On the Generalization Limits of Quantum Generative Adversarial Networks with Pure State Generators

TL;DR

This work analytically derive a lower bound for the discriminator quality given by the fidelity between the pure-state output of the generator and the target data distribution, thereby providing a theoretical explanation for the limitations observed in current models.

Abstract

We investigate the capabilities of Quantum Generative Adversarial Networks (QGANs) in image generations tasks. Our analysis centers on fully quantum implementations of both the generator and discriminator. Through extensive numerical testing of current main architectures, we find that QGANs struggle to generalize across datasets, converging on merely the average representation of the training data. When the output of the generator is a pure-state, we analytically derive a lower bound for the discriminator quality given by the fidelity between the pure-state output of the generator and the target data distribution, thereby providing a theoretical explanation for the limitations observed in current models. Our findings reveal fundamental challenges in the generalization capabilities of existing quantum generative models. While our analysis focuses on QGANs, the results carry broader implications for the performance of related quantum generative models.

Figures (6)

• Figure 1: (a) IQGAN circuit. (b) (i) Generated image after training the IQGAN circuit on class 3 of the MNIST dataset, using PCA, an 8-qubit circuit and angle embedding. (ii) Average image of MNIST class 3. (iii) Generated image after simultaneously training the IQGAN on MNIST classes 3 and 6. (iv) Generated image after training on class 0 of the CIFAR-10 dataset, with PCA, an 8-qubit circuit and angle embedding. (v) Images produced by the inverse PCA model, using a random input (range [0,1] and normalized). (vi) Generated image after training the IQGAN on class 3 of MNIST without PCA, using a 16-qubit circuit, reducing the size of the training samples by 50% and using amplitude embedding. (c) (i) Average over each class of MNIST. (ii) Generated images of IQGAN-784 for each class without embedded noise. (iii) Generated images of IQGAN-784 for each class with embedded noise.
• Figure 2: (a) QuGAN circuit architecture. (b) Generated images under various training settings: (i) QuGAN trained on class 3 of MNIST using PCA, an 8-qubit circuit, and angle embedding. (ii) QuGAN trained simultaneously on classes 3, 6, and 9 of MNIST with PCA, an 8-qubit circuit, and angle embedding. (iii) QuGAN trained on class 0 of CIFAR-10 using PCA, an 8-qubit circuit, and angle embedding. (iv) QuGAN trained on class 3 of MNIST without PCA, using a 16-qubit circuit, 50% downsampling of training data, and amplitude embedding. Note, that each generated image was sampled every 40 batches during the training process.
• Figure 3: FID values as a function of the number of principal components. For each model, samples drawn from the trained generator circuit (blue markers with solid lines) are compared against a classical baseline obtained by uniformly random sampling of latent bitstrings (orange markers with dashed lines).
• Figure 4: We show the leading eigenvectors of $\rho_\text{data}$ (first principal components) for each class separately in the MNIST dataset.
• Figure 5: Loss curves for: (a) IQGAN and (b) QuGAN trained on class 3 of MNIST. (c) QuGAN trained on classes 3,6, and 9 of MNIST. Loss values were sampled every 40 batches throughout the training process.