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Generative modelling is an important unsupervised task in machine learning.In this work, we study a hybrid quantum-classical approach to this task, basedon the use of a quantum circuit Born machine. In particular, we considertraining a quantum circuit Born machine using $f$-divergences. We first discussthe adversarial framework for generative modelling, which enables theestimation of any $f$-divergence in the near term. Based on this capability, weintroduce two heuristics which demonstrably improve the training of the Bornmachine. The first is based on $f$-divergence switching during training. Thesecond introduces locality to the divergence, a strategy which has provedimportant 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 theirestimation. In particular, we generalise existing algorithms for estimating theKullback-Leibler divergence and the total variation distance to obtain afault-tolerant quantum algorithm for estimating another $f$-divergence, namely,the Pearson divergence.

F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits