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Quantum Diffusion Models for Medical Image Analysis

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

arXiv:2609.31070v1eess.IVcs.AIcs.CVcs.ETcs.LGquant-ph

TL;DR

Current quantum-device limits have kept many QML approaches to reduced-dimensional data, while QDM medical-image studies have used quantum components only in the backward process. This paper combines a real-device quantum forward walk with classical U-Net denoising for medical images and volumes. Its hybrid model is competitive with the classical counterpart, with lower KL divergence and FID and equal SSIM on FractureMNIST3D, and higher SSIM across all three datasets.

  • Problem

    Many quantum machine-learning approaches are restricted to reduced-dimensional data or hybrid designs by current noisy-device limitations, and prior QDM medical-image studies used quantum layers only in the backward process.

    Source: §I · pp. 1–2

  • Method

    The method runs a discrete-time quantum walk on a real device for forward diffusion and trains a classical U-Net to reverse the process.

    Source: §I · p. 2, B. Quantum forward process · p. 3

  • Results

    The hybrid model is competitive with its classical counterpart, with lower KL divergence and FID and equal SSIM on FractureMNIST3D, and higher SSIM across all three datasets.

    Source: §III · p. 7

  • Takeaways & Limitations

    The work is a proof of concept for applying available quantum computers to healthcare problems.

    Source: §IV · p. 9

    For BloodMNIST, images were limited to 64 × 64 pixels and 32 intensity levels, and generated images had lower SSIM than reported in [48].

    Source: §IV · p. 9

Abstract

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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.

I. INTRODUCTION

The paper adapts hybrid quantum diffusion to medical images, combining a quantum forward process with classical denoising and comparing its outputs with a classical discrete-state diffusion model. It also reviews diffusion models’ forward-noising and learned reverse-denoising framework.

I. INTRODUCTION

  • Earlier quantum machine-learning algorithms often use reduced-dimensional data or hybrid frameworks because of limitations in current noisy intermediate-scale quantum devices.Prior medical-image applications included generation, spine-volume reconstruction, and skin-cancer data augmentation.Source: §I · pp. 1–2
  • The method uses a quantum forward process and classical backward process, differing from earlier medical QDMs that used quantum layers only in the backward process.The work also extends its earlier algorithm with changes to the quantum noise process, intensity levels, backward architecture, and loss.Source: §I · p. 2
  • The study evaluates grayscale and RGB images and 3D volumes from BloodMNIST, BraTS2020, and FractureMNIST3D using KL divergence, FID, and SSIM.It also simulates the QDM, runs it on a real quantum computer with error mitigation, and reproduces a classical discrete-state diffusion model for comparison.Source: §I · p. 2

A. Diffusion Models

  • Diffusion models progressively corrupt data through a forward process, then train a learning model to reverse the noise and recover samples.The forward chain ends at a tractable prior, such as a Gaussian or uniform distribution.Source: A. Diffusion Models · p. 2
  • Because stochastic forward diffusion cannot be inverted analytically, neural networks approximate reverse conditional probabilities at each step.For discrete-state models, data are categorical variables and the forward transition matrix specifies movement between categories; the prior is discrete uniform.Source: A. Diffusion Models · p. 3

B. Quantum forward process

The forward process associates each image pixel or volume voxel with a discrete-time quantum walk on a real NISQ device. Measurement and reinitialization produce noisy data at successive diffusion steps.

B. Quantum forward process

  • Each pixel or voxel is discretized and assigned a discrete-time quantum walk; RGB data use an independent walk for each channel.The walk’s transitions define a ring-chain Markov process, with Nq qubits representing 2^Nq intensity values and another qubit serving as the coin.Source: B. Quantum forward process · p. 3
  • The theoretical walk applies the circuit unitary for t steps, while the real-device dynamics also account for a quantum noise channel.The circuit’s green-shaded block is repeated for the required steps.Source: B. Quantum forward process · p. 3
  • The real-device forward evolution combines walk-induced diffusion with intrinsic quantum noise, which the authors identify as necessary for convergence to a uniform distribution.A purely unitary walk would not converge to that distribution.Source: B. Quantum forward process · p. 3
  • At each time step, a measurement yields pixel values; the walk is reinitialized from those values and run for one additional step to form training pairs.The figure describes measuring each noisy state after t steps, then reinitializing the position and coin states to obtain the next step.Source: B. Quantum forward process · pp. 3–4, Figure 1 · p. 4
  • The study runs the walk on IBM Torino using six near-connected qubits, with one coin qubit and five intensity qubits, increasing intensity levels from 8 to 64.The device has 133 qubits; circuit optimization is used to reduce gate count while maintaining moderate noise in early steps.Source: B. Quantum forward process · p. 5

