  Physically motivated classical heuristic optimization algorithms such as
simulated annealing (SA) treat the objective function as an energy landscape,
and allow walkers to escape local minima. It has been argued that quantum
properties such as tunneling may give quantum algorithms advantage in finding
ground states of vast, rugged cost landscapes. Indeed, the Quantum Adiabatic
Algorithm (QAO) and the recent Quantum Approximate Optimization Algorithm
(QAOA) have shown promising results on various problem instances that are
considered classically hard. Here, we argue that the type of control strategy
used by the optimization algorithm may be crucial to its success. Along with
SA, QAO and QAOA, we define a new, bang-bang version of simulated annealing,
BBSA, and study the performance of these algorithms on two well-studied problem
instances from the literature. Both classically and quantumly, the successful
control strategy is found to be bang-bang, exponentially outperforming the
quasistatic analogues on the same instances. Lastly, we construct O(1)-depth
QAOA protocols for a class of symmetric cost functions, and provide an
accompanying physical picture.
