  Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm
(QAOA) are two special cases of the following control problem: apply a
combination of two Hamiltonians to minimize the energy of a quantum state.
Which is more effective has remained unclear. Here we analytically apply the
framework of optimal control theory to show that generically, given a fixed
amount of time, the optimal procedure has the pulsed (or "bang-bang") structure
of QAOA at the beginning and end but can have a smooth annealing structure in
between. This is in contrast to previous works which have suggested that
bang-bang (i.e., QAOA) protocols are ideal. To support this theoretical work,
we carry out simulations of various transverse field Ising models,
demonstrating that bang-anneal-bang protocols are more common. The general
features identified here provide guideposts for the nascent experimental
implementations of quantum optimization algorithms.
