  In this paper we give a polynomial-time quantum algorithm for computing
orders of solvable groups. Several other problems, such as testing membership
in solvable groups, testing equality of subgroups in a given solvable group,
and testing normality of a subgroup in a given solvable group, reduce to
computing orders of solvable groups and therefore admit polynomial-time quantum
algorithms as well. Our algorithm works in the setting of black-box groups,
wherein none of these problems can be computed classically in polynomial time.
As an important byproduct, our algorithm is able to produce a pure quantum
state that is uniform over the elements in any chosen subgroup of a solvable
group, which yields a natural way to apply existing quantum algorithms to
factor groups of solvable groups.
