  Generative modelling is an important unsupervised task in machine learning.
In this work, we study a hybrid quantum-classical approach to this task, based
on the use of a quantum circuit Born machine. In particular, we consider
training a quantum circuit Born machine using $f$-divergences. We first discuss
the adversarial framework for generative modelling, which enables the
estimation of any $f$-divergence in the near term. Based on this capability, we
introduce two heuristics which demonstrably improve the training of the Born
machine. The first is based on $f$-divergence switching during training. The
second introduces locality to the divergence, a strategy which has proved
important in similar applications in terms of mitigating barren plateaus.
Finally, we discuss the long-term implications of quantum devices for computing
$f$-divergences, including algorithms which provide quadratic speedups to their
estimation. In particular, we generalise existing algorithms for estimating the
Kullback-Leibler divergence and the total variation distance to obtain a
fault-tolerant quantum algorithm for estimating another $f$-divergence, namely,
the Pearson divergence.
