Reducing QUBO Density by Factoring out Semi-Symmetries

Jonas Nüßlein, Leo Sünkel, Jonas Stein, Tobias Rohe, Daniëlle Schuman, Sebastian Feld, Corey O’Meara, Giorgio Cortiana, Claudia Linnhoff-Popien

2025

Abstract

Quantum Approximate Optimization Algorithm (QAOA) and Quantum Annealing are prominent approaches for solving combinatorial optimization problems, such as those formulated as Quadratic Unconstrained Binary Optimization (QUBO). These algorithms aim to minimize the objective function $x^T Q x$, where $Q$ is a QUBO matrix. However, the number of two-qubit CNOT gates in QAOA circuits and the complexity of problem embeddings in Quantum Annealing scale linearly with the number of non-zero couplings in $Q$, contributing to significant computational and error-related challenges. To address this, we introduce the concept of \textit{semi-symmetries} in QUBO matrices and propose an algorithm for identifying and factoring these symmetries into ancilla qubits. \textit{Semi-symmetries} frequently arise in optimization problems such as \textit{Maximum Clique}, \textit{Hamilton Cycles}, \textit{Graph Coloring}, and \textit{Graph Isomorphism}. We theoretically demonstrate that the modified QUBO matrix $Q_{\text{mod}}$ retains the same energy spectrum as the original $Q$. Experimental evaluations on the aforementioned problems show that our algorithm reduces the number of couplings and QAOA circuit depth by up to $45\%$. For Quantum Annealing, these reductions also lead to sparser problem embeddings, shorter qubit chains and better performance. This work highlights the utility of exploiting QUBO matrix structure to optimize quantum algorithms, advancing their scalability and practical applicability to real-world combinatorial problems.

Paper Citation

in Harvard Style

Nüßlein J., Sünkel L., Stein J., Rohe T., Schuman D., Feld S., O’Meara C., Cortiana G. and Linnhoff-Popien C. (2025). Reducing QUBO Density by Factoring out Semi-Symmetries. In Proceedings of the 17th International Conference on Agents and Artificial Intelligence - Volume 1: QAIO; ISBN 978-989-758-737-5, SciTePress, pages 783-792. DOI: 10.5220/0013395900003890

in Bibtex Style

@conference{qaio25,
author={Jonas Nüßlein and Leo Sünkel and Jonas Stein and Tobias Rohe and Daniëlle Schuman and Sebastian Feld and Corey O’Meara and Giorgio Cortiana and Claudia Linnhoff-Popien},
title={Reducing QUBO Density by Factoring out Semi-Symmetries},
booktitle={Proceedings of the 17th International Conference on Agents and Artificial Intelligence - Volume 1: QAIO},
year={2025},
pages={783-792},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013395900003890},
isbn={978-989-758-737-5},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 17th International Conference on Agents and Artificial Intelligence - Volume 1: QAIO
TI - Reducing QUBO Density by Factoring out Semi-Symmetries
SN - 978-989-758-737-5
AU - Nüßlein J.
AU - Sünkel L.
AU - Stein J.
AU - Rohe T.
AU - Schuman D.
AU - Feld S.
AU - O’Meara C.
AU - Cortiana G.
AU - Linnhoff-Popien C.
PY - 2025
SP - 783
EP - 792
DO - 10.5220/0013395900003890
PB - SciTePress