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In adiabatic quantum annealing the required run-time to reach a givenground-state fidelity is dictated by the size of the minimum gap that appearsbetween the ground and first excited state in the annealing spectrum. Ingeneral the presence of avoided level crossings demands an exponential increasein the annealing time with the system size which has consequences both for theefficiency of the algorithm and the required qubit coherence times. Onepromising avenue being explored to produce more favourable gap scaling is theintroduction of non-stoquastic XX-couplings in the form of a catalyst - ofparticular interest are catalysts which utilise accessible information aboutthe optimisation problem in their construction. Here we show extremesensitivity of the effect of an XX-catalyst to subtle changes in the encodingof the optimisation problem. In particular, we observe that a targeted catalystcontaining a single coupling at constant strength can significantly reduce thegap closing with system size at an avoided level crossing. For slightlydifferent encodings of the same problems however, these same catalysts resultin closing gaps in the annealing spectrum. To understand the origin of theseclosing gaps, we study how the evolution of the ground-state vector is alteredby the presence of the catalyst and find that the negative components of theground-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 diabaticquantum annealing protocols - a promising alternative to adiabatic quantumannealing in which transitions to higher energy levels are exploited to reducethe run time of the algorithm.

Effects of XX-catalysts on quantum annealing spectra with perturbative crossings