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Query a known set of $m$ inputs and store its output table. If no function collision occurs there, each of its $m$ distinct outputs has exactly one partner in the complement, for a two-to-one function. A compute-phase-uncompute construction marks those partners using two function queries and reversible table comparison. Known-subset Grover search then uses $O(\sqrt{(N-m)/m})$ such phase queries. Including table preparation and a final verification query givesChoosing $m\asymp N^{1/3}$ balances both terms and gives $T(m)=O(N^{1/3})$. The Grover rotation angle rounding bound gives success tending to one. This is a query bound; it does not make reversible table lookup free in a gate or physical-memory cost model.

For a known finite set $B$, prepare the uniform superposition state $|s_B⟩$ and implement its reflection operator $2|s_B⟩⟨s_B|-I$. If exactly $k$ basis states in $B$ are marked, the same two-dimensional Grover rotation angle argument uses $O(\sqrt{|B|/k})$ phase queries. Preparation and reflection are known operations with no calls to the unknown predicate. Their gate cost must be accounted for separately. The success guarantee requires $k>0$, and choosing the optimal iteration number uses a known $k$.

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.

Interpret a function collision as two distinct inputs with equal outputs. Without distinctness the request is vacuous, since an input paired with itself always qualifies. Set $m=\lceil N^{1/3}\rceil$ (handling small $N$ by direct queries), choose a known set $A$ of size $m$, and query all its inputs. Store a classical table of their outputs and corresponding inputs. If this table contains a function collision, return it immediately.

Otherwise the table contains $m$ distinct outputs. Because $g$ is exactly two-to-one, each $a\in A$ has exactly one partner in $B$, the known complement of $A$. None lies in $A$, and different table outputs have different partners. Thus exactly $m$ inputs in $B$, of size $L=N-m$, satisfy the known-table membership predicateBuild a marked-state phase oracle for this predicate by a compute-phase-uncompute construction: use $U_g$ on a clean answer quantum register, reversibly compare its output against the classical table, apply the conditional minus sign, undo the comparison, and apply $U_g$ again. Since the answer operation is bitwise exclusive or, $U_g^{-1}=U_g$. All workspace returns to zero, and each phase query costs two calls to $U_g$.

Perform known-subset Grover search in the span of the computational basis states labelled by $B$, starting in their uniform superposition state. Reflection about this known state uses no $U_g$ queries; choosing $A$ as an initial interval makes $B$ a known interval that can also be encoded by its own $L$ labels. Its marked fraction is $m/L$, so the previous Grover rotation angle analysis uses $O(\sqrt{L/m})$ phase queries. On measuring $b$, one further $U_g$ query checks the output and retrieves its partner $a$ from the table. The inputs are distinct because the two sets are disjoint.

Consequently the quantum query complexity isFor large $N$, rounding the optimal Grover search algorithm iteration count gives success at least $1-m/(N-m)>1/2$. An internal table function collision already gives certain success. The balance $m\asymp \sqrt{N/m}$ explains the cube-root choice. This is the Brassard–Høyer–Tapp collision algorithm: its table uses $O(m\log N)$ classical bits. The bound counts calls to $U_g$, as requested; it does not charge table lookup and its reversible implementation as constant-time gates, or claim an overall polynomial time in $\log N$.

Because $H_n|0^n⟩=|{\psi }_0⟩$,which is the reflection operator in the line spanned by $|{\psi }_0⟩$. Meanwhile $I_G$ changes the sign of every good basis state and fixes every bad basis state; onit is the reflection in the $|{\psi }_B⟩$ axis. Both reflections preserve $\mathcal{P}$, so their product $Q$ does too.

WriteIn Euclidean geometry, the composition of reflections in two lines through the origin has a rotation matrix through twice the oriented angle between the lines. Hence $Q$ rotates the good-bad plane by $2\theta$ from $|{\psi }_B⟩$ toward $|{\psi }_G⟩$. In particular, the Grover rotation angle formula is

Use the construction in part (c) for the marked-state phase reflection in each Grover search algorithm iteration. Every such reflection costs one query to $U_f$; the identity-oracle circuit, Hadamard gates, and $I_0$ are independent of $f$.

Starting from $|s⟩$, after $r$ iterations the success probability isby the Grover rotation angle. Choose an integerThen $(2r+1)\theta$ differs from $\pi /2$ by at most $\theta$, soFor every sufficiently large $N$ this is greater than $1/2$, whileThus measuring the search register determines the faulty input with the required constant success probability using $O(\sqrt{N})$ oracle queries.