Solving correlation clustering with QAOA and a Rydberg qudit system: a full-stack approach

TL;DR

This paper develops a full-stack quantum approach using QAOA and Rydberg qudits for correlation clustering, demonstrating superior efficiency and performance bounds on small graphs, with analysis of error impacts.

Contribution

It introduces a qudit-based implementation of QAOA for correlation clustering, including Hamiltonian formulation, performance analysis, and gate design, showing advantages over qubit encoding.

Findings

Qudit implementation reduces gate count compared to qubit encoding.

QAOA surpasses the Swamy bound for approximation ratio at depth p ≥ 2.

Errors significantly limit the algorithm's performance on complete graphs.

Abstract

We study the correlation clustering problem using the quantum approximate optimization algorithm (QAOA) and qudits, which constitute a natural platform for such non-binary problems. Specifically, we consider a neutral atom quantum computer and propose a full stack approach for correlation clustering, including Hamiltonian formulation of the algorithm, analysis of its performance, identification of a suitable level structure for ${}^{87}{\rm Sr}$ and specific gate design. We show the qudit implementation is superior to the qubit encoding as quantified by the gate count. For single layer QAOA, we also prove (conjecture) a lower bound of $0.6367$ ($0.6699$) for the approximation ratio on 3-regular graphs. Our numerical studies evaluate the algorithm's performance by considering complete and Erd\H{o}s-R\'enyi graphs of up to 7 vertices and clusters. We find that in all cases the QAOA surpasses the Swamy bound $0.7666$ for the approximation ratio for QAOA depths $p \geq 2$. Finally, by analysing the effect of errors when solving complete graphs we find that their inclusion severely limits the algorithm's performance.