Quantum optimization on Rydberg-atom arrays with arbitrary connectivity: Gadget limitations and a heuristic approach

Programmable quantum systems based on Rydberg atom arrays have recently emerged as a promising test bed for combinatorial optimization. Indeed, the maximum weighted independent set problem on unit-disk graphs can be efficiently mapped to such systems due to their geometric constraints. However, extending this capability to arbitrary graph instances typically necessitates the use of reduction gadgets, which introduce additional experimental overhead and complexity. Here, we analyze the complexity\penalty1000-\hskip0pttheoretic limits of polynomial reductions from arbitrary graphs to unit\penalty1000-\hskip0ptdisk instances. We prove any such reduction incurs a quadratic blow up in vertex count and degrades solution approximation guarantees. As a practical alternative, we propose a divide\penalty1000-\hskip0ptand\penalty1000-\hskip0ptconquer heuristic with only linear overhead, which leverages precalibrated atomic layouts. We benchmark it on Erd"os-R\'enyi graphs, and demonstrate feasibility on the Orion Alpha processor.