$\zeta$-QVAE: A Quantum Variational Autoencoder utilizing Regularized Mixed-state Latent Representations

A major challenge in quantum computing is its application to large real-world datasets due to scarce quantum hardware resources. One approach to enabling tractable quantum models for such datasets involves finding low-dimensional representations that preserve essential information for downstream analysis. In classical machine learning, variational autoencoders (VAEs) facilitate efficient data compression, representation learning for subsequent tasks, and novel data generation. However, no quantum model has been proposed that captures these features for direct application to quantum data on quantum computers. Some existing quantum models for data compression lack regularization of latent representations. Others are hybrid models with only some internal quantum components, impeding direct training on quantum data. To address this, we present a fully quantum framework, $\zeta$-QVAE, which encompasses all the capabilities of classical VAEs and can be directly applied to map both classical and quantum data to a lower-dimensional space, while effectively reconstructing much of the original state from it. Our model utilizes regularized mixed states to attain optimal latent representations. It accommodates various divergences for reconstruction and regularization. Furthermore, by accommodating mixed states at every stage, it can utilize the full-data density matrix and allow for a training objective defined on probabilistic mixtures of input data. Doing so, in turn, makes efficient optimization possible and has potential implications for private and federated learning. In addition to exploring the theoretical properties of $\zeta$-QVAE, we demonstrate its performance on genomics and synthetic data. Our results indicate that $\zeta$-QVAE learns representations that better utilize the capacity of the latent space and exhibits similar or better performance compared to matched classical models.