  We study the effect of a thermal environment on adiabatic quantum computation
using the Bloch-Redfield formalism. We show that in certain cases the
environment can enhance the performance in two different ways: (i) by
introducing a time scale for thermal mixing near the anticrossing that is
smaller than the adiabatic time scale, and (ii) by relaxation after the
anticrossing. The former can enhance the scaling of computation when the
environment is superohmic, while the latter can only provide a prefactor
enhancement. We apply our method to the case of adiabatic Grover search and
show that performance better than classical is possible with a superohmic
environment, with no a priori knowledge of the energy spectrum.
