  We extend the work in New J. Phys. 19, 103015 (2017) by deriving a lower
bound for the minimum time necessary to implement a unitary transformation on a
generic, closed quantum system with an arbitrary number of classical control
fields. This bound is explicitly analyzed for a specific N-level system similar
to those used to represent simple models of an atom, or the first excitation
sector of a Heisenberg spin chain, both of which are of interest in quantum
control for quantum computation. Specifically, it is shown that the resultant
bound depends on the dimension of the system, and on the number of controls
used to implement a specific target unitary operation. The value of the bound
determined numerically, and an estimate of the true minimum gate time are
systematically compared for a range of system dimension and number of controls;
special attention is drawn to the relationship between these two variables. It
is seen that the bound captures the scaling of the minimum time well for the
systems studied, and quantitatively is correct in the order of magnitude.
