-forward components within the denoising transformer with quantum neural networks, yielding a hybrid quantum transformer that reduces the number of trainable parameters in each replaced component by nearly three orders of magnitude. Evaluated on financial time series from Apple and Amazon, the model generates synthetic data that more accurately reproduces the real distributions, reducing Wasserstein distance by approximately 44% relative to its classical counterpart across both datasets. In a downstream forecasting task, augmentation with the generated data improves predictive performance by up to 71% in RMSE over a baseline trained solely on real data. These results show that quantum enhanced architectures can consistently match and frequently surpass classical performance with substantially fewer parameters, establishing a practical framework towards more efficient and scalable data-driven generative modelling.
The combination of quantum and classical computational resources towards more effective algorithms is one of the most promising research directions in computer science. In such a hybrid framework, existing quantum computers can be used to their fullest extent and for practical applications. Generative modeling is one of the applications that could benefit the most, either by speeding up the underlying sampling methods or by unlocking more general models. In this work, we design a number of hybrid generative models and validate them on real hardware and datasets. The quantum-assisted Boltzmann machine is trained to generate realistic artificial images on quantum annealers. Several challenges in state-of-the-art annealers shall be overcome before one can assess their actual performance. We attack some of the most pressing challenges such as the sparse qubit-to-qubit connectivity, the unknown effective-temperature, and the noise on the control parameters. In order to handle datasets of realistic size and complexity, we include latent variables and obtain a more general model called the quantum-assisted Helmholtz machine. In the context of gate-based computers, the quantum circuit Born machine is trained to encode a target probability distribution in the wavefunction of a set of qubits. We implement this model on a trapped ion computer using low-depth circuits and native gates. We use the generative modeling performance on the canonical Bars-and-Stripes dataset to design a benchmark for hybrid systems. It is reasonable to expect that quantum data, i.e., datasets of wavefunctions, will become available in the future. We derive a quantum generative adversarial network that works with quantum data. Here, two circuits are optimized in tandem: one tries to generate suitable quantum states, the other tries to distinguish between target and generated states.