Controlled quantum search

Quantum searching for one of N marked items in an unsorted database of n items is solved in steps using Grover's algorithm. Using nonlinear quantum dynamics with a Gross-Pitaevskii-type quadratic nonlinearity, Childs and Young (Phys Rev A 93:022314, 2016, discovered an unstructured quantum search algorithm with a complexity , which can be used to find a marked item after repetitions, where g is the nonlinearity strength. In this work we develop an quantum search on a complete graph using a time-dependent nonlinearity which obtains one of the N marked items with certainty. The protocol has runtime , where g is related to the time-dependent nonlinearity. 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 4.