  We propose an approach for learning probability distributions as
differentiable quantum circuits (DQC) that enable efficient quantum generative
modelling (QGM) and synthetic data generation. Contrary to existing QGM
approaches, we perform training of a DQC-based model, where data is encoded in
a latent space with a phase feature map, followed by a variational quantum
circuit. We then map the trained model to the bit basis using a fixed unitary
transformation, coinciding with a quantum Fourier transform circuit in the
simplest case. This allows fast sampling from parametrized distributions using
a single-shot readout. Importantly, simplified latent space training provides
models that are automatically differentiable, and we show how samples from
distributions propagated by stochastic differential equations (SDEs) can be
accessed by solving stationary and time-dependent Fokker-Planck equations with
a quantum protocol. Finally, our approach opens a route to multidimensional
generative modelling with qubit registers explicitly correlated via a (fixed)
entangling layer. In this case quantum computers can offer advantage as
efficient samplers, which perform complex inverse transform sampling enabled by
the fundamental laws of quantum mechanics. On a technical side the advances are
multiple, as we introduce the phase feature map, analyze its properties, and
develop frequency-taming techniques that include qubit-wise training and
feature map sparsification.
