An improved formalism for the Grover search algorithm

TL;DR

This paper introduces a more efficient geometric algebra formalism for Grover's quantum search algorithm, enabling simpler visualization and solving of exact and general search problems.

Contribution

It demonstrates that Clifford's geometric algebra offers a superior mathematical framework for Grover's algorithm compared to traditional notation.

Findings

Geometric algebra provides a more efficient representation of Grover's algorithm.

The formalism allows simple visualization of the search as spin precession.

The approach enables solving exact and general search problems more easily.

Abstract

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is of significant interest to describe it in the most efficient mathematical formalism. We show firstly, that Clifford's formalism of geometric algebra, provides a significantly more efficient representation than the conventional Bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search as the precession of a spin-1/2 particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations.