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Quantum Mathematics

tensor decompositions and tensor-based algorithms; operator theory (transfer operators); numerical methods PDE; data-driven and kernel-based techniques

Developing quantum algorithms and quantum-inspired methods to push the boundaries in dynamical systems, machine learning, and information theory.

Patrick Gelß

What we are interested in

We are interested in exploring the multifaceted aspects of quantum computation and quantum information theory. The goal is to the development and application of new algorithms and quantum-inspired methods for solving problems in various fields such as dynamical systems, quantum nonlocality, and supervised learning. Our research encompasses a wide range of topics and utilizes a variety of mathematical tools, including but not limited to: tensor decompositions and tensor-based algorithms, convex optimization methods, operator theory (in particular transfer operators), numerical methods for partial differential equations, data-driven and kernel-based techniques.

Through our work, we aim to advance knowledge and understanding in the field of quantum science and technology by developing mathematically profound methods which exploit the potential of quantum mechanical concepts.

Members

Patrick Gelß
gelss (at) zib.de
Arwed Steuer
steuer (at) zib.de
Gabriele Iommazzo
iommazzo (at) zib.de
Hugo Abreu
abreu (at) zib.de
Sebastian Knebel
knebel (at) zib.de
Steven-Marian Stengl
stengl (at) zib.de
Sébastien Designolle
designolle (at) zib.de
Thomas Bake
bake (at) zib.de
Ye-Chao Liu
liu (at) zib.de
Zarin Shakibaei
shakibaei (at) zib.de

Projects

Publications

  1. Designolle, S., Vértesi, T., and Pokutta, S. (2024). Symmetric Multipartite Bell Inequalities Via Frank-Wolfe Algorithms. Physics Review A. [arXiv]
    [BibTeX]
    @article{symmetricBell2023,
  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}
}
  2. 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{dibkgp_bell_23,
  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}
}
  3. Gelß, P., Issagali, A., and Kornhuber, R. (2023). Fredholm Integral Equations for Function Approximation and the Training of Neural Networks. [arXiv]
    [BibTeX]
    @misc{gik_fredholm_23,
  archiveprefix = {arXiv},
  eprint = {2303.05262},
  primaryclass = {math.NA},
  year = {2023},
  author = {Gelß, Patrick and Issagali, Aizhan and Kornhuber, Ralf},
  title = {Fredholm Integral Equations for Function Approximation and the Training of Neural Networks}
}
  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{gkms_tdse_23,
  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}
}
  5. Klus, S., and Gelß, P. (2023). Continuous Optimization Methods for the Graph Isomorphism Problem. [arXiv]
    [BibTeX]
    @misc{kg_graph_isomorphism,
  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}
}
  6. 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{rgks_wavetrain_23,
  year = {2023},
  journal = {The Journal of Chemical Physics},
  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}
}
  7. Stengl, S.-M. (2023). An Alternative Formulation of the Quantum Phase Estimation Using Projection-based Tensor Decompositions. [arXiv]
    [BibTeX]
    @misc{stengl_qpe_23,
  archiveprefix = {arXiv},
  eprint = {2303.05894},
  primaryclass = {quant-ph},
  year = {2023},
  author = {Stengl, Steven-Marian},
  title = {An Alternative Formulation of the Quantum Phase Estimation Using Projection-based Tensor Decompositions}
}
  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{gkms_tise_22,
  year = {2022},
  journal = {The Journal of Chemical Physics},
  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. Gelß, P., Klus, S., Shakibaei, Z., and Pokutta, S. (2022). Low-rank Tensor Decompositions of Quantum Circuits. [arXiv]
    [BibTeX]
    @misc{gksp_lowrankqc_22,
  archiveprefix = {arXiv},
  eprint = {2205.09882},
  primaryclass = {quant-ph},
  year = {2022},
  author = {Gelß, Patrick and Klus, Stefan and Shakibaei, Zarin and Pokutta, Sebastian},
  title = {Low-rank Tensor Decompositions of Quantum Circuits}
}
  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{gkss_fsa_21,
  year = {2021},
  journal = {Knowledge-Based Systems},
  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{kgnn_antisymmetric_21,
  year = {2021},
  journal = {Machine Learning: Science and Technology},
  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}
}
  12. 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{ngkc_metastable_2021,
  year = {2021},
  journal = {Physica D: Nonlinear Phenomena},
  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}
}