 Solution

ID: 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:

U = ∣1 ⟩ ⟨ 0∣ + ∣0 ⟩ ⟨ 1∣ + ∣2 ⟩ ⟨ 2∣ = 010100001 .

This permutation matrix has orthonormal columns; equivalently,

U^{†} = U

and

U^{2} = I

, proving unitarity. Its first column gives

U ∣0 ⟩ = ∣1 ⟩ = ∣ ϕ_{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.

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