Description of the Grover algorithm based on geometric considerations

TL;DR

This paper offers a geometric and tensor-based description of Grover's quantum search algorithm, providing an alternative to matrix-based explanations and validating it through numerical experiments with Qiskit.

Contribution

It introduces a geometric, tensor product approach to describe Grover's algorithm, enhancing understanding beyond traditional matrix methods.

Findings

Numerical validation of the geometric description using Qiskit.

Confirmation of the quadratic speedup in unstructured search.

Detailed tensor-based and geometric system state representations.

Abstract

This paper concerns the Grover algorithm that permits to make amplification of quantum states previously tagged by an Oracle. Grover's algorithm allows searches in an unstructure database of n entries finding a marked element with a quadratic speedup. The algorithm requires a predefined number of runs to succeed with probability close to one.This article provides a description of the amplitude amplification quantum algorithm mechanism in a very short computational way, based on tensor products and provides a geometric presentation of the successive system states. All the basis changes are fully described to provide an alternative to the wide spread Grover description based only on matrices and complex tensor computation. Our experiments encompass numerical evaluations of circuit using the Qiskit library of IBM that meet the theoretical considerations