C. Classical backward process

A classical, time-conditioned U-Net learns the reverse process from pairs generated by the quantum forward dynamics. The paper uses distinct architectures for 2D images and 3D volumes, then iteratively denoises uniform-noise samples to generate images.

C. Classical backward process

  • The backward model replaces the earlier multilayer perceptron with a U-Net and tailored time embeddings to capture the diffusion process.The 2D version embeds time in encoder layers; the more complex 3D version also embeds time in the decoder and uses a deeper bottleneck.Source: C. Classical backward process · p. 5
  • The 2D U-Net uses five downsampling encoders and four upsampling decoders connected by skip connections.The 3D model uses four encoding blocks, a three-block bottleneck, four decoding blocks, group batch normalization, and SiLU activations.Source: C. Classical backward process · p. 5
  • Training selects a random time step and teaches the U-Net to predict xt from xt+1, conditioned on time, using independently corrupted pixel or voxel values.The models use ADAM; 2D training runs for 20,000 epochs, while 3D training uses early stopping at about 4,000 epochs because of overfitting susceptibility.Source: C. Classical backward process · p. 5
  • Generation begins with uniform-noise samples and repeatedly applies the trained, time-conditioned U-Net from T−1 down to 0.At each step, the model takes the current noisier sample as input and produces the preceding sample.Source: C. Classical backward process · p. 5

D. Losses

The paper uses dataset-dependent losses for training and several complementary measures to assess generated data. These include distribution comparisons and structural or perceptual similarity measures.

D. Losses

  • Training uses cross-entropy for categorical data; BraTS2020 adds KL divergence with a time-dependent weight that shifts from image reconstruction at low t toward matching pixel-value distributions at high t.The added KL term is intended to improve the generated pixel-value distribution.Source: D. Losses · pp. 5–6

E. Metrics

  • KL divergence compares original and generated pixel-intensity distributions at each forward and backward time step.The metric is asymmetric, with the original distribution as reference and the generated distribution as the comparison.Source: E. Metrics · p. 6
  • FID compares generated-image and ground-truth distributions, with lower values indicating better generative performance; SSIM assesses structural similarity, with higher values indicating greater fidelity.FID uses feature means and covariances from a pretrained Inception-v3 network; SSIM ranges from −1 to 1.Source: E. Metrics · p. 6
  • PCA projects samples onto the two components with the highest eigenvalues, and KDE estimates overlap between the projected original and generated distributions.The paper uses this analysis to visualize, compare, and quantify the data distributions.Source: E. Metrics · p. 6

F. Medical datasets

The paper evaluates its hybrid model on three medical datasets and finds it competitive with a classical counterpart, with metric-specific strengths on 3D data and SSIM. The reported comparisons use 100 generated samples, and image resolution and quantization affect SSIM.

F. Medical datasets

  • The evaluation covers 3,329 RGB BloodMNIST images, 484 cropped grayscale BraTS2020 slices, and 1,370 grayscale FractureMNIST3D volumes.The datasets include 64 × 64 images, 190 × 190 slices, and 28 × 28 × 28 volumes, respectively.Source: F. Medical datasets · p. 6

III. RESULTS

  • The forward probabilities come from DTQWs run on real IBM NISQ devices; a classical model learns to reconstruct corrupted data and generates samples iteratively from noise.Generation proceeds from pure noise through successive reverse steps.Source: §III · p. 6
  • Metrics are computed on 100 generated samples, and the authors attribute differing SSIM values partly to image resolution and intensity quantization.For 3D volumes, the reported FID averages slices at z = 5, 10, 15, 20, and 25.Source: §III · p. 7, Table I · p. 7
  • The quantum hybrid model is competitive with its classical counterpart; on FractureMNIST3D it has lower KL divergence and FID and equal SSIM, while SSIM is higher on all three datasets.The authors caution that FID may be a poor metric for specialized medical data because it uses a model pretrained on generic high-quality data.Source: §III · p. 7, Table I · p. 7

A. Quantum diffusion on RGB BloodMNIST images

On BloodMNIST, generated images preserve the main visual features and represent both modes of the dataset distribution. On BraTS2020, the model reproduces varied brain-slice features and comparable variance without model collapse.

