Quantum Mathematics

📝 Publications and preprints

Preprints

1. Sander, A., Fröhlich, M., Eigel, M., Eisert, J., Gelß, P., Hintermüller, M., Milbradt, R., Wille, R., and Mendl, C. (2025). Large-scale Stochastic Simulation of Open Quantum Systems. [arXiv]
   [BibTeX]

   @misc{2025_SanderEtAl_Largescalesimulation,
     archiveprefix = {arXiv},
     eprint = {2501.17913},
     primaryclass = {quant-ph},
     year = {2025},
     author = {Sander, Aaron and Fröhlich, Maximilian and Eigel, Martin and Eisert, Jens and Gelß, Patrick and Hintermüller, Michael and Milbradt, Richard and Wille, Robert and Mendl, Christian},
     title = {Large-scale Stochastic Simulation of Open Quantum Systems},
     date = {2025-01-29}
   }

2. Designolle, S., Vértesi, T., and Pokutta, S. (2024). Better Bounds on Grothendieck Constants of Finite Orders. [arXiv]
   [BibTeX]

   @misc{2023_DesignolleVertesiPokutta_Grothendieckconstants,
     archiveprefix = {arXiv},
     eprint = {2409.03739},
     primaryclass = {math.OC},
     year = {2024},
     author = {Designolle, Sébastien and Vértesi, Tamás and Pokutta, Sebastian},
     title = {Better Bounds on Grothendieck Constants of Finite Orders},
     date = {2024-09-05}
   }

3. Liu, Y.-C., and Shang, J. (2024). Beating the Optimal Verification of Entangled States Via Collective Strategies. [arXiv]
   [BibTeX]

   @misc{2024_LiuShang_Collectiveverification,
     archiveprefix = {arXiv},
     eprint = {2410.00554},
     primaryclass = {quant-ph},
     year = {2024},
     author = {Liu, Ye-Chao and Shang, Jiangwei},
     title = {Beating the Optimal Verification of Entangled States Via Collective Strategies},
     date = {2024-10-01}
   }

4. Gelß, P., Klein, R., Matera, S., and Schmidt, B. (2023). Quantum Dynamics of Coupled Excitons and Phonons in Chain-like Systems: Tensor Train Approaches and Higher-order Propagators. [arXiv]
   [BibTeX]

   @misc{2023_GelssKleinMateraSchmidt_QuantumdynamicsTensortrain,
     archiveprefix = {arXiv},
     eprint = {2302.03568},
     primaryclass = {quant-ph},
     year = {2023},
     author = {Gelß, Patrick and Klein, Rupert and Matera, Sebastian and Schmidt, Burkhard},
     title = {Quantum Dynamics of Coupled Excitons and Phonons in Chain-like Systems: Tensor Train Approaches and Higher-order Propagators},
     date = {2023-02-07}
   }

5. Klus, S., and Gelß, P. (2023). Continuous Optimization Methods for the Graph Isomorphism Problem. [arXiv]
   [BibTeX]

   @misc{2023_KlusGelss_Continuousgraphisomorphism,
     archiveprefix = {arXiv},
     eprint = {2311.16912},
     primaryclass = {cs.DM},
     year = {2023},
     author = {Klus, Stefan and Gelß, Patrick},
     title = {Continuous Optimization Methods for the Graph Isomorphism Problem},
     date = {2023-11-28}
   }

6. Gelß, P., Klus, S., Knebel, S., Shakibaei, Z., and Pokutta, S. (2022). Low-rank Tensor Decompositions of Quantum Circuits. [arXiv]
   [BibTeX]

   @misc{2022_GelssKlusShakibaeiPokutta_Lowranktensordecompositions,
     archiveprefix = {arXiv},
     eprint = {2205.09882},
     primaryclass = {quant-ph},
     year = {2022},
     author = {Gelß, Patrick and Klus, Stefan and Knebel, Sebastian and Shakibaei, Zarin and Pokutta, Sebastian},
     title = {Low-rank Tensor Decompositions of Quantum Circuits},
     date = {2022-05-19}
   }

