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

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.