A. Quantum diffusion on RGB BloodMNIST images

B. Quantum diffusion on BraTS2020 images

C. Quantum diffusion on FractureMNIST3D volumes

The FractureMNIST3D experiment extends the hybrid model to 3D volumes, while the discussion frames real-device quantum forward diffusion and classical reverse denoising as its broader contribution.

C. Quantum diffusion on FractureMNIST3D volumes

  • The 3D U-Net generates FractureMNIST3D volumes with characteristics like the originals, although some samples are not generated as expected.Figure 6 shows five original and generated volume renderings.Source: Figure 6 · p. 8, C. Quantum diffusion on… · p. 9
  • FID uses an image-pretrained Inception V3 model and averages measurements across slices z = 5, 10, 15, 20, 25 because it is not pretrained on volumes.KL divergence and SSIM are calculated as for the other datasets.Source: C. Quantum diffusion on… · p. 9

IV. DISCUSSION AND CONCLUSIONS

  • The hybrid quantum forward dynamics handle RGB images, large grayscale images, and 3D volumes on current NISQ devices without error-correcting codes.The forward process exploits intrinsic quantum noise, which the authors say is necessary for convergence to the uniform distribution.Source: §IV · p. 9
  • Moderate quantum noise may accelerate the forward diffusion, but it can also make measurement results harder to distinguish as noise complexity increases.The authors describe this as a possible effect, not a demonstrated outcome in this work.Source: §IV · p. 9
  • The DTQW uses a cyclic topology and a coin register with three-degree connections to drive the walk forward or backward across tunable time steps.The measured position probabilities are added to the remaining pixels to sample the image distribution.Source: §IV · p. 9
  • A classical neural network reversely denoises data by learning how unknown noise samples act on pixel values, while the forward dynamics use a real quantum device.The authors describe the study as a proof of concept for available quantum computers in healthcare.Source: §IV · p. 9
  • FID is slightly higher than [48] [49], while SSIM is comparable or higher for BraTS2020 and FractureMNIST3D but lower for BloodMNIST.The authors attribute BloodMNIST's lower SSIM to its 64 × 64 resolution and 32 intensity levels, compared with 1024 × 1024 and 256 levels in [48].Source: §IV · p. 9

A. Quantum computing

The section introduces qubits as quantum information units, describes superposition and measurement, and explains how gates and noise shape computation on current devices.

A. Quantum computing

  • A qubit state can be a superposition of |0⟩ and |1⟩, with complex probability amplitudes whose squared magnitudes sum to 1.Multiple qubits can represent superpositions of 2^n bit-strings.Source: A. Quantum computing · p. 13
  • Measurement yields a classical bit-string with probability equal to the squared magnitude of its amplitude, so repeated runs estimate the output distribution.Measurement collapses the qubit state to the observed basis state.Source: A. Quantum computing · p. 13
  • Quantum gates are unitary transformations, and the Hadamard gate is given as an example that transforms basis states into a superposition.A gate acting on m qubits is represented by a 2^m × 2^m unitary matrix.Source: A. Quantum computing · p. 13
  • NISQ devices coexist with quantum noise and errors because fragile superpositions are difficult to preserve and current devices cannot implement fault-tolerant operations.Hybrid quantum-classical algorithms use classical optimization to adjust circuit parameters against device noise.Source: A. Quantum computing · pp. 13–14
  • A density matrix represents a quantum state; its diagonal elements give measurement probabilities, while its off-diagonal elements encode coherence.The stated matrix properties ensure real eigenvalues, nonnegative probabilities, and probabilities summing to 1.Source: A. Quantum computing · p. 14

B. Classical random walks and discrete-state space diffusion models

The section contrasts classical random-walk transition schemes on discrete states: a ring chain with a stay probability and a fully connected chain with a time-varying scheduler.