Full articles

1. Abbas, A., Ambainis, A., Augustino, B., Bärtschi, A., Buhrman, H., Coffrin, C., Cortiana, G., Dunjko, V., Egger, D. J., Elmegreen, B. G., Franco, N., Fratini, F., Fuller, B., Gacon, J., Gonciulea, C., Gribling, S., Gupta, S., Hadfield, S., Heese, R., … Zoufal, C. (2023). Challenges and Opportunities in Quantum Optimization. Nature Reviews Physics. DOI: https://doi.org/10.1038/s42254-024-00770-9 [arXiv]
   [BibTeX]

   @article{2023_AmiraEtAl_Quantumoptimization,
     year = {2024},
     journal = {Nature Reviews Physics},
     month = sep,
     doi = {https://doi.org/10.1038/s42254-024-00770-9},
     archiveprefix = {arXiv},
     eprint = {2312.02279},
     primaryclass = {quant-ph},
     author = {Abbas, Amira and Ambainis, Andris and Augustino, Brandon and Bärtschi, Andreas and Buhrman, Harry and Coffrin, Carleton and Cortiana, Giorgio and Dunjko, Vedran and Egger, Daniel J. and Elmegreen, Bruce G. and Franco, Nicola and Fratini, Filippo and Fuller, Bryce and Gacon, Julien and Gonciulea, Constantin and Gribling, Sander and Gupta, Swati and Hadfield, Stuart and Heese, Raoul and Kircher, Gerhard and Kleinert, Thomas and Koch, Thorsten and Korpas, Georgios and Lenk, Steve and Marecek, Jakub and Markov, Vanio and Mazzola, Guglielmo and Mensa, Stefano and Mohseni, Naeimeh and Nannicini, Giacomo and O'Meara, Corey and Peña Tapia, Elena and Pokutta, Sebastian and Proissl, Manuel and Rebentrost, Patrick and Sahin, Emre and Symons, Benjamin C. B. and Tornow, Sabine and Valls, Victor and Woerner, Stefan and Wolf-Bauwens, Mira L. and Yard, Jon and Yarkoni, Sheir and Zechiel, Dirk and Zhuk, Sergiy and Zoufal, Christa},
     title = {Challenges and Opportunities in Quantum Optimization},
     date = {2023-12-04}
   }

2. Gelß, P., Issagali, A., and Kornhuber, R. (2024). Fredholm Integral Equations for Function Approximation and the Training of Neural Networks. SIAM Journal on Mathematics of Data Science. DOI: 10.1137/23M156642X [URL] [arXiv]
   [BibTeX]

   @article{2023_GelssIssagaliKornhuber_Fredholmneuralnetworks,
     year = {2024},
     journal = {SIAM Journal on Mathematics of Data Science},
     date = {2024-10-30},
     doi = {10.1137/23M156642X},
     url = {https://epubs.siam.org/doi/abs/10.1137/23M156642X},
     archiveprefix = {arXiv},
     eprint = {2303.05262},
     primaryclass = {math.NA},
     author = {Gelß, Patrick and Issagali, Aizhan and Kornhuber, Ralf},
     title = {Fredholm Integral Equations for Function Approximation and the Training of Neural Networks}
   }

3. Stengl, S.-M., Gelß, P., Klus, S., and Pokutta, S. (2024). Existence and Uniqueness of Solutions of the Koopman–von Neumann Equation on Bounded Domains. Journal of Physics A: Mathematical and Theoretical. DOI: 10.1088/1751-8121/ad6f7d [URL] [arXiv]
   [BibTeX]

   @article{2023_StenglGelssKlusPokutta_Koopmanvonneumann,
     year = {2024},
     journal = {Journal of Physics A: Mathematical and Theoretical},
     date = {2024-08-14},
     doi = {10.1088/1751-8121/ad6f7d},
     url = {https://iopscience.iop.org/article/10.1088/1751-8121/ad6f7d},
     archiveprefix = {arXiv},
     eprint = {2306.13504},
     primaryclass = {math.AP},
     author = {Stengl, Steven-Marian and Gelß, Patrick and Klus, Stefan and Pokutta, Sebastian},
     title = {Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains}
   }

