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A good item must exist, so assume $1\le k<N$; if $k=0$, the requested search is impossible, while $k=N$ makes every output good. Put $p=k/N$ and $\theta =\arcsin \sqrt{p}$. Normalize the good and bad components of the uniform superposition state asThe Boolean phase oracle negates $|g⟩$ and fixes $|b⟩$. The Grover diffusion operator is $-I_{{\psi }_0}=2|{\psi }_0⟩⟨{\psi }_0|-I$. Its implementation is independent of $f$: conjugate the reflection about $|0^n⟩$ by Hadamard gates. One Grover search algorithm iteration isHence, by the Grover rotation angle formula,Choose the nonnegative integer $j$ nearest to $\pi /(4\theta )-1/2$. Its final angle differs from $\pi /2$ by at most $\theta$, so a computational-basis measurement succeeds with probability at least ${\cos }^2\theta =1-p$. The allowed small-density regime includes $p\le 1/3$, giving the requested $2/3$ bound. Each iteration uses one Boolean phase oracle query, and $j=O(1/\sqrt{p})=O(\sqrt{N/k})$.

For $k=N/4$, $\theta =\pi /6$ and one iteration reaches $3\theta =\pi /2$ exactly. Thus one query gives a good outcome with certainty:This is exact Grover search on four entries applied to a marked fraction of one quarter, not only to a four-element register.

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.

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.