Controlled Quantum Search

TL;DR

This paper develops a structured quantum search algorithm on complete graphs using time-dependent nonlinearity, achieving certainty in finding marked items with optimized runtime, extending to general symmetric graphs with simplified analysis for small diameters.

Contribution

It introduces a novel structured quantum search method employing time-dependent nonlinearity, improving runtime efficiency and certainty in finding marked items, and extends the approach to general symmetric graphs.

Findings

Achieves certainty in finding marked items with structured nonlinearity.

Provides runtime formulas depending on marked and unmarked items.

Extends analysis to symmetric graphs with diameter less than 5.

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 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.