Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 10C (Quantum Information and Computation)

The selective phase rotation $R_g^{\varphi }$ multiplies the component parallel to $|g⟩$ by $e^{i\varphi }$ and fixes its orthogonal complement. Therefore

Conjugating the reflection operator by a unitary operator reflects about the conjugated state:Thus one may take $|\psi ⟩=|{\psi }_{\mathrm{in}}⟩=U|00⟩$.

For $p=1/4$, put $|{\psi }_{\mathrm{in}}⟩=\frac{1}{2}|g⟩+\frac{\sqrt{3}}{2}|b⟩$. After the first phase rotation the state isSince $U{R}_{00}^{\pi }{U}^{†}=I-2|{\psi }_{\mathrm{in}}⟩⟨{\psi }_{\mathrm{in}}|$, the requested state isIt is proportional to $|g⟩$ exactly when $e^{i\varphi }=-1$, hence when $\varphi =\pi$. The state is then $-|g⟩$, whose global phase has no physical effect.

Choose $U$ to prepare the uniform superposition, so the marked-state probability is $p=1/4$. Apply the marked-state phase oracle $R_g^{\pi }$, followed by the reflection $U{R}_{00}^{\pi }{U}^{†}$ about the prepared state. Part (iii) shows that this single Grover iteration produces $-|g⟩$. A projective measurement in the four-item basis therefore returns the marked element with probability one. This is exact Grover search on four entries.