  We have studied the decoherence properties of adiabatic quantum computation
(AQC) in the presence of in general non-Markovian, e.g., low-frequency, noise.
The developed description of the incoherent Landau-Zener transitions shows that
the global AQC maintains its properties even for decoherence larger than the
minimum gap at the anticrossing of the two lowest energy levels. The more
efficient local AQC, however, does not improve scaling of the computation time
with the number of qubits $n$ as in the decoherence-free case. The scaling
improvement requires phase coherence throughout the computation, limiting the
computation time and the problem size n.
