Role of nonstoquastic catalysts in quantum adiabatic optimization

The viability of nonstoquastic 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 nonstoquastic catalysts over its stoquastic counterpart. We revisit this model and note how the incremental changes in the ground-state wave function give an indication of how the nonstoquastic catalyst provides an advantage. We then construct two new examples that exhibit an advantage for nonstoquastic 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.