05 Fakultät Informatik, Elektrotechnik und Informationstechnik

Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/6

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Subject: search.filters.subject.510Subject: search.filters.subject.004Start date: 2020End date: 2023

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    Circuit complexity of group theoretic problems
    (2021) Weiß, Armin; Diekert, Volker (Prof. Dr. rer. nat.)
    In dieser kumulativen Habilitationsschrift werden sechs Arbeiten zum Thema "Schaltkreiskomplexität von Gruppentheoretischen Problemen" zusammengefasst. An vorderster Stelle steht hierbei das Wortproblem: Gegeben ein Wort über den Erzeugern einer Gruppe, ist die Frage, ob das Wort das Einselement der Gruppe darstellt. Daneben werden noch weitere Probleme, wie das Konjugationsproblem, das Power-Wortproblem (wie das Wortproblem, aber die Eingabe wird in komprimierter Form gegeben) und das Lösen von Gleichungen betrachtet. Die meisten der hier zusammengefassten Arbeiten betrachten die genannten Probleme für spezielle Klassen von Gruppen und klassifizieren deren Komplexität mit Methoden der Schaltkreiskomplexität. Eine Ausnahme bildet die letzte Arbeit zum Thema Gleichungen: hier liegt der Zusammenhang zur Schaltkreiskomplexität darin, dass sich das Erfüllbarkeitsproblem für Gleichungen in endlichen auslösbaren Gruppen ähnlich verhält wie das Erfüllbarkeitsproblem für CC^0 Schaltkreise.
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    Equation satisfiability in solvable groups
    (2022) Idziak, Paweł; Kawałek, Piotr; Krzaczkowski, Jacek; Weiß, Armin
    The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell in (Inf. Comput. 178 (1), 253-262, 10 ) where they showed that this problem is in P for nilpotent groups while it is NP -complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups G having a nilpotent normal subgroup H with nilpotent factor G / H . This paper shows that such a normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless the Exponential Time Hypothesis fails.
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    Quantum support vector machines of high-dimensional data for image classification problems
    (2023) Vikas Singh, Rajput
    This thesis presents a comprehensive investigation into the efficient utilization of Quantum Support Vector Machines (QSVMs) for image classification on high-dimensional data. The primary focus is on analyzing the standard MNIST dataset and the high-dimensional dataset provided by TRUMPF SE + Co. KG. To evaluate the performance of QSVMs against classical Support Vector Machines (SVMs) for high-dimensional data, a benchmarking framework is proposed. In the current Noisy Intermediate Scale Quantum (NISQ) era, classical preprocessing of the data is a crucial step to prepare the data for classification tasks using NISQ machines. Various dimensionality reduction techniques, such as principal component analysis (PCA), t-distributed stochastic neighbor embedding (tSNE), and convolutional autoencoders, are explored to preprocess the image datasets. Convolutional autoencoders are found to outperform other methods when calculating quantum kernels on a small dataset. Furthermore, the benchmarking framework systematically analyzes different quantum feature maps by varying hyperparameters, such as the number of qubits, the use of parameterized gates, the number of features encoded per qubit line, and the use of entanglement. Quantum feature maps demonstrate higher accuracy compared to classical feature maps for both TRUMPF and MNIST data. Among the feature maps, one using 𝑅𝑧 and 𝑅𝑦 gates with two features per qubit, without entanglement, achieves the highest accuracy. The study also reveals that increasing the number of qubits leads to improved accuracy for the real-world TRUMPF dataset. Additionally, the choice of the quantum kernel function significantly impacts classification results, with the projected type quantum kernel outperforming the fidelity type quantum kernel. Subsequently, the study examines the Kernel Target Alignment (KTA) optimization method to improve the pipeline. However, for the chosen feature map and dataset, KTA does not provide significant benefits. In summary, the results highlight the potential for achieving quantum advantage by optimizing all components of the quantum classifier framework. Selecting appropriate dimensionality reduction techniques, quantum feature maps, and quantum kernel methods is crucial for enhancing classification accuracy. Further research is needed to address challenges related to kernel optimization and fully leverage the capabilities of quantum computing in machine learning applications.