Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 10D / b

The marked-state phase oracle changes only the amplitude of $|x_0⟩$. Since that amplitude in $|\psi ⟩$ is $1/2$,Also,It follows that $A|\psi ⟩=-|\psi ⟩$ and $A|x_0⟩=|x_0⟩-|\psi ⟩$, soThe global minus sign has no observable effect. Prepare $|\psi ⟩=H^{\otimes 2}|00⟩$, query $I_{x_0}$ once, apply the fixed operator $A$, and perform a quantum measurement in the computational basis. The outcome is $x_0$ with probability one. This is exact Grover search on four entries.