permutation matrix [@CurtisReiner62], the output qubit will be a mixture of $\ket{0}$ or $\ket{1}$: this we call the [*classical special case*]{}. This way the quantum neuron can simulate any classical neuron as defined above. The generalisation recipe summarised in FIG. \[fig:neuron\_generalisation\] also illustrates how any irreversible classical computation can be recovered as a special case from reversible classical computation (by ignoring the dummy and copied bits), which in turn can be recovered as a special case from quantum computation.

![\[fig:neuron\_generalisation\]Diagram summarising our method of generalising the classical irreversible neuron with Heaviside activation function, first to a reversible neuron represented by a permutation matrix (P), and finally to a quantum reversible computation, represented by a unitary operator (U).](neuron_generalisation_greybox_small.png){width="\linewidth"}

### The network {#the-network .unnumbered}

In order to form a neural network, classical neurons are connected together in various configurations. Here, we consider feedforward classical networks, where neurons are arranged in layers and each neuron in the $l^{th}$ layer is connected to every neuron in the $(l-1)^{th}$ and $(l+1)^{th}$ layers, but with no connections within the same layer. For an example of such a classical network, see FIG. \[fig:cl\_autoencoder\]. Note that in this case the same output of a single neuron is sent 