Efficient Quantum Algorithms for Estimating Gauss Sums

TL;DR

This paper introduces a quantum algorithm that efficiently estimates Gauss sums over finite fields and rings, providing a quantum advantage and insights into the problem's classical hardness.

Contribution

It develops a novel quantum algorithm for Gauss sum estimation and links its complexity to the discrete logarithm problem, highlighting potential quantum advantages.

Findings

Quantum algorithm efficiently estimates Gauss sums.

Classical hardness linked to discrete logarithm problem.

Provides evidence of quantum advantage in number-theoretic problems.

Abstract

We present an efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings. This is a natural problem as the description of a Gauss sum can be done without reference to a black box function. With a reduction from the discrete logarithm problem to Gauss sum estimation we also give evidence that this problem is hard for classical algorithms. The workings of the quantum algorithm rely on the interaction between the additive characters of the Fourier transform and the multiplicative characters of the Gauss sum.