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

Abstract

The efficiency of Adiabatic Quantum Annealing is limited by the scaling with system size of the minimum gap that appears between the ground and first excited state in the annealing energy spectrum. In general the algorithm is unable to find the solution to an optimisation problem in polynomial time due to the presence of avoided level crossings at which the gap size closes exponentially with system size. 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. We observe that catalysts designed to enhance the minimum gap at an avoided level crossing can, under certain conditions, result in closing gaps in the 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. We find that the negative components of the ground state vector are key to understanding the response of the gap spectrum. In particular the closing gaps correspond to changes in which vector components become negative over the course of the evolution. 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.