  In adiabatic quantum annealing, the speed with which an anneal can be run,
while still achieving a high final ground state fidelity, is dictated by the
size of the minimum gap that appears between the ground and first excited state
in the annealing spectrum. To avoid the exponential slowdown associated with
exponentially closing gaps, diabatic transitions to higher energy levels may be
exploited in such a way that the system returns to the ground state before the
end of the anneal. In certain cases, this is facilitated by the original
annealing spectrum. However, there are also examples where careful manipulation
of the annealing Hamiltonian has been used to alter the spectrum to create a
diabatic path to the ground state. Since diabatic transitions depend on the
evolution rate and the gap sizes in the spectrum, it is important to consider
the sensitivity of any potential enhancement to changes in the anneal time as
well as any parameters involved in the manipulation of the spectrum. We explore
this sensitivity using annealing spectra containing an exponentially closing
gap and an additional, tuneable, small gap created by a catalyst. We find that
there is a trade-off between the precision needed in the catalyst strength and
the anneal time in order to maintain the enhancement to the final ground state
fidelity.
