On the Existence of certain Quantum Algorithms

TL;DR

This paper explores whether quantum algorithms can efficiently compute maxima of conjugated elements in number fields, linking their existence to an open number theory conjecture.

Contribution

It establishes a connection between the potential of quantum algorithms for specific number-theoretic problems and an unresolved conjecture, highlighting a new research direction.

Findings

No definitive quantum algorithm proven for the problem.

The existence of such algorithms is related to an open conjecture.

Provides a theoretical framework connecting quantum computing and number theory.

Abstract

We investigate the question if quantum algorithms exist that compute the maximum of a set of conjugated elements of a given number field in quantum polynomial time. We will relate the existence of these algorithms for a certain family of number fields to an open conjecture from elementary number theory.