nonlinear, dissipative dynamics of neural network computing and the linear, reversible dynamics of quantum computing [@Schuld14]. A key reason for wanting well-functioning quantum neural networks is that these could do for quantum inputs what classical networks can do for classical inputs, e.g. compressing data encoded in quantum superpositions to a minimal number of qubits.

We here accordingly focus on creating quantum generalisations of classical neural networks, which can take quantum inputs and process them coherently. Our networks contribute to a research direction known as [ *quantum learning*]{}  [@Bisio10; @Sasaki02; @Sentis15; @Banchi16; @Palittapongarnpim16] which concerns learning and optimising with truly quantum objects. The networks provide a route to harnessing the powerful neural network paradigm for this purpose. Moreover they are strict generalisations of the classical networks, providing a clear framework for comparing the power of quantum and classical neural networks.

The networks generalise classical neural networks to the quantum case in a similar sense to how quantum computing generalises classical computing. We start with a common classical neural network family: feedforward perceptron networks. We make the invidual neurons reversible and then naturally generalise them to being quantum reversible (unitary). This resolves the classical-quantum clash mentioned above from [@Schuld14]. An efficient training 