  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.
