Past exam of the computer science course of the University of Oxford / 2020 / Quantum information / 1 / c / Solution

Swap the first two computational vectors and fix the third:This permutation matrix has orthonormal columns; equivalently, $U^{†}=U$ and $U^2=I$, proving unitarity. Its first column gives $U|0⟩=|1⟩=|{\varphi }_1⟩$, as required. A unitary must also specify what happens to the rest of the basis, which is why a single outer product alone would not suffice.