Quantum Hamiltonian Algorithms for Maximum Independent Sets

With qubits encoded into atomic ground and Rydberg states and situated on the vertexes of a graph, the conditional quantum dynamics of Rydberg blockade, which inhibits simultaneous excitation of nearby atoms, has been employed recently to find maximum independent sets following an adiabatic evolution algorithm hereafter denoted by HV [Science 376, 1209 (2022)]. An alternative algorithm, short named the PK algorithm, reveals that the independent sets diffuse over a media graph governed by a non-abelian gauge matrix of an emergent PXP model. This work shows the above two algorithms are mathematically equivalent, despite of their seemingly different physical implementations. More importantly, we demonstrated that although the two are mathematically equivalent, the PK algorithm exhibits more efficient and resource-saving performance. Within the same range of experimental parameters, our numerical studies suggest that the PK algorithm performs at least 25% better on average and saves at least $6\times10^6$ measurements ($\sim 900$ hours of continuous operation) for each graph when compared to the HV algorithm. We further consider the measurement error and point out that this may cause the oscillations in the performance of the HV's optimization process.