  Neural networks enjoy widespread success in both research and industry and,
with the imminent advent of quantum technology, it is now a crucial challenge
to design quantum neural networks for fully quantum learning tasks. Here we
propose the use of quantum neurons as a building block for quantum feed-forward
neural networks capable of universal quantum computation. We describe the
efficient training of these networks using the fidelity as a cost function and
provide both classical and efficient quantum implementations. Our method allows
for fast optimisation with reduced memory requirements: the number of qudits
required scales with only the width, allowing the optimisation of deep
networks. We benchmark our proposal for the quantum task of learning an unknown
unitary and find remarkable generalisation behaviour and a striking robustness
to noisy training data.
