Quantum Algorithms for Unit Group and principal ideal problem

TL;DR

This paper introduces efficient quantum algorithms for computing the unit group and solving the principal ideal problem in constant-degree number fields, improving upon previous methods by using a novel period function and lattice basis computation.

Contribution

It presents new quantum algorithms that enhance existing solutions for algebraic number theory problems, especially with non-one-to-one period functions and lattice basis determination.

Findings

Quantum algorithms outperform classical methods in specific tasks

New period function approach improves algorithm efficiency

Method for lattice basis computation from encoded lattice data

Abstract

Computing the unit group and solving the principal ideal problem for a number field are two of the main tasks in computational algebraic number theory. This paper proposes efficient quantum algorithms for these two problems when the number field has constant degree. We improve these algorithms proposed by Hallgren by using a period function which is not one-to-one on its fundamental period. Furthermore, given access to a function which encodes the lattice, a new method to compute the basis of an unknown real-valued lattice is presented.