B. Classical random walks and discrete-state space diffusion models

  • The first classical chain uses a ring adjacency matrix with β = 0.9 and a stay probability of 1 − β; the identity-matrix term leaves the walker's position unchanged.Figure S1 contrasts this first-neighbor ring with a fully connected network.Source: B. Classical random walks and… · p. 14, Figure S1 · p. 14
  • The first chain evolves by multiplying its transition matrix Q t times from an initial distribution concentrated at the first state.The initial distribution is a vector of zeros except for its first element, which is 1.Source: B. Classical random walks and… · p. 15
  • The second chain uses a time-dependent scheduler and a fully connected adjacency matrix, allowing the walker to diffuse to any other site and accelerating diffusion.Its transition matrix uses distinct probabilities for staying at a site and moving to another site.Source: Figure S1 · p. 14, B. Classical random walks and… · p. 15

C. Discrete-time quantum walk

The discrete-time quantum walk applies unitary steps to a coin-and-position state, combining coin operations and shifts to evolve the walker across positions.

C. Discrete-time quantum walk

  • The global quantum state in the example has four available walker positions: |00⟩, |01⟩, |10⟩, and |11⟩.The walk's unitary operation is repeated to preserve quantum correlations and allow position states to interfere.Source: C. Discrete-time quantum walk · p. 15
  • The walk's shift operator acts on the entangled coin-position state, with two cases determined by the coin state.The operation sequence combines the Hadamard gate and the shift operator.Source: C. Discrete-time quantum walk · p. 15
  • The example applies periodic boundary conditions when the walker shifts from the |00⟩ position.The unitary operator defines a single time step of the walk.Source: C. Discrete-time quantum walk · pp. 15–16

D. Additional results

The additional results show diffusion and denoising examples for BloodMNIST and BraTS2020, alongside classical Markov-chain configurations. The visualizations track sample evolution and pixel-value distributions, with some reported diagnostics.

1. BloodMnist

  • BloodMNIST examples show DTQW forward diffusion reaching a uniform distribution by T = 30, followed by U-Net denoising and generation of novel samples.The figures track nine samples and the three RGB-channel distributions; KL divergence and final FID are also reported.Source: §1 · pp. 16–17, Figure S3 · p. 17
  • For BloodMNIST, classical comparisons include a fixed β = 0.9 nearest-neighbor ring and a fully connected graph with a time-dependent β scheduler.The corresponding figures show RGB-image diffusion and U-Net backward processing.Source: §1 · p. 17, Figure S4 · p. 18, Figure S5 · p. 19

2. BraTS2020

  • For BraTS2020, the paper displays 100 original and reproduced images using DTQW and a classical Markov chain, with β fixed for DTQW and time-dependent for the chain.Source: §2 · p. 18

3. PCA on both models

The PCA results present projections for BloodMNIST, BraTS2020, and FractureMNIST, comparing quantum and classical forward dynamics. The text specifies the plotted data sample counts but does not state which model performs better.

3. PCA on both models

  • PCA projections compare quantum dynamics on IBM Torino with classical nearest-neighbor and fully connected Markov-chain dynamics across BloodMNIST, BraTS2020, and FractureMNIST.The text identifies the classical nearest-neighbor setup as using β = 0.99 and the fully connected setup as using time-dependent β.Source: §3 · p. 19, Figure S10 · p. 23, Figure S11 · p. 24
  • BloodMNIST PCA projections include 3,000 original-data samples; Fig. S10 compares projections for the two distinct versions of the classical diffusion models.Source: Figure S10 · p. 23
  • BraTS2020 PCA projections include 450 generated-data samples and 450 original-data samples.The projections for the two classical diffusion-model versions are shown in Fig. S11.Source: Figure S11 · p. 24
  • The FractureMNIST PCA projections include 1,370 original-data samples, and Fig. S12 compares projections on this dataset between two distinct versions of classical diffusion models.Source: Figure S12 · p. 24
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