# Quantum Algorithms for Optimization ongoing

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.

## 🧑🎓 Project Members (excluding external)

## 🪙 Funding

This project is being funded by the EU ERA-LEARN QuantERA II Call 2021 from May 2022 to April 2025.

## 🔬 Project Description

Computational devices and closely-related information-processing technologies are among the most revolutionary inventions of the past century. Another groundbreaking discovery of the 20th century was quantum mechanics. Developed as a theoretical model to describe physics at the atomic level, it fundamentally changed our understanding of the world around us. The field of quantum computation stems from both of them. Its main objective is to understand how quantum mechanics changes our understanding of computation, especially the division between feasible and infeasible problems. Recent developments in quantum algorithms indicates that various optimization problems can be solved much faster on a quantum computer. Optimization problems permeate our society, they are key for the efficient operation of industry, logistics, and for countless other tasks that are crucial to the functioning of our modern society. Also, optimization problems are notoriously hard, and solving many of them precisely is out of reach of the most powerful modern computers even for modestly-sized instances. The radical vision of this project is to push the use of quantum computers for optimization tasks much further, developing new quantum algorithms which go well beyond the capability of even the best classical computers we have today. We aim at both general-purpose algorithms that can be used for a large variety of applications as well as more application- focused algorithms. We consider both continuous and discrete optimization, quantum algorithms for mixed-integer programs, as well as applications for machine learning, logistics, big data and physics. Our approach includes recent and exciting developments like quantum dynamic programming and graph sparsification. We are also interested in studying the QAOA-type algorithms, which can be executed even on really small quantum computers like the ones available today. The QOPT project is aimed at producing quantum algorithms for a variety of different flavours of optimization problems and attains a significant impact on state-of-the-art of quantum algorithms in the following ways: identification of new opportunities and applications fostered through quantum technologies, enhancing interdisciplinarity and crossing traditional boundaries between disciplines in order to enlarge the community involved in tackling these new challenges, creating diverse and inclusive quantum community, and spreading excellence throughout Europe by involving partners from the widening countries.

## 💬 Talks and posters

### Conference and workshop talks

- Feb 2024
*Fredholm Integral Equations for the Training of Shallow Neural Networks*by Patrick Gelß

7th SIAM UQ Conference- Sep 2023
*Fredholm Integral Equations for the Training of Shallow Neural Networks*by Patrick Gelß

7th RIKEN-MODAL Workshop, Berlin- Jun 2023
*Fredholm Integral Equations for the Training of Shallow Neural Networks*by Patrick Gelß

1st Workshop on Quantum Computation and Optimization, Berlin- Sep 2022
*Low-rank Tensor Decompositions of Quantum Circuits*by Patrick Gelß

6th RIKEN-MODAL Workshop, Tokyo / Fukuoka- May 2022
*Low-rank Tensor Decompositions of Quantum Circuits*by Patrick Gelß

3rd Workshop on Quantum Algorithms and Applications, Brussels

### Research seminar talks

- Jul 2023
*Fredholm Integral Equations for the Training of Shallow Neural Networks*by Patrick Gelß

IBM NOIP seminar Seminar, Berlin- Jun 2023
*Fredholm Integral Equations for the Training of Shallow Neural Networks*by Patrick Gelß

DESY CQTA seminar Seminar, Zeuthen- Oct 2022
*The Tensor-train Format and Its Applications*by Patrick Gelß

Tensor Learning Team Seminar

## 📝 Publications and preprints

- 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]

- 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]