Controlled Quantum Search

Quantum searching for one of $N$ marked items in an unsorted database of $n$ items is solved in $\mathcal{O}(\sqrt{n/N})$ steps using Grover's algorithm. Using nonlinear quantum dynamics with a Gross-Pitaevskii type quadratic nonlinearity, Childs and Young discovered an unstructured quantum search algorithm with a complexity $\mathcal{O}( \min \{ 1/g \, \log (g n), \sqrt{n} \} )$, which can be used to find a marked item after $o(\log(n))$ repetitions, where $g$ is the nonlinearity strength [PhysRevA.93.022314]. In this work we develop a structured search on a complete graph using a time dependent nonlinearity which obtains one of the $N$ marked items with certainty. The protocol has runtime $\mathcal{O}((N^{\perp} - N) / (G \sqrt{N N^{\perp}}) ) if N^{\perp} > N$, where $N^{\perp}$ denotes the number of unmarked items and $G$ is related to the time dependent nonlinearity. If $N^{\perp} \leq N$, we obtain a runtime $\mathcal{O}( 1 )$. We also extend the analysis to a quantum search on general symmetric graphs and can greatly simplify the resulting equations when the graph diameter is less than 5.