4. Stengl, S.-M. (2024). An Alternative Formulation of the Quantum Phase Estimation Using Projection-based Tensor Decompositions. Quantum Information Processing, 23. DOI: 10.1007/s11128-024-04347-4 [URL] [arXiv]
   [BibTeX]

   @article{2023_Stengl_Quantumphaseestimation,
     year = {2024},
     journal = {Quantum Information Processing},
     date = {2024-04-09},
     volume = {23},
     doi = {10.1007/s11128-024-04347-4},
     url = {https://link.springer.com/article/10.1007/s11128-024-04347-4},
     archiveprefix = {arXiv},
     eprint = {2303.05894},
     primaryclass = {quant-ph},
     author = {Stengl, Steven-Marian},
     title = {An Alternative Formulation of the Quantum Phase Estimation Using Projection-based Tensor Decompositions}
   }

5. Designolle, S., Vértesi, T., and Pokutta, S. (2023). Symmetric Multipartite Bell Inequalities Via Frank-Wolfe Algorithms. Physics Review A. [arXiv]
   [BibTeX]

   @article{2024_DesignolleVertesiPokutta_SymmetricBellinequalities,
     year = {2024},
     journal = {Physics Review A},
     archiveprefix = {arXiv},
     eprint = {2310.20677},
     primaryclass = {quant-ph},
     author = {Designolle, Sébastien and Vértesi, Tamás and Pokutta, Sebastian},
     title = {Symmetric Multipartite Bell Inequalities Via Frank-Wolfe Algorithms},
     date = {2023-10-31}
   }

6. Designolle, S., Iommazzo, G., Besançon, M., Knebel, S., Gelß, P., and Pokutta, S. (2023). Improved Local Models and New Bell Inequalities Via Frank-Wolfe Algorithms. Physical Review Research, 5(4). DOI: 10.1103/PhysRevResearch.5.043059 [arXiv] [slides] [code]
   [BibTeX]

   @article{2023_DesignolleEtAl_LocalmodelsBellinequalities,
     year = {2023},
     journal = {Physical Review Research},
     month = oct,
     volume = {5},
     number = {4},
     doi = {10.1103/PhysRevResearch.5.043059},
     archiveprefix = {arXiv},
     eprint = {2302.04721},
     primaryclass = {quant-ph},
     author = {Designolle, Sébastien and Iommazzo, Gabriele and Besançon, Mathieu and Knebel, Sebastian and Gelß, Patrick and Pokutta, Sebastian},
     title = {Improved Local Models and New Bell Inequalities Via Frank-Wolfe Algorithms},
     code = {https://github.com/ZIB-IOL/BellPolytopes.jl},
     slides = {https://www.pokutta.com/slides/20230808-tokyo-bell.pdf},
     date = {2023-02-09}
   }

7. Riedel, J., Gelß, P., Klein, R., and Schmidt, B. (2023). WaveTrain: A Python Package for Numerical Quantum Mechanics of Chain-like Systems Based on Tensor Trains. The Journal of Chemical Physics, 158(16), 164801. DOI: 10.1063/5.0147314 [URL] [arXiv]
   [BibTeX]

   @article{2023_RiedelGelssKleinSchmidt_Wavetrain,
     year = {2023},
     journal = {The Journal of Chemical Physics},
     date = {2023-04-28},
     volume = {158},
     number = {16},
     pages = {164801},
     doi = {10.1063/5.0147314},
     url = {https://pubs.aip.org/aip/jcp/article/158/16/164801/2887212/WaveTrain-A-Python-package-for-numerical-quantum},
     archiveprefix = {arXiv},
     eprint = {2302.03725},
     primaryclass = {quant-ph},
     author = {Riedel, Jerome and Gelß, Patrick and Klein, Rupert and Schmidt, Burkhard},
     title = {WaveTrain: A Python Package for Numerical Quantum Mechanics of Chain-like Systems Based on Tensor Trains}
   }

