On quantum computation of Kloosterman sums

TL;DR

This paper introduces two quantum algorithms for efficiently computing Kloosterman sums over finite fields, significantly improving computational speed for these mathematical objects.

Contribution

The paper presents novel quantum algorithms that compute all or individual Kloosterman sums over finite fields with improved efficiency.

Findings

First algorithm computes all Kloosterman sums in polynomial time in log q.

Second algorithm computes a single Kloosterman sum in quasi-linear time in sqrt q.

Algorithms outperform classical methods for large finite fields.

Abstract

We give two quantum algorithms for computing (twisted) Kloosterman sums attached to a finite field $\mathbf{F}$ of $q$ elements. The first algorithm computes a quantum state containing, as its coefficients with respect to the standard basis, all Kloosterman sums for $\mathbf{F}$ twisted by a given multiplicative character, and runs in time polynomial in $\log q$. The second algorithm computes a single Kloosterman sum to a prescribed precision, and runs in time quasi-linear in $\sqrt{q}$.