Ground state geometry of random Rydberg atom graphs

Networks of Rydberg atoms provide a natural way of encoding maximum independent set (MIS) problems. At zero Rabi frequency, there is a regime in the detuning parameter where the ground state of a Rydberg atom graph is the maximum independent set of the graph. A quantum computer that solves MIS problems can then be designed in the following way. An apparatus applies an adiabatic protocol to a system of Rydberg atoms that brings it from some initial ground state to another ground state in the MIS regime, thereby solving the foregoing MIS problem. To optimize the adiabatic drive, we look at the Fubini-Study (FS) metric of the system. In this study, we wish to investigate the geometry arising from random graphs of Rydberg atoms. We will look for universality among these graphs and how phase transitions shape the geometry of the parameter space. An average metric field of the ensemble will then be taken to extract what a typical geodesic would look like for a graph of Rydberg atoms.