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

Quantum Annealing: a journey through Digitalization, Control, and hybrid Quantum Variational schemes