 Solution

An oracle can recognize a solution without disclosing it in advance. If an efficiently evaluable or black-box predicate has the promised unique marked input, write $f(x)=1$ exactly for that input. A Boolean quantum oracle computes this predicate reversibly asIts diagonal projector on the search register is ${\sum }_{x}f(x)|x⟩⟨x|=|w⟩⟨w|$. To implement a selectable phase, compute $f(x)$ into a zero quantum ancilla, apply a known phase to its $1$ component, and uncompute. The search register then undergoesThis requires evaluation access to the predicate, not classical knowledge of which input passes it. The $P_{\psi }$ term is known from the Hadamard transform. With Hamiltonian-oracle access, or by controlled short-time simulations of the two terms, their sum can realize continuous-time quantum search.

The unknown projector is supplied through an oracle or a physically encoded recognition rule. It cannot be manufactured for an arbitrary hidden $w$ with no such input-dependent resource. The oracle assumption and the fixed energy scale $\Delta$ are essential when interpreting the square-root search time as a computational advantage.