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Is it possible to make any gate reversible merely by retaining the input bits in the output and introducing ancilla bits as necessary? That is, given an irreversible gate with $k$ inputs and $l$ outputs, can we always find a reversible gate with $k + l$ inputs and outputs? Why or why not?

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Yes, this is always possible. To do this you need to implement, given a function $f:\mathbb{Z}^k\rightarrow\mathbb{Z}^l$, a unitary evolution $U_f$ which will take the register bits to themselves and the ancilla bits to the function: $$U_f|x\rangle|0\rangle=|x\rangle|f(x)\rangle.$$ This is part of a more general problem: is it always possible to execute an arbitrary unitary $U$ in $n$ qubits? This is possible if you know what $f$ is and you can perform a universal set of quantum gates (single-qubit gates and CNOTs being the typical example).

In terms of classical gates, though, if you can evaluate the function $f$, then there is nothing stopping you from writing the result on $l$ previously empty ancilla bits, and keeping the argument, thus making your gate reversible. The "irreversibility" of standard classical gates is simply a matter of information loss; this does not typically mean irreversibility of a thermodynamical process.

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