Symmetries in Quantum Algorithms: Researchers from Jülich and the U.S. Analyze the Structure of QAOA

25 November 2025

How can quantum algorithms be better understood and more effectively improved? Researchers from Forschungszentrum Jülich (Germany) and Los Alamos National Laboratory (USA) have taken a closer mathematical look at the well-known Quantum Approximate Optimization Algorithm (QAOA). They demonstrate that the algorithm operates according to clearly defined symmetries – a crucial step toward assessing its performance on a theoretical basis. The findings open new perspectives for analyzing quantum algorithms and deepen our understanding of the fundamental principles of quantum optimization.

An international team of scientists has gained new insights into the mathematical structure of the promising Quantum Approximate Optimization Algorithm (QAOA). In their recent study, the researchers show that the algorithm can be precisely described using methods from Lie algebra and symmetry analysis. This mathematical framework provides new opportunities for improving and theoretically grounding quantum algorithms.

Understanding data rather than just computing it

QAOA is regarded as a relatively easy-to-implement approach for solving complex optimization problems with quantum computers and quantum simulators. It alternates between quantum operations and classical optimization steps.

The team examined how these quantum processes can be described mathematically – in particular, how symmetries and algebraic structures determine the algorithm’s behavior.

Symmetries as the key

The analysis reveals that QAOA operates within a restricted subspace of the overall quantum state space. By describing this subspace in terms of Lie algebras, the corresponding symmetries can be precisely characterized. This approach makes it possible to understand the underlying mechanisms analytically rather than relying solely on numerical simulations – enabling researchers to better predict and control the algorithm’s behavior.

Based on these findings, the authors propose a targeted selection of QAOA variants. This strategy allows for a more efficient use of quantum resources and lays the groundwork for new, symmetry-adapted optimization methods on quantum computers.

Outlook

For now, the approach is mainly applicable to systems with clearly identifiable symmetries.
Future work aims to extend the method to additional QAOA variants and to explore how similar mathematical concepts might be applied to other quantum algorithms.

The study represents an important step toward a deeper theoretical understanding of QAOA.
It demonstrates that symmetries and algebraic structures are key tools for making quantum algorithms more efficient and for explaining their behavior systematically – a decisive move toward a more rigorous foundation of quantum algorithmics.

The research was supported by the European projects HPCQS (High-Performance Computer and Quantum Simulator) and PASQuanS2.1 (Programmable Atomic Large-Scale Quantum Simulation).

Original publication: Analyzing the Quantum Approximate Optimization Algorithm: Ansätze, Symmetries, and Lie Algebras, Sujay Kazi, Martín Larocca, Marco Farinati, Patrick J. Coles, M. Cerezo, and Robert Zeier, PRX Quantum 2025
DOI: 10.1103/yfwq-yqmk