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

We establish a number of connections between a digital version of Quantum Annealing (QA) with the Quantum Approximate Optimization Algorithm (QAOA) introduced by Fahri et al. (arXiv:1411.4028) as an alternative hybrid quantum-classical variational scheme for quantum-state preparation and optimization. We prove a rigorous bound concerning the performance of QAOA for MaxCut on a $2$-regular graph, equivalent to an unfrustrated antiferromagnetic Ising chain, which shows that the optimal variational error is generally bound to be $\epsilon^{\mathrm{res}}_{\mathrm{P}}\ge (2\mathrm{P}+2)^{-1}$. In particular, we explicitly demonstrate that among the $2^{\mathrm{P}}$ degenerate minima which can be found, all strictly satisfying the equality $\epsilon^{\mathrm{res}}_{\mathrm{P}}=(2\mathrm{P}+2)^{-1}$, there is a special regular optimal solution which can be interpreted as an optimized digital-QA schedule. These findings help elucidating the intimate relation between digital-QA, QAOA, and optimal Quantum Control.