  We have studied numerically the evolution of an adiabatic quantum computer in
the presence of a Markovian ohmic environment by considering Ising spin glass
systems with up to 20 qubits independently coupled to this environment via two
conjugate degrees of freedom. The required computation time is demonstrated to
be of the same order as that for an isolated system and is not limited by the
single-qubit decoherence time $T_2^*$, even when the minimum gap is much
smaller than the temperature and decoherence-induced level broadening. For
small minimum gap, the system can be described by an effective two-state model
coupled only longitudinally to environment.
