  A fundamental problem in quantum engineering is determining the lowest time
required to ensure that all possible unitaries can be generated with the tools
available, which is one of a number of possible quantum speed limits. We
examine this problem from the perspective of quantum control, where the system
of interest is described by a drift Hamiltonian and set of control
Hamiltonians. Our approach uses a combination of Lie algebra theory, Lie groups
and differential geometry, and formulates the problem in terms of geodesics on
a differentiable manifold. We provide explicit lower bounds on the quantum
speed limit for the case of an arbitrary drift, requiring only that the control
Hamiltonians generate a topologically closed subgroup of the full unitary
group, and formulate criteria as to when our expression for the speed limit is
exact and not merely a lower bound. These analytic results are then tested and
confirmed using a numerical optimization scheme. Finally we extend the analysis
to find a lower bound on the quantum speed limit in the common case where the
system is described by a drift Hamiltonian and a single control Hamiltonian.
