  Tunneling is often claimed to be the key mechanism underlying possible
speedups in quantum optimization via quantum annealing (QA), especially for
problems featuring a cost function with tall and thin barriers. We present and
analyze several counterexamples from the class of perturbed Hamming-weight
optimization problems with qubit permutation symmetry. We first show that, for
these problems, the adiabatic dynamics that make tunneling possible should be
understood not in terms of the cost function but rather the semi-classical
potential arising from the spin-coherent path integral formalism. We then
provide an example where the shape of the barrier in the final cost function is
short and wide, which might suggest no quantum advantage for QA, yet where
tunneling renders QA superior to simulated annealing in the adiabatic regime.
However, the adiabatic dynamics turn out not be optimal. Instead, an evolution
involving a sequence of diabatic transitions through many avoided
level-crossings, involving no tunneling, is optimal and outperforms adiabatic
QA. We show that this phenomenon of speedup by diabatic transitions is not
unique to this example, and we provide an example where it provides an
exponential speedup over adiabatic QA. In yet another twist, we show that a
classical algorithm, spin vector dynamics, is at least as efficient as diabatic
QA. Finally, in a different example with a convex cost function, the diabatic
transitions result in a speedup relative to both adiabatic QA with tunneling
and classical spin vector dynamics.
