Quantum Algorithms for Unit Group and principal ideal problem

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.