Lattices with special properties have been used to make lattice-based cryptosystems more efficient.  One example is a recent proposal by Peikert and Rosen to use ideal lattices from families of number fields with bounded root discriminant.  Computing such families appears to be computationally very difficult.  Class field theory, which classifies all finite abelian extensions of a number field, can be used to show that these families exist.  Less is known about the complexity of computing them.  We explore some number theoretic problems which could lead to efficient quantum algorithms for computing such families of number fields with bounded root discriminant.  First we show how to compute the ray class group of constant degree number fields.  Then we show that some subfields of Hilbert class fields of small degree can be computed efficiently.  Our algorithms build on previous quantum algorithms for the unit group and class group of a number field and results and machinery from class field theory.