Exact quantum lower bound for Grover's problem

TL;DR

This paper establishes an exact quantum lower bound for Grover's search algorithm, proving its optimality in the query model through an angle-based information bound, using elementary mathematics.

Contribution

It introduces a simple, self-contained proof technique that precisely determines the optimality of Grover's algorithm in the query complexity model.

Findings

Grover's algorithm is proven to be exactly optimal.

The proof uses an angle-based information bound.

The method is elementary and self-contained.

Abstract

One of the most important quantum algorithms ever discovered is Grover's algorithm for searching an unordered set. We give a new lower bound in the query model which proves that Grover's algorithm is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound.