The answer to this question begins with a statement of my interest:
|⟩=|()⟩ where :2→2 is a reversible Boolean function. These are exactly the permutations of bitstrings
Assuming one has to perform $U$ followed by measuring the system. If $f$ is also a permutation function, I am led to think that $U$ can be computed classically, and accordingly re-arrange the measurement bases.
Is this intuition correct?
Notice that the statement regards only $U$; it should stay valid for any input and any measurement.
