Universität Stuttgart
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Item Open Access Long wave approximation over and beyond the natural time scale(2024) Hofbauer, Sarah; Schneider, Guido (Prof. Dr.)Item Open Access Rigorous compilation for near-term quantum computers(2024) Brandhofer, Sebastian; Polian, Ilia (Prof.)Quantum computing promises an exponential speedup for computational problems in material sciences, cryptography and drug design that are infeasible to resolve by traditional classical systems. As quantum computing technology matures, larger and more complex quantum states can be prepared on a quantum computer, enabling the resolution of larger problem instances, e.g. breaking larger cryptographic keys or modelling larger molecules accurately for the exploration of novel drugs. Near-term quantum computers, however, are characterized by large error rates, a relatively low number of qubits and a low connectivity between qubits. These characteristics impose strict requirements on the structure of quantum computations that must be incorporated by compilation methods targeting near-term quantum computers in order to ensure compatibility and yield highly accurate results. Rigorous compilation methods have been explored for addressing these requirements as they exactly explore the solution space and thus yield a quantum computation that is optimal with respect to the incorporated requirements. However, previous rigorous compilation methods demonstrate limited applicability and typically focus on one aspect of the imposed requirements, i.e. reducing the duration or the number of swap gates in a quantum computation. In this work, opportunities for improving near-term quantum computations through compilation are explored first. These compilation opportunities are included in rigorous compilation methods to investigate each aspect of the imposed requirements, i.e. the number of qubits, connectivity of qubits, duration and incurred errors. The developed rigorous compilation methods are then evaluated with respect to their ability to enable quantum computations that are otherwise not accessible with near-term quantum technology. Experimental results demonstrate the ability of the developed rigorous compilation methods to extend the computational reach of near-term quantum computers by generating quantum computations with a reduced requirement on the number and connectivity of qubits as well as reducing the duration and incurred errors of performed quantum computations. Furthermore, the developed rigorous compilation methods extend their applicability to quantum circuit partitioning, qubit reuse and the translation between quantum computations generated for distinct quantum technologies. Specifically, a developed rigorous compilation method exploiting the structure of a quantum computation to reuse qubits at runtime yielded a reduction in the required number of qubits of up to 5x and result error by up to 33%. The developed quantum circuit partitioning method optimally distributes a quantum computation to distinct separate partitions, reducing the required number of qubits by 40% and the cost of partitioning by 41% on average. Furthermore, a rigorous compilation method was developed for quantum computers based on neutral atoms that combines swap gate insertions and topology changes to reduce the impact of limited qubit connectivity on the quantum computation duration by up to 58% and on the result fidelity by up to 29%. Finally, the developed quantum circuit adaptation method enables to translate between distinct quantum technologies while considering heterogeneous computational primitives with distinct characteristics to reduce the idle time of qubits by up to 87% and the result fidelity by up to 40%.Item Open Access Data-efficient and safe learning with Gaussian processes(2020) Schreiter, Jens; Toussaint, Marc (Prof. Dr. rer. nat.)Data-based modeling techniques enjoy increasing popularity in many areas of science and technology where traditional approaches are limited regarding accuracy and efficiency. When employing machine learning methods to generate models of dynamic system, it is necessary to consider two important issues. Firstly, the data-sampling process should induce an informative and representative set of points to enable high generalization accuracy of the learned models. Secondly, the algorithmic part for efficient model building is essential for applicability, usability, and the quality of the learned predictive model. This thesis deals with both of these aspects for supervised learning problems, where the interaction between them is exploited to realize an exact and powerful modeling. After introducing the non-parametric Bayesian modeling approach with Gaussian processes and basics for transient modeling tasks in the next chapter, we dedicate ourselves to extensions of this probabilistic technique to relevant practical requirements in the subsequent chapter. This chapter provides an overview on existing sparse Gaussian process approximations and propose some novel work to increase efficiency and model selection on particularly large training data sets. For example, our sparse modeling approach enables real-time capable prediction performance and efficient learning with low memory requirements. A comprehensive comparison on various real-world problems confirms the proposed contributions and shows