Step through Grover's quantum search algorithm on a 16-item register. Watch amplitude amplification boost the target item's probability in O(√N) iterations versus O(N) classically.
Grover's algorithm uses two operations per iteration: an oracle that flips the sign of the target amplitude, and a diffusion operator that reflects amplitudes about their mean. After ~√N iterations, the target probability approaches 1.
Select a target item in the 16-element register. Step through iterations and watch the amplitude histogram. The target bar grows while others shrink. Compare quantum O(√N) = 3 iterations with classical O(N) = 16.
Grover's algorithm provides a provably optimal quadratic speedup for unstructured search — no quantum algorithm can do better. For a database of 1 million items, it finds the answer in ~1000 queries instead of 500,000.