to all the neurons in the next layer [@Nielsen15; @Azoff94].

To make the copying reversible, in line with our approach of firstly making the classical neural network reversible, we propose the recipe:\
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[*Irreversible $\rightarrow$ reversible:*]{} For a classical irreversible copying operation of a bit $b \rightarrow (b, b)$, create a classical reversible gate, which can be represented by a permutation matrix [@Feynman86], taking $(b, 0) \rightarrow (b, b)$.\
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In the quantum case the no-cloning theorem shows one cannot do this in the most naive way [@NielsenChuang00]. For a 2-qubit case, one can use a CNOT for example to copy in the classical computational basis [@Feynman86]: $\ket{b}\ket{0}\rightarrow \ket{b}\ket{b}$, if $\ket{b}\in \{\ket{0}, \ket{1} \}$. Thus one may consider replacing the copying with a CNOT. However when investigating applications of the network we realised that there are scenarios (the autoencoder in particular) where entanglement between different neurons is needed to perform the task. We have therefore chosen the following definition:\
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[*Reversible $\rightarrow$ unitary:*]{} The classical CNOT is generalised to a general 2-qubit ‘fan-out’ unitary $U_F$, with one dummy input set to $\ket{0}$, such that $\ket{b}\ket{0}\rightarrow U_F\ket{b}\ket{0}$. As this unitary does not in general copy quantum states that are non-orthogonal we 