## 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

## 🔬 Projects

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.

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.

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 Universität Berlin (FU Berlin), the Weierstrass Institute for Applied Analysis and Stochastics (WIAS), and the Zuse Institute Berlin (ZIB).

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.

## 💬 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

- Designolle, S., Vértesi, T., and Pokutta, S. (2024). Symmetric Multipartite Bell Inequalities Via Frank-Wolfe Algorithms.
*Physics Review A*. [arXiv]## [BibTeX]

- 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]

- Gelß, P., Issagali, A., and Kornhuber, R. (2023).
*Fredholm Integral Equations for Function Approximation and the Training of Neural Networks*. [arXiv]## [BibTeX]

- 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]

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

- 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]

- Stengl, S.-M. (2023).
*An Alternative Formulation of the Quantum Phase Estimation Using Projection-based Tensor Decompositions*. [arXiv]## [BibTeX]

- 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]

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

- 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]

- 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]

- 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]