Necessary adiabatic run times in quantum optimization

Quantum annealing is guaranteed to find the ground state of optimization problems provided it operates in the adiabatic limit. Recent work [S. Muthukrishnan et al., Phys. Rev. X 6, 031010 ( 2016)] has found that for some barrier tunneling problems, quantum annealing can be run much faster than is adiabatically required. Specifically, an n-qubit optimization problem was presented for which a nonadiabatic, or diabatic, annealing algorithm requires only a constant run time, while an adiabatic annealing algorithm requires a run-time polynomial in n. Here we show that this nonadiabatic speedup is the direct result of a specific symmetry in the studied problem. In the more general case, no such nonadiabatic speedup occurs and we show why the special case achieves this speedup compared to the general case. We also prove that the adiabatic annealing algorithm has a necessary and sufficient run time that is quadratically better than the standard quantum adiabatic condition suggests. We conclude with an observation about the required precision in timing for the diabatic algorithm.