An Efficient Deterministic Quantum Algorithm for the Integer Square-free Decomposition Problem

TL;DR

This paper introduces an efficient, deterministic quantum algorithm for integer square-free decomposition, leveraging Gauss sums and quantum Fourier transform, expanding quantum algorithm capabilities beyond classical limitations.

Contribution

The paper presents a novel quantum algorithm for square-free decomposition that is both efficient and deterministic, utilizing new quantum concepts not previously applied.

Findings

Algorithm efficiently finds square-free parts of large integers.

Uses properties of Gauss sums and quantum Fourier transform.

Introduces new quantum methods potentially applicable to other problems.

Abstract

Quantum computers are known to be qualitatively more powerful than classical computers, but so far only a small number of different algorithms have been discovered that actually use this potential. It would therefore be highly desirable to develop other types of quantum algorithms that widen the range of possible applications. Here we propose an efficient and deterministic quantum algorithm for finding the square-free part of a large integer - a problem for which no efficient classical algorithm exists. The algorithm relies on properties of Gauss sums and uses the quantum Fourier transform. We give an explicit quantum network for the algorithm. Our algorithm introduces new concepts and methods that have not been used in quantum information processing so far and may be applicable to a wider class of problems.