  In adiabatic quantum annealing the required run-time to reach a given
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. In
general the presence of avoided level crossings demands an exponential increase
in the annealing time with the system size which has consequences both for the
efficiency of the algorithm and the required qubit coherence times. One
promising avenue being explored to produce more favourable gap scaling is the
introduction of non-stoquastic XX-couplings in the form of a catalyst - of
particular interest are catalysts which utilise accessible information about
the optimisation problem in their construction. Here we show extreme
sensitivity of the effect of an XX-catalyst to subtle changes in the encoding
of the optimisation problem. In particular, we observe that a targeted catalyst
containing a single coupling at constant strength can significantly reduce the
gap closing with system size at an avoided level crossing. For slightly
different encodings of the same problems however, these same catalysts result
in closing gaps in the annealing spectrum. To understand the origin of these
closing gaps, we study how the evolution of the ground-state vector is altered
by the presence of the catalyst and find that the negative components of the
ground-state vector are key to understanding the response of the gap spectrum.
We also consider how and when these closing gaps could be utilised in diabatic
quantum annealing protocols - a promising alternative to adiabatic quantum
annealing in which transitions to higher energy levels are exploited to reduce
the run time of the algorithm.
