Harnessing the potential of near-term quantum computers for practical applications remains a significant challenge. Hybrid quantum-classical machine learning approaches offer a promising pathway. This research explores the use of parameterized quantum circuits (PQCs) as a novel approach to enhance deep generative models. While sophisticated quantum algorithms like Shor's and Grover's offer theoretical speedups, they require specific problem structures, and current quantum hardware limitations, including qubit fidelity and coherence time, restrict their applicability. Quantum Machine Learning (QML) leverages quantum phenomena like superposition and entanglement to improve learning and generalization. Existing QML methods include quantum kernel-based machine learning and PQC-based neural networks. PQCs have been successfully used in Quantum Generative Adversarial Networks (QGANs), showing advantages in learning discrete patterns. However, scaling PQCs in current hardware is difficult, limiting their use to low-dimensional datasets. Previous work has demonstrated that quantum-circuit-based generative models can learn and sample the prior distribution of a GAN, improving image generation quality. This study further investigates leveraging PQCs as programmable quantum priors to enhance deep generative algorithms for practical applications, specifically focusing on ghost imaging, image reconstruction, and manipulation. The paper introduces QDGP, utilizing programmable quantum priors as the latent space for data distributions and a classical neural network for data generation, making it suitable for near-term quantum devices. The authors use the pre-trained BigGAN generator as the backbone, with PQC-generated distributions forming the quantum latent space. By jointly optimizing the PQC and BigGAN using a problem-specific loss function, they examine high-resolution ghost imaging reconstruction under undersampling conditions, and image restoration and manipulation in computer vision tasks. This hybrid approach aims to overcome the limitations of classical methods and demonstrate the practical benefits of near-term quantum devices in large-scale generative AI.

The authors review existing classical and quantum approaches to generative modeling. Classical methods like Deep Image Prior (DIP) utilize the inherent image statistics captured by convolutional neural networks. Generative Adversarial Networks (GANs) model complex data by integrating a learnable generator and a latent space. DIP fixes the latent space and optimizes the generator, while Deep Generative Prior (DGP) concurrently optimizes both, extending the GAN manifold. Existing quantum approaches often focus on low-dimensional data or utilize non-programmable quantum devices. The authors highlight the gap in research utilizing PQCs as programmable quantum priors for high-dimensional data generation and practical applications. They compare their proposed method (QDGP) to existing approaches, emphasizing its unique capabilities in handling high-dimensional data and its suitability for near-term quantum hardware.

The QDGP algorithm uses a pre-trained BigGAN generator as its backbone. The latent space of the BigGAN generator is replaced with a parameterized quantum circuit (PQC). The PQC, denoted as U(θ), acts on an initial quantum state ρ(0) to produce a learned quantum state ρ(θ) = U(θ)ρ(0), where θ represents the trainable quantum parameters. By measuring some observable M of the quantum state, a quantum latent code is obtained as ⟨M⟩ = Tr(ρ(θ)M). This code is then fed into the BigGAN generator. The PQC structure, described in detail in Supplementary Note 1, includes variational quantum layers and entangling layers. The authors use a hardware-efficient circuit structure, which is more practical for near-term quantum devices. The data-reuploading encoding style is employed for the quantum encoding layer to enhance expressivity. In ghost imaging experiments, a physical forward model is incorporated into the loss function, allowing for joint fine-tuning of the PQC and the BigGAN generator. The loss function incorporates the L2-norm difference between experimental and calculated bucket signals, along with a total variation (TV) constraint to enhance image smoothness. The authors use both randomly initialized and pre-trained BigGAN generators to compare the performance. For computer vision tasks (category transfer, inpainting, colorization, and super-resolution), the QDGP uses a pre-trained BigGAN model, utilizing feature loss and mean squared error (MSE) loss to jointly fine-tune the PQC and the generator. Gradients for the classical parameters of the generator are obtained via automatic differentiation, while gradients for the quantum parameters of the PQC are calculated using the parameter-shift rule. The optimization process uses the Adam optimizer. The authors use various metrics to evaluate the performance of QDGP, including Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM).

The QDGP algorithm shows superior performance in several key areas: **Ghost Imaging:** QDGP consistently outperforms both DGP and Differential Ghost Imaging (DGI) across various sampling rates, achieving higher PSNR and SSIM values, especially when reconstructing objects not present in the training data. This highlights the improved generalization ability of QDGP. **High-Resolution Ghost Imaging:** In experiments with higher-resolution images (128x128), QDGP demonstrates better reconstruction quality than DGP and DGI, particularly with the SSIM metric, even when the target image (wasp's wing) is significantly different from the training data. The pretrained QDGP shows improved PSNR over the randomly initialized version but a lower SSIM value indicating a trade-off between detail and noise reduction. **Computer Vision Tasks:** QDGP shows significant advantages in image inpainting and colorization over DGP, outperforming it by a substantial margin in PSNR and SSIM. In category transfer and super-resolution tasks, it achieves comparable or slightly better performance. **Latent Space Analysis:** The probability density visualization of the latent space shows QDGP's quantum latent space is more flexible and decentralized than the Gaussian latent space in DGP, which explains the enhanced out-of-distribution generalization capability. The larger variation in density between the start and end iterations of QDGP aids in finding optimal reconstruction paths for samples outside the training data. **Noise Robustness:** Experiments with noisy PQCs show that the performance of QDGP degrades gracefully with increasing noise, suggesting robustness of the proposed approach.

The results demonstrate that incorporating a programmable quantum latent space significantly enhances the capabilities of deep generative models. The QDGP algorithm leverages the strengths of both classical deep learning and quantum computation. The superior performance in ghost imaging, particularly under undersampling conditions, indicates the potential for quantum-enhanced computational imaging techniques. The improved generalization in computer vision tasks highlights QDGP's ability to handle unseen data and adapt to different image manipulation tasks. The more flexible quantum latent space, compared to the classical Gaussian prior, is a crucial factor contributing to this improved generalization. The algorithm's success in handling both natural images and biological samples which are outside the typical training dataset demonstrates its versatility. The findings are relevant to various fields, including computer vision, medical imaging, and other applications requiring high-quality image reconstruction from limited data.

This research presents a novel hybrid quantum-classical algorithm, QDGP, showcasing the potential of near-term quantum devices for enhancing deep generative models. The improved performance in ghost imaging, particularly in low-sampling regimes, and the superior results in image inpainting and colorization demonstrate the algorithm's practical advantages. The increased generation diversity and enhanced out-of-distribution generalization are key contributions. Future research could explore scaling the algorithm to even larger datasets and more complex quantum circuits, as well as investigating further applications in various fields where data-limited inverse problems are crucial.

The study's findings are based on a specific quantum hardware architecture and the use of a pre-trained BigGAN model. The performance might vary with different hardware and model architectures. While the noise robustness was tested, the impact of more significant noise levels and different noise models needs further investigation. The current implementation uses a limited number of qubits, and scalability to larger quantum systems remains a future challenge. The evaluation of category transfer primarily relies on visual inspection, lacking a quantitative metric for objective comparison between QDGP and DGP.