Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 57 / 2 / a / Solution

Put $L=2^N$ and define $\theta =\arcsin \sqrt{M/L}$. The normalized good and bad vectors have disjoint computational basis supports, so they are orthogonal. Counting the terms in the uniform quantum superposition givesSince $V_0=I-2|0^N⟩⟨0^N|$, the Hadamard transform gives $-H^{\otimes N}{V}_0{H}^{\otimes N}=2|\varphi ⟩⟨\varphi |-I$, the Grover diffusion operator. On the ordered orthonormal basis $(|{\Psi }_g⟩,|{\Psi }_b⟩)$, the two reflection operators have matricesTheir product isThis is a plane rotation through angle $-2\theta$ in the displayed coordinate convention, with the invariant good-bad plane preserved by both operators. In particular its action sends $(\sin a,\cos a)$ to $(\sin (a+2\theta ),\cos (a+2\theta ))$. Induction therefore gives the Grover rotation angle formula