Quantum Diffusion Models for Medical Image Analysis

Francesco Aldo Venturelli; Stefano Martina; Marco Parigi; Filippo Caruso; Alba Cervera-Lierta; Miguel A. González Ballester

eess.IVSep 25, 2026

Quantum Diffusion Models for Medical Image Analysis

Authors: Francesco Aldo Venturelli, Stefano Martina, Marco Parigi, Filippo Caruso, Alba Cervera-Lierta, Miguel A. González Ballester

Organizations: BCN Medtech, Department of Engineering, Universitat Pompeu Fabra, Barcelona, Spain · Barcelona Supercomputing Center, Barcelona, Spain · Department of Physics and Astronomy, University of Florence, Sesto Fiorentino, Florence, Italy · LENS - European Laboratory for Non-Linear Spectroscopy, University of Florence, Sesto Fiorentino, Florence, Italy · ICREA, Barcelona, Spain

Abstract

Quantum Machine Learning is a novel field of research aimed at devising machine learning approaches exploiting principles of quantum mechanics, such as superposition, entanglement and interference. In this context, we present a scalable hybrid Quantum Diffusion Model, and evaluate its use for medical image analysis. Specifically, our method is based on a Discrete-Time Quantum Walk algorithm, executed on a real quantum device, to model the forward dynamics of the diffusion model. For the backward step of the diffusion model, we devise and evaluate a classical learning model, which is used to reversely denoise the data. In contrast with other existing attempts at applying quantum machine learning for image analysis tasks, severely limited by the size of existing quantum devices, our method allows to process real-world large size medical data. In particular, we present results on grayscale and RGB images, as well as 3D volumes of moderate sizes. We benchmark our results by reproducing an alternative classical counterpart model, based on diffusion models on discrete state spaces. By doing so, we compare the generation capabilities of both models in terms of three distinct state-of-the-art metrics in the field of image generation, showing the competitive, promising results of our approach.

Figures & tables

Figure 1: Hybrid quantum diffusion model. Given an initial image $\mathbf{x}_{0}$ at time $t=0$ , every single pixel is associated with a quantum walker whose corresponding color $i_{0}$ is initially encoded in the quantum state $\ket{i_{0}}$ . The depicted quantum circuit visualizes only one of the multiple time steps of the dtqw algorithm, where $\ket{\psi_{pos}}\equiv\ket{i_{0}}$ , $\ket{\psi_{coin}}\equiv\ket{0}$ , $H$ are Hadamard gates, $R_{k}$ are gates $\mathrm{diag}(1,exp(2\pi i/2^{k}))$ (and their controlled versions $\mathrm{diag}(1,1,1,exp(2\pi i/2^{k}))$ ), $\mathcal{F}^{\dagger}$ is the inverse Quantum Fourier Transform [ 57 ] , and $C$ is the coin operation (implemented in our case with a Hadamard gate). The green-shaded block is repeated for the needed time steps. To obtain the pixel values of $\mathbf{x}_{t}$ at time $t$ we perform a single measurement of every noisy state $\rho_{t}$ after $t$ steps of evolution of each walker. Then, for every pixel we reinitialize $\ket{\psi_{pos}}$ with the obtained value, $\ket{\psi_{coin}}$ with $\ket{0}$ , and we perform a further single step to obtain the states $\rho_{t+1}$ and the corresponding $\mathbf{x}_{t+1}$ . To train the classical backward model, we minimize for all values of $t$ the difference between the pixel-value distribution of $\mathbf{x}_{t}$ and the output $\mathbf{\hat{x}}_{t}$ of a U-Net model that takes $\mathbf{x}_{t+1}$ as input. Finally, to generate new images $\mathbf{\hat{x}}_{0}$ we start from samples $\mathbf{\hat{x}}_{T}$ of uniform noise and we iteratively apply the trained model.

Model evaluation on 2D and 3D images
Dataset	KL			FID			SSIM
	Classical		Quantum	Classical		Quantum	Classical		Quantum
	$\beta=0.9$	$\beta=\beta_{t}$	Quantum	$\beta=0.9$	$\beta=\beta_{t}$	Quantum	$\beta=0.9$	$\beta=\beta_{t}$	Quantum
BloodMNIST	0.098	0.046	0.090	274.4	166.6	243.3	0.220	0.336	0.377
BraTS2020	0.396	0.192	0.124	280.3	266.8	297.7	0.646	0.569	0.676
FractureMNIST3D	0.047	0.013	0.011	169.6	100.1	89.70	0.564	0.550	0.564

Table 1: Quantitative evaluation across the three datasets using kl divergence, fid , and ssim . Lower kl divergence and fid indicate better distributional alignment, while higher ssim means good structural similarity. The metrics are calculated for

100

generated samples and, regarding the 3D volumes, the reported fid value corresponds to the average of different volume slices (

z=5,10,15,20,25

Panel of Figure 2 — Figure 2: Five samples of the original set of BloodMNIST images Fig. 2a and five samples of images generated by our model Fig. 2b . Each image has a shape of $64\times 64$ with $32$ intensity levels for each RGB channel.

