Expressivity of parameterized quantum circuits for generative modeling of continuous multivariate distributions

Parameterized quantum circuits have been extensively used as the basis for machine learning models in regression, classification, and generative tasks. For supervised learning their expressivity has been thoroughly investigated and several universality properties have been proven. However, in the case of quantum generative modeling, the situation is less clear, especially when the task is to model distributions over continuous variables. In this work, we focus on expectation value sampling-based models; models where random variables are sampled classically, encoded with a parametrized quantum circuit, and the expectation value of fixed observables is measured and returned as a sample. We prove the universality of such variational quantum algorithms for the generation of multivariate distributions. Additionally, we provide a detailed analysis of these models, including fundamental upper bounds on the dimensionality of the distributions these models can represent. We further present a tight trade-off result connecting the needed number of measurements and qubit numbers in order to have universality for a desired dimension of output distribution within an error tolerance. Finally we also show that the data encoding strategy relates to the so-called polynomial chaos expansion, which is an analog of the Fourier expansion. Our results may help guide the design of future quantum circuits in generative modeling tasks.