  Most investigations devoted to the conditions for adiabatic quantum computing
are based on the first-order correction ${\bra{\Psi_{\rm ground}(t)}\dot
H(t)\ket{\Psi_{\rm excited}(t)} /\Delta E^2(t)\ll1}$. However, it is
demonstrated that this first-order correction does not yield a good estimate
for the computational error. Therefore, a more general criterion is proposed,
which includes higher-order corrections as well and shows that the
computational error can be made exponentially small -- which facilitates
significantly shorter evolution times than the above first-order estimate in
certain situations. Based on this criterion and rather general arguments and
assumptions, it can be demonstrated that a run-time $T$ of order of the inverse
minimum energy gap $\Delta E_{\rm min}$ is sufficient and necessary, i.e.,
$T=\ord(\Delta E_{\rm min}^{-1})$. For some examples, these analytical
investigations are confirmed by numerical simulations. PACS: 03.67.Lx,
03.67.-a.
