General error estimate for adiabatic quantum computing

Most investigations devoted to the conditions for adiabatic quantum computing are based on the first-order correction $⟨{\ensuremath{\Psi}}_{\mathrm{ground}}(t)\ensuremath{\mid}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}(t)\ensuremath{\mid}{\ensuremath{\Psi}}_{\mathrm{excited}}(t)⟩∕\mathrm{\ensuremath{\Delta}}{E}^{2}(t)⪡1$. 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 $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{min}}$ is sufficient and necessary, i.e., $T=O(\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{min}}^{\ensuremath{-}1})$. For some examples, these analytical investigations are confirmed by numerical simulations.