Grover unstructured search

For an unstructured search space of size $N$ with a single marked item, Grover's algorithm reaches the marked state with high probability using $O(\sqrt N)$ oracle queries, versus $O(N)$ expected queries for classical exhaustive search — a quadratic query-complexity separation, not a claim about wall-clock time on any given hardware. The construction alternates an oracle $U_f$ that flips the sign of the marked amplitude with the diffusion operator $2|s\rangle\langle s|-I$; each pass rotates the state by $2\theta$ (where $\sin\theta=1/\sqrt N$) toward the marked subspace, and the success amplitude after $k$ iterations is $\sin\big((2k+1)\theta\big)$, so overshooting past the optimal $k\approx\frac{\pi}{4}\sqrt N$ actually reduces success probability. This record's two-qubit ($N=4$) toy oracle and diffusion layer are checked against the ideal circuit — small-instance evidence for the construction — while the $O(\sqrt N)$ asymptotic claim rests on the cited literature and assumes a coherent oracle where oracle calls, not classical readout, dominate cost.