Scaling Quantum Machine Learning without Tricks: Full-Resolution and Diverse Image Generation

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.

Generative models have become central across science and industry, from image and text synthesis to the design of molecules and materials. Quantum generative models are considered one of the most promising applications for quantum computers, since a quantum circuit naturally produces samples from the distribution it encodes, and for suitable circuits that distribution is believed to be hard for any classical computer to reproduce. A leading strategy trains these models on a classical computer and reserves the quantum device for generating samples at deployment. This is possible when the training loss can be evaluated on a classical computer. A prime example is the maximum mean discrepancy (MMD$^2$), a moment-matching loss that compares the model and the data through their Pauli-$Z$ correlations. Research so far has asked whether such models can be trained and whether their sampling is hard; whether minimizing such an objective yields a model that generalizes, rather than one that merely reproduces the training statistics, remains poorly understood. We benchmark a broad set of quantum and classical generative models by direct sampling and show that models trained with a moment-matching loss generally show worse generalization than the likelihood-trained models. We show this on two application-inspired datasets: first a cardinality-constrained dataset at up to $30$ qubits and second a dataset of genomic single-nucleotide variants, whose valid set is the observed data. These results indicate that a converged moment-matching loss is not a reliable measure of generalization, and that train-classical, deploy-quantum workflows will need approaches that target generalization directly, leaving open whether better training objectives suffice or whether the model architectures themselves must change.

Quantum generative adversarial networks (QGANs) have attracted increasing attention for image generation using parameterized quantum circuits. Existing amplitude-based approaches face two key limitations: pixel locations are typically encoded by computational-basis indices or address qubits, causing quantum resources to grow with image resolution; meanwhile, jointly decoding many pixels from normalized quantum states introduces probability competition among pixels and limits precise pixel-wise control. To address these issues, we reformulate quantum image generation as coordinate-conditioned implicit function learning. Our method takes spatial coordinates and latent variables as inputs, uses a classical embedding network to generate input-dependent circuit parameters, and evaluates a variational quantum circuit at each coordinate. Pixel intensities are directly obtained from the expectation value of a dedicated color qubit, and a complete image is generated by querying all spatial coordinates. This design decouples image resolution from address-qubit requirements and avoids shared probability-normalization constraints across pixels. We further design a specialized variational quantum circuit to provide structural inductive bias for coordinate-conditioned generation. Simulated experiments on two benchmark datasets show that our method outperforms FRQI-based generation and PQWGAN in visual and quantitative quality while using fewer qubits, and also achieves better generation quality than the corresponding classical baseline.