Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 2 / b / ii / Solution

Set $|\Psi ⟩=U|0^m⟩$. Unitary conjugation givesThe operator negates $|\Psi ⟩$ and fixes every vector in $\mathcal{S}$ orthogonal to it. It is therefore the reflection in a hyperplane whose normal is $|\Psi ⟩$.