Systems theory of quantum algorithms

Existing techniques for addressing noise typically separate the error handling from the algorithm analysis and design, e.g., via subsequent error correction or mitigation steps. This ignores significant potential for robustness improvements on the algorithmic side. In particular, from systems and control theory, it is well-known that the robustness of a system can and should be influenced at the design stage. In our research, we develop methods for analyzing the robustness of quantum algorithms based on tools from classical control theory such as Lipschitz bounds, set-membership uncertainty models, and robust control. For a given algorithm and noise model, our goal is to compute bounds on the worst-case error, which provide insights into robustness mechanisms and which can be used to design algorithms that are inherently more robust against hardware imperfections.

Variational quantum algorithms (VQAs) are a promising class of quantum algorithms, containing a parameterized quantum algorithm with iterative parameter adaptation via classical optimization. They can be used to tackle challenging computational problems, e.g., in optimization, simulation, and machine learning. Mathematically, VQAs are feedback interconnections consisting of a discrete-time dynamical system (implemented on a classical computer) with a static nonlinear function (implemented on a quantum computer).

In our research, we study systems-theoretic properties of VQAs, including stability, convergence, and robustness, to improve their theoretical understanding and practical reliability. To this end, we employ powerful tools from classical robust control such as dissipativity and integral quadratic constraints. Further, we address limitations of VQAs, e.g., their unfavorable optimization landscape, through alternative quantum algorithm classes with feedback structure.

Tutorial and perspective article

• J. Berberich, D. Fink - "Quantum computing through the lens of control: A tutorial introduction" - IEEE Control Systems Magazine, 2024. Link

Robustness of quantum algorithms

• J. Berberich, T. Fellner, R. L. Kosut, C. Holm - "Robustness of quantum algorithms: Worst-case fidelity bounds and implications for design" - arXiv:2509.08481. Link
• M. Legnini, J. Berberich - "Robust feedback-based quantum optimization: Analysis of coherent control errors" - IEEE International Conference on Quantum Control, Computing, and Learning (qCCL) 2025. Link
• N. Funcke, J. Berberich - "Robustness of optimal quantum annealing protocols" - New Journal of Physics, 2024. Link
• J. Berberich, D. Fink, C. Holm - "Robustness of quantum algorithms against coherent control errors" - Physical Review A, 2024. Link

Variational quantum algorithms

• M. Legnini, J. Berberich - "Noise resilience and robust convergence guarantees for the variational quantum eigensolver" - arXiv:2601.16758. Link

C. Holm (Institute for Computational Physics, University of Stuttgart, DE)

R. L. Kosut (SC Solutions, Quantum Elements, Inc., and Princeton University, US)

Planned:

T. Pfau (5th Institute of Physics, University of Stuttgart, DE)

M. Gachechiladze (TU Darmstadt, DE)