Role of Non-stoquastic Catalysts in Quantum Adiabatic Optimization

The viability of non-stoquastic catalyst Hamiltonians to deliver consistent quantum speedups in quantum adiabatic optimization remains an open question. The infinite-range ferromagnetic $p$-spin model is a rare example exhibiting an exponential advantage for non-stoquastic catalysts over its stoquastic counterpart. We revisit this model and note how the incremental changes in the ground state wavefunction give an indication of how the non-stoquastic catalyst provides an advantage. We then construct two new examples that exhibit an advantage for non-stoquastic catalysts over stoquastic catalysts. The first is another infinite range model that is only 2-local but also exhibits an exponential advantage, and the second is a geometrically local Ising example that exhibits a growing advantage up to the maximum system size we study.