Quantum-Noise-Driven Generative Diffusion Models

TL;DR

This work investigates how diffusion-based generative modeling can be extended into the quantum realm by leveraging quantum noise as a constructive resource. It formalizes three quantum-noise-driven diffusion models: $cqgdm$ (classical forward diffusion with quantum denoising), $qcgdm$ (quantum forward diffusion with classical denoising), and $qqgdm$ (fully quantum diffusion and denoising). Through targeted simulations, the authors demonstrate that $cqgdm$ can reconstruct classical data distributions via a quantum denoiser with a decreasing KL loss, while $qcgdm$ and $qqgdm$ achieve high fidelity in reconstructing quantum states under depolarizing diffusion (approximately $F \approx 0.997$ and $F \approx 0.996$, respectively). The results emphasize the potential of quantum-inspired or quantum-based diffusion schemes to sample richer distributions on NISQ hardware, with future directions including alternative noise channels, loss functions, and real-world quantum-data applications.

Abstract

Generative models realized with machine learning techniques are powerful tools to infer complex and unknown data distributions from a finite number of training samples in order to produce new synthetic data. Diffusion models are an emerging framework that have recently overcome the performance of the generative adversarial networks in creating synthetic text and high-quality images. Here, we propose and discuss the quantum generalization of diffusion models, i.e., three quantum-noise-driven generative diffusion models that could be experimentally tested on real quantum systems. The idea is to harness unique quantum features, in particular the non-trivial interplay among coherence, entanglement and noise that the currently available noisy quantum processors do unavoidably suffer from, in order to overcome the main computational burdens of classical diffusion models during inference. Hence, we suggest to exploit quantum noise not as an issue to be detected and solved but instead as a very remarkably beneficial key ingredient to generate much more complex probability distributions that would be difficult or even impossible to express classically, and from which a quantum processor might sample more efficiently than a classical one. An example of numerical simulations for an hybrid classical-quantum generative diffusion model is also included. Therefore, our results are expected to pave the way for new quantum-inspired or quantum-based generative diffusion algorithms addressing more powerfully classical tasks as data generation/prediction with widespread real-world applications ranging from climate forecasting to neuroscience, from traffic flow analysis to financial forecasting.

Figures (4)

• Figure 1: Depiction of the diffusion (from left to right) and denoising (from right to left) processes within a diffusion probabilistic model framework. The original image $\mathbf{x}_0$ sampled from the unknown data distribution $p(\mathbf{x}_0)$ is progressively perturbed ($t \rightarrow t+1$) by adding noise to obtain a latent variable $\mathbf{x}_T$ from a known and tractable distribution where the information is completely destroyed. In our framework the diffusion process can be implemented with a classical or a quantum stochastic dynamics. The denoising process is trained to approximate the structure of the data distribution in order to generate new samples. The latter is implemented step by step, using a classical (on the left in orange) or quantum (on the right in green) parameterized model $\hat{U}(\theta)$ in order to approximate the backward mapping. The standard diffusion models implement both the diffusion and the denoising processes in a classical framework. We propose three different new approaches for the other cases: i) classical diffusion and quantum denoising (cqgdm); ii) quantum diffusion and classical denoising (qcgdm); iii) quantum diffusion and quantum denoising (qqgdm). A similar picture can be applied to time series.
• Figure 2: (a) Evolution of the data distribution for a trained simulated cqgdm and (b) its kl divergence loss function $\mathcal{L}$ during the training, averaged every $1\,000$ iterations. The initial data distribution consists of two-dimensional points distributed in a line segment between $-1$ and $1$. The diffusion process is implemented via a classical diffusion process that transforms the initial data distribution $p(\mathbf{x}_0)$ at time $t=0$ to the prior $p(\mathbf{x}_{T})$ that is a normalized Gaussian distribution at the final time $t=40$. Meanwhile, the denoising is implemented via a (noiseless) simulated pqc to reconstruct the initial data distribution ($t=0$) from the Gaussian prior ($t=40$). In the top row of (a), we show the forward process (from left to right) for a sample of $1\,000$ points at different discrete time steps $t=0,8,\dots,40$. In the bottom row, we display the denoising (from right to left) of a different sample of $1\,000$ points. Under the figure is reported also the kl divergence between the data distribution $p(\mathbf{x}_0)$ and the reconstructed distribution at the corresponding time.
• Figure 3: Reconstruction of 10 random one-qubit pure states for trained (a) qcgdm and (b) qqgdm, and (c) the evolution of the quantum infidelity loss for both models for one of the 10 states. In (a) and (b), the blue segments represent the evolution of the forward dynamic implemented with a depolarizing quantum channel. In (a) the red segments are the evolutions of the backward process implemented with a neural network and in (b) with a parameterized quantum circuit. The green points are the final reconstructed states. The solid lines in (c) report the evolution of the loss used during the training while the dashed lines are the same loss but calculated only on the final reconstructed states. The red lines refers to the qcgdm and the blue lines to qqgdm.
• Figure 4: Relationship between the space of the probability distributions that are tractable with classical computation ($\mathbf{C}$) and instead only with quantum computation ($\mathbf{Q}$). We show the trajectories arising from the mappings between probability distributions (colored and white shapes) during the diffusion (blue wavy arrows) and denoising (red arrows) processes for the four different combination: $\mathbf{CC}$, $\mathbf{CQ}$, $\mathbf{QC}$ and $\mathbf{CC}$ indicating whether the diffusion (first letter) and the denoising (second letter) are classical or quantum. The initial data distribution (squares) is progressively transformed during diffusion (changing color and shape) to an uninformative distribution represented by the white circles, and vice versa during denoising. Completely classical models are limited to operate within the space of classically-tractable probability distributions, while completely quantum models can manipulate quantum-tractable probabilities. Models that have classical diffusion and quantum denoising are forced to work only with classical probabilities, but during the denoising phase they can exploit quantum properties within each step. Finally, models that have quantum diffusion and classical denoising can manipulate quantum probabilities during the forward, but in that case, it is not possible to train the classical backward to map those probability distributions.