  Fault-tolerant quantum computers offer the promise of dramatically improving
machine learning through speed-ups in computation or improved model
scalability. In the near-term, however, the benefits of quantum machine
learning are not so clear. Understanding expressibility and trainability of
quantum models-and quantum neural networks in particular-requires further
investigation. In this work, we use tools from information geometry to define a
notion of expressibility for quantum and classical models. The effective
dimension, which depends on the Fisher information, is used to prove a novel
generalisation bound and establish a robust measure of expressibility. We show
that quantum neural networks are able to achieve a significantly better
effective dimension than comparable classical neural networks. To then assess
the trainability of quantum models, we connect the Fisher information spectrum
to barren plateaus, the problem of vanishing gradients. Importantly, certain
quantum neural networks can show resilience to this phenomenon and train faster
than classical models due to their favourable optimisation landscapes, captured
by a more evenly spread Fisher information spectrum. Our work is the first to
demonstrate that well-designed quantum neural networks offer an advantage over
classical neural networks through a higher effective dimension and faster
training ability, which we verify on real quantum hardware.