8. Gelß, P., Klein, R., Matera, S., and Schmidt, B. (2022). Solving the Time-independent Schrödinger Equation for Chains of Coupled Excitons and Phonons Using Tensor Trains. The Journal of Chemical Physics, 156, 024109. DOI: 10.1063/5.0074948 [URL] [arXiv]
   [BibTeX]

   @article{2021_GelssKleinMateraSchmidt_Tensortrainschrdinger,
     year = {2022},
     journal = {The Journal of Chemical Physics},
     date = {2022-01-12},
     volume = {156},
     pages = {024109},
     doi = {10.1063/5.0074948},
     url = {https://pubs.aip.org/aip/jcp/article/156/2/024109/2839835/Solving-the-time-independent-Schrodinger-equation},
     archiveprefix = {arXiv},
     eprint = {2109.15104},
     primaryclass = {physics.comp-ph},
     author = {Gelß, Patrick and Klein, Rupert and Matera, Sebastian and Schmidt, Burkhard},
     title = {Solving the Time-independent Schrödinger Equation for Chains of Coupled Excitons and Phonons Using Tensor Trains}
   }

9. Nüske, F., Gelß, P., Klus, S., and Clementi, C. (2021). Tensor-based Computation of Metastable and Coherent Sets. Physica D: Nonlinear Phenomena, 427, 133018. DOI: 10.1016/j.physd.2021.133018 [URL] [arXiv]
   [BibTeX]

   @article{2019_NueskeGelssKlusClementi_Tensorkoopman,
     year = {2021},
     journal = {Physica D: Nonlinear Phenomena},
     date = {2021-09-20},
     volume = {427},
     pages = {133018},
     doi = {10.1016/j.physd.2021.133018},
     url = {https://sciencedirect.com/science/article/abs/pii/S0167278921001755},
     archiveprefix = {arXiv},
     eprint = {1908.04741},
     primaryclass = {math.NA},
     author = {Nüske, Feliks and Gelß, Patrick and Klus, Stefan and Clementi, Cecilia},
     title = {Tensor-based Computation of Metastable and Coherent Sets}
   }

10. Gelß, P., Klus, S., Schuster, I., and Schütte, C. (2021). Feature Space Approximation for Kernel-based Supervised Learning. Knowledge-Based Systems, 221, 106935. DOI: 10.1016/j.knosys.2021.106935 [URL] [arXiv]
    [BibTeX]

    @article{2020_GelssKlusSchusterSchuette_Featureapproximation,
      year = {2021},
      journal = {Knowledge-Based Systems},
      date = {2021-03-20},
      volume = {221},
      pages = {106935},
      doi = {10.1016/j.knosys.2021.106935},
      url = {https://sciencedirect.com/science/article/abs/pii/S0950705121001982},
      archiveprefix = {arXiv},
      eprint = {2011.12651},
      primaryclass = {stat.ML},
      author = {Gelß, Patrick and Klus, Stefan and Schuster, Ingmar and Schütte, Christof},
      title = {Feature Space Approximation for Kernel-based Supervised Learning}
    }

11. Klus, S., Gelß, P., Nüske, F., and Noé, F. (2021). Symmetric and Antisymmetric Kernels for Machine Learning Problems in Quantum Physics and Chemistry. Machine Learning: Science and Technology, 2(4), 18958. DOI: 10.1088/2632-2153/ac14ad [URL] [arXiv]
    [BibTeX]

    @article{2021_KlusGelssNueskeNoe_Symmetricantisymmetrickernels,
      year = {2021},
      journal = {Machine Learning: Science and Technology},
      date = {2021-08-06},
      volume = {2},
      number = {4},
      pages = {18958},
      doi = {10.1088/2632-2153/ac14ad},
      url = {https://iopscience.iop.org/article/10.1088/2632-2153/ac14ad},
      archiveprefix = {arXiv},
      eprint = {2103.17233},
      primaryclass = {quant-ph},
      author = {Klus, Stefan and Gelß, Patrick and Nüske, Feliks and Noé, Frank},
      title = {Symmetric and Antisymmetric Kernels for Machine Learning Problems in Quantum Physics and Chemistry}
    }