  The notion of quantum speed limit (QSL) refers to the fundamental fact that
two quantum states become completely distinguishable upon dynamical evolution
only after a finite amount time, called the QSL time. A different, but related
concept is that of minimum control time (MCT), which is the minimum evolution
time needed for a state to be driven (by suitable, generally time-dependent,
control fields) to a given target state. While the QSL can give information
about the MCT, it usually imposes little restrictions to it, and is thus
unpractical for control purposes. In this work we revisit this issue by first
presenting a theory of geometrical QSL for unitary transformations, rather than
for states, and discuss its implications and limitations. Then, we propose a
framework for bounding the MCT for realizing unitary transformations that goes
beyond the QSL results and gives much more meaningful information to understand
the controlled dynamics of the system at short times.
