Fully solvable example of an all-to-all coupling implemented in a local architecture

Both quantum simulation and quantum optimization tasks benefit greatly from the increase in connectivity of the underlying hardware graph. A complete graph, which we refer to as all-to-all coupling, is traditionally embedded via ferromagnetic chains. The length of these chains has to be proportional to the system size in any planar local architecture. Flipping a single chain by tunneling induced by a local transverse field takes time exponential in the chain length, thus also in the system size. This is known to slow down quantum optimization, making the scaling the time to solution of the embedded problems much worse than the corresponding scaling of the classical algorithms working with the complete graph directly. It also impedes quantum simulation by exponentially reducing the range of available effective transverse fields. We propose a way to improve from exponential to polynomial in both of those cases. This is achieved by replacing ferromagnetic chains by chains close to the critical point of the transverse field Ising model. The specific schedules for such embedding have at least 4 groups of terms with different dependences on the anneal parameter, compared to 2 in the traditional annealing. Next generation annealers are planned to be capable of implementing such schedules.