The parallel Grover as dynamic system

TL;DR

This paper models the parallel Grover algorithm as a dynamic system, revealing its geometric interpretation and proposing an iteration count for solving the 3-item search problem efficiently.

Contribution

It provides a new dynamic system representation of the parallel Grover algorithm and extends understanding to the case of three items, with an optimized iteration count.

Findings

Parallel Grover can be represented as a dynamic system.

For k=2, it is interpretable as a rotation in 3D space.

Proposes approximately 1.51√N iterations for k=3.

Abstract

A sequential application of the Grover algorithm to solve the iterated search problem has been improved by Ozhigov by parallelizing the application of the oracle. In this work a representation of the parallel Grover as dynamic system of inversion about the mean and Grover operators is given. Within this representation the parallel Grover for $k = 2$ can be interpreted as rotation in three-dimensional space and it can be shown that the sole application of the parallel Grover operator does not lead to a solution for $k > 2$. We propose a solution for $k = 3$ with a number of approximately $1.51\sqrt{N}$ iterations.