Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 15D / a / i / Solution

Writewhere $|\beta ⟩$ is the normalized component orthogonal to $|x_0⟩$. On this real plane, $I_{x_0}$ is reflection in the line perpendicular to $|x_0⟩$, whileis reflection in the line spanned by $|{\psi }_0⟩$. Their product $Q=-J{I}_{x_0}$ is a rotation through $2\theta =\pi /3$ toward $|x_0⟩$. This is the geometric core of Grover search.