classical neuron to a quantum mechanical one. In the absence of measurement, quantum mechanical processes are required to be reversible, and more specifically, unitary, in a closed quantum system [@Feynman86; @NielsenChuang00]. This suggests the following procedure for generalising the neuron first to a reversible gate and finally to a unitary gate:

[*Irreversible $\rightarrow$ reversible:*]{} For an $n$-input classical neuron having $(in_1, in_2, ..., in_n )\rightarrow out$, create a classical reversible gate taking ($in_1, in_2, ..., in_n, 0) \rightarrow (in_1, in_2, ..., in_n, out)$. Such an operation can always be represented by a permutation matrix [@Muthukrishnan99]. This is a clean way of rendering the classical neuron reversible. The extra ‘dummy’ input bit is used to make it reversible [@Feynman86]; in particular, some of the ‘2 bits in – 1 bit out’ functions the neuron can implement require 3 bits to be made reversible in this manner.

[*Reversible $\rightarrow$ unitary:*]{} Generalise the classical reversible gate to a quantum unitary taking input ($\ket{\psi_{in}}_{1,2,...,n}\ket{0}) \rightarrow  \ket{\psi_{out}}_{1,2,...,n,out}$, such that the final output qubit is the output of interest. This is the natural way of making a permutation matrix unitary.

If the input is a mixture of states in the computational basis and the unitary a 