Reducibility of native weighted graphs on Rydberg Arrays

TL;DR

This paper examines how classical preprocessing can simplify native weighted graph problems on Rydberg arrays, identifying regimes where quantum advantage may be most relevant.

Contribution

It systematically analyzes the effectiveness of kernelisation techniques on native Rydberg graph problems, highlighting limitations and implications for quantum benchmarking.

Findings

Small or sparse instances can be fully reduced by classical preprocessing.

Dense graphs often retain irreducible kernels after reduction.

Extending interaction range suppresses reduction efficiency.

Abstract

We investigate the classical reducibility of random unit-disk graph (UDG) instances of the maximum independent set (MIS) and maximum weighted independent set (MWIS) problems, which can be natively realised in Rydberg atom quantum processors. Using state-of-the-art kernelisation techniques, we systematically probe how far classical preprocessing can simplify such native optimisation problems of varying size and connectivity. While many small or sparse instances can be fully reduced, dense graphs often retain finite irreducible kernels even after extensive reductions. Introducing vertex weights tends to increase reducibility, whereas extending the interaction range in the underlying UDG connectivity suppresses the reduction efficiency. By exploring where classical reductions cease to be effective, we aim to delineate the regime of problem instances that remain computationally demanding - those most relevant for testing and benchmarking near-term quantum optimisation hardware. We find that for the remaining finite kernels, quantum execution would require non-native embeddings with substantial resource overheads, suggesting that directly running native instances may be more practical than embedding a reduced kernel.