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

Jordi R. Weggemans1,2, Alexander Urech2,3, Alexander Rausch4, Robert Spreeuw2,3, Richard Boucherie5, Florian Schreck2,3, Kareljan Schoutens2,6, Jiří Minář2,6, and Florian Speelman2,7

1CWI, Science Park 123, 1098 XG Amsterdam, The Netherlands
2QuSoft, Science Park 123, 1098 XG Amsterdam, The Netherlands
3Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands
4Robert Bosch GmbH, Corporate Research, Robert-Bosch-Campus 1, 71272 Renningen, Germany
5Stochastic Operations Research, Department of Applied Mathematics, University of Twente, 7500 AE, Enschede, The Netherlands.
6Institute for Theoretical Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands
7Informatics Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands

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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 Erdős-Rényi 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.

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