Figure 3: BloodMNIST distribution of the two largest eigenvalues of the pca for

3\,000

original samples (purple) and 100 generated samples (orange). The contours represent the kde .

Panel of Figure 4 — Figure 4: Five samples of the original set of BraTS2020 images Fig. 4a and five samples of the brain images generated by our model Fig. 4b . Each image has a shape of $190\times 190$ with $64$ intensity levels.

Figure 5: BraTS2020 images distribution of the two largest eigenvalues of the pca for 450 original samples (purple) and 100 generated samples (orange). The contours represent the kde .

Panel of Figure 6 — Figure 6: Volume rendering of $5$ distinct samples from Fig. 6a the original dataset FractureMNIST3D and Fig. 6b generated volumes produced by the 3D U-Net. Renderings generated using 3D Slicer.

Figure 7: FractureMNIST3D distribution of the two largest eigenvalues of the pca for 1370 original samples (purple) and 100 randomly generated samples (orange). The contours represent the kde .

Panel of Figure 9 — Figure S1: Comparison between two classical Markov chains. The network in Fig. S1a shows a first-neighbor chain, while the network in Fig. S1b is fully connected, allowing the walker to move to any of the $N-1$ remaining nodes.

Panel of Figure 10 — Figure S2: Batch of 100 random BloodMNIST images.

Figure S3: Diffusion process executed on IBM Torino for $T=30$ steps using the dtqw on BloodMNIST with 32 states (top) and backward process performed by the U-Net trained for $\sim 20\,000$ epochs (bottom). We show the evolution for 9 random combined samples of original and reconstructed images and the evolution of the pixel-value distribution for the three channels (in red, green and blue). The visualized target distribution is the Uniform for the forward and the one of the original dataset for the backward. We also report the kl divergence between the data and target distributions and the final fid of the final generated images.

Figure S4: Diffusion process performed with the classical Markov chain on a 32-node ring network embedding pixel intensity levels with nearest-neighbor connection of RGB BloodMNIST images (Top) and backward process performed by the U-Net trained for $\sim 20\ 000$ epochs (Down). The $\beta$ scheduler is kept fixed $\beta=0.9.$

Figure S5: Diffusion process performed with the classical Markov chain on a 32-node ring network embedding pixel intensity levels with full connection of RGB BloodMNIST images (Top) and backward process performed by the U-Net trained for $\sim 20\ 000$ epochs (Down). The $\beta$ scheduler varies at each time step according to Eq. S9 .

Panel of Figure 14 — Figure S6: Batch of 100 random BraTS2020 images.

Figure S7: Diffusion process executed on IBM Torino for $T=30$ steps using the dtqw on grayscale BraTS2020 brain images of 64 states. The top row shows the forward process, while the bottom row shows the backward process performed by the U-Net trained for $\sim 20\,000$ epochs. We show the evolution of 9 random combined samples of original and reconstructed images and the evolution of the pixel-value distribution compared with the target distribution.

Figure S8: Diffusion process performed with the classical Markov chain on a 64-node ring network embedding pixel intensity levels with nearest-neighbour connections. The top row shows the forward process, while the bottom row shows the backward process performed by the U-Net trained for $\sim 20\,000$ epochs. The $\beta$ scheduler is set to 0.9 with $T=900$ steps, reaching a uniform distribution.

Figure S9: Diffusion process performed with the classical Markov chain on a 64-nodes ring fully-connected network embedding pixel intensity levels (Top) and backward process performed by the U-Net trained for $\sim 20\,000$ epochs (Down). The $\beta$ scheduler is $\beta_{t}$ and requires only 30 time steps to converge.

Panel of Figure 18 — Figure S10: PCA projections on the BloodMNIST dataset comparing the two distinct versions of the classical DMs.

Panel of Figure 19 — Figure S11: PCA projections on the BraTS2020 dataset comparing the two distinct versions of the classical DMs.

Panel of Figure 20 — Figure S12: PCA projections on the FractureMNIST dataset comparing the two distinct versions of the classical DMs.

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An Hybrid Quantum-Classical Diffusion Model for Image Generation

Qipeng Qian, Keli Deng, Yuntao Qian

Program of Applied Mathematics, Department of Mathematics, University of Arizona, Tucson, USA · College of Computer Science and Technology, Zhejiang University, Hangzhou, China

Latent Diffusion Models Quantum Generative Modeling Quantum Machine Learning Hybrid Quantum-Classical ML Diffusion-Based Image Generation Diffusion Models

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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

ρ_0

at timestep

t

and computing the one-step reverse update via an analytic backward propagation rule, rather than learning an explicit predictor for

ρ_{t-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.

Abstract

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