  Quantum generative models, in providing inherently efficient sampling
strategies, show promise for achieving a near-term advantage on quantum
hardware. Nonetheless, important questions remain regarding their scalability.
In this work, we investigate the barriers to the trainability of quantum
generative models posed by barren plateaus and exponential loss concentration.
We explore the interplay between explicit and implicit models and losses, and
show that using implicit generative models (such as quantum circuit-based
models) with explicit losses (such as the KL divergence) leads to a new flavour
of barren plateau. In contrast, the Maximum Mean Discrepancy (MMD), which is a
popular example of an implicit loss, can be viewed as the expectation value of
an observable that is either low-bodied and trainable, or global and
untrainable depending on the choice of kernel. However, in parallel, we
highlight that the low-bodied losses required for trainability cannot in
general distinguish high-order correlations, leading to a fundamental tension
between exponential concentration and the emergence of spurious minima. We
further propose a new local quantum fidelity-type loss which, by leveraging
quantum circuits to estimate the quality of the encoded distribution, is both
faithful and enjoys trainability guarantees. Finally, we compare the
performance of different loss functions for modelling real-world data from the
High-Energy-Physics domain and confirm the trends predicted by our theoretical
results.
