QuDDPM: Quantum Denoising Diffusion Model

• QuDDPM is a quantum denoising diffusion model that employs variational quantum circuits and amplitude embedding to implement a noise-driven Markov process for generative modeling.
• It introduces quantum convolutional architectures such as QU-Net and a one-step consistency model, which drastically reduce computational complexity and accelerate sampling.
• Benchmark results show improved generative fidelity, reduced parameter count, and superior speed compared to classical diffusion models on standard image datasets.

A Quantum Denoising Diffusion Probabilistic Model (QuDDPM) generalizes classical denoising diffusion models by leveraging quantum variational circuits for the generative modeling of data—classical or quantum—via a noise-driven Markov process and reverse quantum denoising architecture. This framework exploits amplitude and angle embedding, unitarity, and the expressive power of quantum entanglement to truncate parameter count, accelerate sampling, and address limitations in classical DDPMs on high-dimensional or structured data. Key advances include the construction of quantum convolutional architectures (QU-Net), a unity-based consistency model for one-step sampling, and performance superiority on standard image-generation benchmarks with dramatically reduced computational resources (Kölle et al., 2024).

1. Mathematical Foundations and Quantum Embedding

QuDDPM applies a $T$-step Markov chain to an initial data sample $x_0 \in \mathbb{R}^N$ via a forward noising process:

$$q(x_t \mid x_{t-1}) = \mathcal{N}\left(x_t;\sqrt{1-\beta_t} x_{t-1}, \beta_t I\right),\quad t=1,\dots,T$$

with closed-form marginal for any time $t$: $$q(x_t|x_0) = \mathcal{N}\left(x_t; \sqrt{\bar\alpha_t}x_0, (1-\bar\alpha_t)I\right),\quad \bar\alpha_t=\prod_{s=1}^t(1-\beta_s)$$ Quantum implementation proceeds by amplitude-embedding the normalized classical vector $x_t$ into $n=\lceil \log_2 N \rceil$ qubits: $$|x_t\rangle = \sum_{i=0}^{2^n-1} \frac{(x_t)_i}{\|x_t\|} |i\rangle$$ At each Markov step, the classically noised $x_t$ is reloaded into the quantum register for further processing.

Reverse denoising is parameterized by a variational quantum circuit $U_\theta^{(t)}$, designed to approximate $p_\theta(x_{t-1}|x_t) \approx q(x_{t-1}|x_t,x_0)$ by measuring amplitudes from $U_\theta^{(t)}|x_t\rangle$ and minimizing a mean-squared-error (MSE) loss: $$\mathcal{L}_t(\theta) = \mathbb{E}_{x_0,\varepsilon}\Big\| \varepsilon - \varepsilon_\theta(\sqrt{\bar\alpha_t}x_0 + \sqrt{1-\bar\alpha_t}\varepsilon, t)\Big\|^2$$ where $\varepsilon \sim \mathcal{N}(0, I)$ and $\varepsilon_\theta$ is read from the quantum circuit output.

2. Quantum Circuit Architecture: Dense, Convolutional, and Consistency Models

QuDDPM exploits flexible circuit architectures:

• Q-Dense blocks: $n$ data qubits ($8\times8$ MNIST: $n=6$) with $L$ strongly entangling layers, each layer adding $3n$ learnable parameters ($\sim1000$ for optimal MNIST layout). Circuits use amplitude embedding via Möttönen decomposition, and optionally angle-embedded class labels via ancilla qubit $R_x(2\pi c/|\mathcal{C}|)$.
• Quantum U-Net (QU-Net): Quantum convolutional units embed $c_{\text{in}} \times k \times k$ classical patches into $\lceil \log_2(c_{\text{in}} k^2)\rceil$ data qubits, process via Q-Dense networks, decode amplitudes back to classical slices, and integrate skip connections analogous to classical U-Net architectures.
• Consistency Model: Leveraging circuit unitarity, all denoising steps $t=1,\dots,T$ are composed into a single circuit $U_{\text{single}} = \prod_{t=T}^1 U_\theta^{(t)}$, enabling fast one-shot image generation. Training targets mean-absolute-error on amplitudes: $$\mathcal{L}_{\text{single}}(\theta) = \|U_{\text{single}}(|x_T\rangle) - |x_0\rangle\|_1$$

3. Training Protocols, Gradient Evaluation, and Loss Functions

Optimization uses Adam with learning rates found by Bayesian search (typ. $10^{-4}$–$10^{-2}$), batch sizes $10$–$20$, and backpropagation through PennyLane–PyTorch interfaces for rotation angles. Hardware execution employs the parameter-shift rule for exact gradients, requiring two evaluations per circuit parameter. Multi-step diffusion uses MSE; single-step (one-shot) employs MAE. Training efficiency is illustrated by substantially reduced simulation runtimes: $2$ s/epoch for Q-Dense (MNIST $8\times8$) versus $134$ s/epoch for similarly sized U-Net.

4. Performance Benchmarks, Parameter Efficiency, and Sampling Speed

QuDDPM demonstrates superior or competitive performance on benchmark datasets with $10\times$ fewer parameters and significant runtime speedup:

Model	FID (CIFAR-10, 32×32)	PSNR	SSIM
U-Net	395.5	8.95	0.026
QU-Net	271.0	9.60	0.086
Q-Dense	399.4	10.06	0.061

Q-Dense achieves FID $\approx100$ (MNIST $8\times8$, $T=5$) compared to classical U-Net or Dense (120–140) at $\sim1000$ parameters. QU-Net surpasses classical U-Net by $>100$ FID points at equal parameter count for CIFAR-10. Simulator sampling is $50$–$100\times$ faster. Demonstrated single-step sampling on IBMQ hardware (10,000 shots) yields recognizable digits.

5. Quantum Advantage Mechanisms and Expressivity

Key mechanisms confer quantum advantage over classical DDPMs:

• Amplitude embedding: Compact representation of high-dimensional features in exponential Hilbert space—$n$ qubits model $2^n$ dimensions.
• Strong entanglement: Every circuit layer globally mixes amplitude space, yielding a highly expressive function approximator with minimal parameters.
• Unitarity-driven collapse: The consistency model exploits circuit unitarity for single-pass sampling, reducing the sampling path from $T$ steps to one.
• Parameter efficiency: Competitive or superior image generation is achieved in much smaller circuits due to quantum encoding richness.

6. Open Challenges and Future Research Directions

Ongoing research addresses:

• Measurement normalization constraints for non-uniform datasets (e.g., CIFAR-10 heterogeneity).
• Alternative data encoding (patching, hybrid amplitude-angle methods).
• Problem-specific entangler optimization to further reduce circuit depth and qubit number.
• Caching strategies and low-precision simulation tools for scaling to larger images.
• Hybrid post-processing to correct for normalization artifacts in quantum outputs.

Potential future developments include hybrid quantum-classical protocols, deployment on noisy intermediate-scale quantum (NISQ) devices, and the extension to structured generative tasks such as image latents, quantum state ensembles, and high-dimensional function modeling (Kölle et al., 2024).

7. Comparative Analysis and Limitations

Compared to classical models, QuDDPM achieves lower FID scores and higher SSIM/PSNR at fixed parameter count, evidencing improved generative fidelity. However, normalization constraints and quantum measurement artifacts present open limitations for certain datasets. Theoretical questions regarding generalization bounds, analytic ELBO construction, and optimal loss selection (MMD, Wasserstein, MAE/MSE) remain active research areas.

QuDDPM establishes a blueprint for quantum-enhanced diffusion-based generative modeling by unifying classical Markov noise frameworks with expressive, unitary quantum circuit denoising, and it substantiates increased efficiency and accuracy on machine learning benchmarks through empirical study (Kölle et al., 2024).