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Amplitude amplification increases the probability of obtaining an answer accepted by a specified Boolean function. It requires a coherent preparation $A$, its inverse, and a marked-state phase oracle that negates accepted components. Alternating that sign change with a reflection operator about the prepared state directs interference toward the accepted subspace. Brassard, Høyer, Mosca, and Tapp developed the general framework, extending Grover's algorithm beyond a uniform quantum superposition. ^[ref].

If a normalized quantum state has good amplitude $\sin \theta$, the product of the reflection in the bad axis and the reflection in the state line rotates its good-bad plane by $2\theta$. After $k$ iterations the good probability isso $O(1/\sin \theta )$ iterations raise a small good amplitude close to one. Success subsequently oscillates, making the stopping time part of the algorithm.

Compute the exact Grover search on four entries directly. Order the computational basis as $00,01,10,11$ and mark $11$. The Hadamard transform prepares amplitudes $(1/2,1/2,1/2,1/2)$. The marked-state phase oracle changes these to $(1/2,1/2,1/2,-1/2)$, whose arithmetic mean is $\overline{a}=1/4$. The Grover diffusion operator performs inversion about the mean:The three unmarked amplitudes become $1/2-1/2=0$, while the marked amplitude becomes $1/2-(-1/2)=1$. All three stages have squared norm one. Here $\sin \theta =1/2$, so $\theta =\pi /6$ and one iteration gives ${\sin }^2(3\theta )=1$, agreeing with the explicit calculation.

Grover's original search paper was submitted on 29 May 1996; its algorithm combines selective sign changes with inversion about the mean. The example exposes the beauty of that construction: a change initially invisible to computational-basis measurement becomes complete cancellation of every wrong answer. ^[ref].

The marked-state phase oracle is an assumed operation, whose implementation cost must be counted. Computational-basis measurement yields a candidate label, here $11$; using it to retrieve a record requires the relevant data access. Preparing a quantum superposition does not make all its entries available as classical outputs. The improvement concerns the number of oracle queries needed to obtain an accepted label.

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.