  A quantum system subject to external fields is said to be controllable if
these fields can be adjusted to guide the state vector to a desired destination
in the state space of the system. Fundamental results on controllability are
reviewed against the background of recent ideas and advances in two seemingly
disparate endeavors: (i) laser control of chemical reactions and (ii) quantum
computation. Using Lie-algebraic methods, sufficient conditions have been
derived for global controllability on a finite-dimensional manifold of an
infinite-dimensional Hilbert space, in the case that the Hamiltonian and
control operators, possibly unbounded, possess a common dense domain of
analytic vectors. Some simple examples are presented. A synergism between
quantum control and quantum computation is creating a host of exciting new
opportunities for both activities. The impact of these developments on
computational many-body theory could be profound.
