Quantum Optimization on Rydberg Atom Arrays with Arbitrary Connectivity: Gadgets Limitations and a Heuristic Approach

TL;DR

This paper investigates the limitations of using Rydberg atom arrays for quantum optimization on arbitrary graphs, proving quadratic complexity limits for reductions and proposing a practical heuristic approach with promising experimental results.

Contribution

It establishes the complexity bounds of graph reductions for Rydberg-based quantum optimization and introduces a linear-overhead heuristic for practical problem solving.

Findings

Quadratic blow-up in vertex count for reductions limits scalability.

The proposed divide-and-conquer heuristic achieves linear overhead.

Feasibility demonstrated on the Orion Alpha quantum processor.

Abstract

Programmable quantum systems based on Rydberg atom arrays have recently emerged as a promising testbed 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-theoretic limits of polynomial reductions from arbitrary graphs to unit-disk 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-and-conquer 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.