Description
We study the quantum dynamics of the encoding scheme proposed in [Nguyen et al., PRX Quantum 4, 010316 (2023)], which is designed to encode optimization problems on graphs with arbitrary connectivity into Rydberg atom arrays.
Here, a graph vertex is represented by a wire of atoms, and the (crossing) crossing-with-edge gadget is placed at the intersection of two wires to (de)couple the wire degrees of freedom, reproducing the graph connectivity. We consider the exemplified geometry of two wires intersecting via a single gadget and look at minimum gap scaling with system size along annealing protocols. We find that both polynomial and exponential scaling are possible and, by means of perturbation theory, we relate the exponential closure of the minimum gap to an unfavorable localization of the wavefunction. We then experimentally observe the occurrence of such localization and eventually propose possible strategies to circumvent it, leading to an exponential improvement of the annealing performance.
