  Quantum computation and quantum control operate by building unitary
transformations out of sequences of elementary quantum logic operations or
applications of control fields. This paper puts upper bounds on the minimum
time required to implement a desired unitary transformation on a d-dimensional
Hilbert space when applying sequences of Hamiltonian transformations. We show
that strategies of building up a desired unitary out of non-infinitesimal and
infinitesimal unitaries, or equivalently, using power and band limited
controls, can yield the best possible scaling in time $O(d^2)$.
