Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 10D / b / Solution

When $k=N/4$,One Grover search algorithm iteration therefore givesThe selective sign change $I_G$ is implemented with one application of the Boolean quantum oracle: prepare its answer qubit in $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$ and use quantum phase kickback, obtaining $(-1{)}^{f(x)}$ on each $|x⟩$. The remaining operations $H_n$, $I_0$, and the final computational-basis measurement are independent of $f$. Measurement of $|{\psi }_G⟩$ returns some $x\in G$ with probability one, so exactly one query to $U_f$ suffices.