  We derive a version of the adiabatic theorem that is especially suited for
applications in adiabatic quantum computation, where it is reasonable to assume
that the adiabatic interpolation between the initial and final Hamiltonians is
controllable. Assuming that the Hamiltonian is analytic in a finite strip
around the real time axis, that some number of its time-derivatives vanish at
the initial and final times, and that the target adiabatic eigenstate is
non-degenerate and separated by a gap from the rest of the spectrum, we show
that one can obtain an error between the final adiabatic eigenstate and the
actual time-evolved state which is exponentially small in the evolution time,
where this time itself scales as the square of the norm of the time-derivative
of the Hamiltonian, divided by the cube of the minimal gap.
