Scaling of computational order parameters in Rydberg-atom graph states

Graph states are computationally powerful quantum states with many applications, including use as resource states for measurement-based quantum computing (MBQC). We demonstrate construction of graph states on a Rydberg atom quantum analog simulator. We show how an always-on interaction can be used to simultaneously entangle all Rydberg atoms into a graph state. We construct and implement many-body computational order parameters for graph states using nonlocal measurement-based logic operations in the Clifford group. The order parameters measure the efficacy of entanglement to allow MBQC on graph states of any size. We parametrize finite-size scaling of these order parameters. Our results define a route to efficiently test computational power in quantum devices.