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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; Bell nonlocality; entanglement quantification

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What we are interested in

Exploring the multifaceted aspects of quantum computation and quantum information theory. The goal is to the develop application of new algorithms and quantum-inspired methods for solving problems in various fields such as dynamical systems, Bell 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

Sebastian Pokutta
Department Head
pokutta (at) zib.de
Patrick Gelß
Research Area Lead
gelss (at) zib.de
Arwed Steuer
steuer (at) zib.de
Steven-Marian Stengl
stengl (at) zib.de
Sebastian Knebel
knebel (at) zib.de
SĂ©bastien Designolle
designolle (at) zib.de
Thomas Bake
bake (at) zib.de
Ye-Chao Liu
liu (at) zib.de
Hugo Abreu
abreu (at) zib.de
Jung Nguyen
nguyen (at) zib.de

🔬 Projects

Entanglement Detection Via Frank-Wolfe Algorithms

We will apply a Frank-Wolfe-based approach for separability certification and entanglement detection of multipartite quantum states. The method will be further exploited to derive entanglement witnesses in the case, often encountered in experiments, of incomplete characterisation of the quantum state.

MATH+ AA2-19
Jan 2024 to Dec 2025
3

Quantum Algorithms for Optimization

Computational devices and quantum mechanics revolutionized the 20th century. Quantum computation merges these fields to solve optimization problems faster than classical computers. This project aims to develop new quantum algorithms for general-purpose and specific applications, including optimization, mixed-integer programs, and uses in machine learning, logistics, big data, and physics. We will explore quantum dynamic programming, graph sparsification, and QAOA-type algorithms for small quantum computers.

QOPT
May 2022 to Apr 2025
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Approximating Combinatorial Optimization Problems

This project will explore to what extent near-term quantum computing may potentially provide better approximations to combinatorial optimization problems, bringing together expertise from quantum computing and applied mathematics. It is a collaborative effort within the Einstein Research Unit on Quantum Devices, involving the Freie UniversitÀt Berlin (FU Berlin), the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), and the Zuse Institute Berlin (ZIB).

ACOP
Jan 2022 to Dec 2024
3

Quantum Computing and Integer Programming

Quantum computing has the potential to occupy an important position in the field of computing in the long term. In the short to medium term, however, we can expect the emergence of so-called "Noisy Intermediate-Scale Quantum algorithms" and "Quantum Approximate Optimization Algorithms" in an initial phase. These algorithms can often solve a special subclass of problems approximately and very quickly using quantum computing and are significantly easier to implement technically. In this pilot project, a workflow will be tested, ranging from the design of hybrid quantum-classical algorithms to their simulation using quantum simulators on HPC hardware.

QCIP
Apr 2021 to Dec 2021
3

💬 Talks and posters

Conference and workshop talks

Feb 2024
Patrick Gelß: Fredholm Integral Equations for the Training of Shallow Neural Networks, 7th SIAM UQ Conference
Sep 2023
Patrick Gelß: Fredholm Integral Equations for the Training of Shallow Neural Networks, 7th RIKEN-MODAL Workshop, Berlin
Jun 2023
Patrick Gelß: Fredholm Integral Equations for the Training of Shallow Neural Networks, 1st Workshop on Quantum Computation and Optimization, Berlin
Sep 2022
Patrick Gelß: Low-rank Tensor Decompositions of Quantum Circuits, 6th RIKEN-MODAL Workshop, Tokyo / Fukuoka
May 2022
Patrick Gelß: Low-rank Tensor Decompositions of Quantum Circuits, 3rd Workshop on Quantum Algorithms and Applications, Brussels

Research seminar talks

Jul 2023
Patrick Gelß: Fredholm Integral Equations for the Training of Shallow Neural Networks, IBM NOIP seminar Seminar, Berlin
Jun 2023
Patrick Gelß: Fredholm Integral Equations for the Training of Shallow Neural Networks, DESY CQTA seminar Seminar, Zeuthen
Oct 2022
Patrick Gelß: The Tensor-train Format and Its Applications, Tensor Learning Team Seminar

📝 Publications and preprints

  1. Designolle, S., VĂ©rtesi, 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, SĂ©bastien and VĂ©rtesi, TamĂĄs 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, SĂ©bastien 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 SchĂŒtte, 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 SchĂŒtte, Christof},
      title = {Feature Space Approximation for Kernel-based Supervised Learning}
    }
  11. Klus, S., Gelß, P., NĂŒske, F., and NoĂ©, 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 NĂŒske, Feliks and NoĂ©, Frank},
      title = {Symmetric and Antisymmetric Kernels for Machine Learning Problems in Quantum Physics and Chemistry}
    }
  12. NĂŒske, 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 = {NĂŒske, Feliks and Gelß, Patrick and Klus, Stefan and Clementi, Cecilia},
      title = {Tensor-based Computation of Metastable and Coherent Sets}
    }