Controlled quantum search

Abstract

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 (Phys Rev A 93:022314, 2016, https://doi.org/10.1103/PhysRevA.93.022314 ) 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. 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 $$\mathcal {O}(n /(g \sqrt{N(n-N)}))$$ , 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.