  We establish and discuss a number of connections between a digitized version
of Quantum Annealing (QA) with the Quantum Approximate Optimization Algorithm
(QAOA) introduced by Farhi et al. (arXiv:1411.4028) as an alternative hybrid
quantum-classical variational scheme for quantum-state preparation and
optimization. We introduce a technique that allows to prove, for instance, a
rigorous bound concerning the performance of QAOA for MaxCut on a $2$-regular
graph, equivalent to an unfrustrated antiferromagnetic Ising chain. The bound
shows that the optimal variational error of a depth-$\mathrm{P}$ quantum
circuit has to satisfy $\epsilon^\mathrm{res}_{\mathrm{P}}\ge
(2\mathrm{P}+2)^{-1}$. In a separate work (Mbeng et al., arXiv:1911.12259) we
have explicitly shown, exploiting a Jordan-Wigner transformation, that among
the $2^{\mathrm{P}}$ degenerate variational minima which can be found for this
problem, all strictly satisfying the equality
$\epsilon^\mathrm{res}_{\mathrm{P}}=(2\mathrm{P}+2)^{-1}$, one can construct a
special {\em regular} optimal solution, which is computationally optimal and
does not require any prior knowledge about the spectral gap. We explicitly
demonstrate here that such a schedule is adiabatic, in a digitized sense, and
can therefore be interpreted as an optimized digitized-QA protocol. We also
discuss and compare our bound on the residual energy to well-known results on
the Kibble-Zurek mechanism behind a continuous-time QA. These findings help
elucidating the intimate relation between digitized-QA, QAOA, and optimal
Quantum Control.
