  A key open question in quantum computation is what advantages quantum neural
networks (QNNs) may have over classical neural networks (NNs), and in what
situations these advantages may transpire. Here we address this question by
studying the memory capacity $C$ of QNNs, which is a metric of the expressive
power of a QNN that we have adapted from classical NN theory. We present a
capacity inequality showing that the capacity of a QNN is bounded by the
information $W$ that can be trained into its parameters: $C \leq W$. One
consequence of this bound is that QNNs that are parameterized classically do
not show an advantage in capacity over classical NNs having an equal number of
parameters. However, QNNs that are parametrized with quantum states could have
exponentially larger capacities. We illustrate our theoretical results with
numerical experiments by simulating a particular QNN based on a Gaussian Boson
Sampler. We also study the influence of sampling due to wavefunction collapse
during operation of the QNN, and provide an analytical expression connecting
the capacity to the number of times the quantum system is measured.
