Exactness of the Original Grover Search Algorithm

TL;DR

This paper investigates the conditions under which Grover's search algorithm is exact, providing a complete characterization based on trigonometric rationality results, and clarifies when it is not exact for other search sizes.

Contribution

It offers a complete mathematical characterization of the exactness of Grover's algorithm beyond the known case of searching one out of four.

Findings

Grover's algorithm is exact only when searching one out of four items.

The paper provides a rationality-based criterion for the exactness of Grover's algorithm.

It concludes that for other search sizes, Grover's algorithm is not exact.

Abstract

It is well-known that when searching one out of four, the original Grover's search algorithm is exact; that is, it succeeds with certainty. It is natural to ask the inverse question: If we are not searching one out of four, is Grover's algorithm definitely not exact? In this article we give a complete answer to this question through some rationality results of trigonometric functions.