Latent-Conditioned Parameterized Quantum Circuits as Universal Approximators for Distributions over Quantum States

Abstract: Many applications in quantum simulation, quantum chemistry, and quantum machine learning require not a single quantum state but an ensemble of states characterizing the heterogeneity of a target system. Preparing such ensembles state-by-state is prohibitive in both variational and fault-tolerant settings, motivating a generative-modeling approach. We introduce latent-conditioned parameterized quantum circuits (LPQCs), a hybrid quantum-classical framework in which classical neural networks map a latent variable sampled from a prior distribution to the parameters of a parameterized quantum circuit. We prove that LPQCs are universal approximators for probability measures over density operators in the $1$-Wasserstein distance, extending classical universal approximation theorems to the quantum-distribution setting. We additionally introduce a multimodal latent prior and a mixture-of-experts circuit architecture, and show that it empirically alleviates the barren plateau problem during optimization. Numerical experiments validate the framework on a synthetic multi-cluster ensemble of mixed quantum states and on a QM9-derived ensemble of 3-D molecular structures. In these tasks, LPQC outperforms recent quantum generative baselines while remaining competitive with typical classical baselines at substantially reduced output dimensionality. By leveraging classical expressivity in the latent space, LPQCs offer a tractable route to quantum generative modeling.