Multi-phase matching in the Grover algorithm

TL;DR

This paper introduces a multi-phase matching technique for the Grover algorithm that achieves near-perfect success probability across a wide range of marked state fractions with only a few iterations.

Contribution

It proposes a novel multi-phase matching rule that maintains high success probability in the Grover algorithm over many iterations, improving efficiency.

Findings

Success probability remains almost constant and near unity over a wide range of marked state fractions.

Six iterations of the multi-phase operation achieve over 99.8% success probability for fractions of 1/10 or larger.

The method reduces oscillations in success probability seen in previous phase matching approaches.

Abstract

Phase matching has been studied for the Grover algorithm as a way of enhancing the efficiency of the quantum search. Recently Li and Li found that a particular form of phase matching yields, with a single Grover operation, a success probability greater than 25/27 for finding the equal-amplitude superposition of marked states when the fraction of the marked states stored in a database state is greater than 1/3. Although this single operation eliminates the oscillations of the success probability that occur with multiple Grover operations, the latter oscillations reappear with multiple iterations of Li and Li's phase matching. In this paper we introduce a multi-phase matching subject to a certain matching rule by which we can obtain a multiple Grover operation that with only a few iterations yields a success probability that is almost constant and unity over a wide range of the fraction of marked items. As an example we show that a multi-phase operation with six iterations yields a success probability between 99.8% and 100% for a fraction of marked states of 1/10 or larger.