a variety of modeling tasks, where approximate Gaussian processes can be successfully applied. Further experiments provide more insight about the whole learning process, and thus a profound understanding of the presented work. In the fourth chapter, we focus on active learning schemes for safe and information-optimal generation of meaningful data sets. In addition to the exploration behavior of the active learner, the safety issue is considered in our work, since interacting with real systems should not result in damages or even completely destroy it. Here we propose a new model-based active learning framework to solve both tasks simultaneously. As basis for the data-sampling process we employ the presented Gaussian process techniques. Furthermore, we distinguish between static and transient experimental design strategies. Both problems are separately considered in this chapter. Nevertheless, the requirements for each active learning problem are the same. This subdivision into a static and transient setting allows a more problem-specific perspective on the two cases, and thus enables the creation of specially adapted active learning algorithms. Our novel approaches are then investigated for different applications, where a favorable trade-off between safety and exploration is always realized. Theoretical results maintain these evaluations and provide respectable knowledge about the derived model-based active learning schemes. For example, an upper bound for the probability of failure of the presented active learning methods is derived under reasonable assumptions. Finally, the thesis concludes with a summary of the investigated machine learning problems and motivate some future research directions.Item Open Access Schaustücke : Einblicke in wissenschaftliche Sammlungen der Universität Stuttgart(Stuttgart : Universität Stuttgart, 2022) Wiatrowski, Frank (Gestaltung, Fotograf); Engstler, Katja Stefanie (Gestaltung); Ceranski, Beate (Vorwort); Rambach, Christiane (Vorwort)Die wissenschaftlichen Sammlungen der Universität zeugen von einer langen Lehr- und Forschungstradition. In Fakultäten und Instituten, in der Universitätsbibliothek und im Universitätsarchiv sind vielfältige Sammlungen beheimatet, zum Teil mit ungewöhnlichen oder gar einzigartigen Objekten. Die Broschüre gibt erste Einblicke in diese vielfach versteckte Welt der universitären Sammlungen in Stuttgart.Item Open Access From simplicial groups to crossed squares(2022) Asiki, Natalia-MariaSimplicial groups are defined to be contravariant functors from the simplex category to the category of groups. The truncation functor maps a simplicial group to a [2,0]-simplicial group, satisfying the Conduché condition. A functor from the category of [2,0]-simplicial groups to the category of crossed squares is constructed, following Porter. It is shown that the latter functor is not an equivalence of categories. In addition, Loday's variant of the resulting crossed square is constructed and shown to be isomorphic to Porter's variant.Item Open Access Stability of compact symmetric spaces(2022) Semmelmann, Uwe; Weingart, GregorIn this article, we study the stability problem for the Einstein-Hilbert functional on compact symmetric spaces following and completing the seminal work of Koiso on the subject. We classify in detail the irreducible representations of simple Lie algebras with Casimir eigenvalue less than the Casimir eigenvalue of the adjoint representation and use this information to prove the stability of the Einstein metrics on both the quaternionic and Cayley projective plane. Moreover, we prove that the Einstein metrics on quaternionic Grassmannians different from projective spaces are unstable.Item Open Access Gendo-Frobenius algebras and comultiplication(2022) Yırtıcı, ÇiğdemGendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras.Item Open Access General mathematical model for the period chirp in interference lithography(2023) Bienert, Florian; Graf, Thomas; Abdou Ahmed, MarwanItem Open Access On a stochastic Camassa-Holm type equation with higher order nonlinearities(2020) Rohde, Christian; Tang, HaoThe subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.Item Open Access Learning with high-dimensional data(2022) Fischer, Simon; Steinwart, Ingo (Prof. Dr.)This thesis is divided into three parts. In the first part we introduce a framework that allows us to investigate learning scenarios with restricted access to the data. We use this framework to model high-dimensional learning scenarios as an infinite-dimensional one in which the learning algorithm has only access to some finite-dimensional projections of the data. Finally, we provide a prototypical example of such an infinite-dimensional classification problem in which histograms can achieve polynomial learning rates. In the second part we present some individual results that might by useful for the investigation of kernel-based learning methods using Gaussian kernels in high- or infinite-dimensional learning problems. To be more precise, we present log-covering number bounds for Gaussian reproducing kernel Hilbert spaces on general bounded subsets of the Euclidean space. Unlike previous results in this direction we focus on small explicit constants and their dependence on crucial parameters such as the kernel width as well as the size and dimension of the underlying space. Afterwards, we generalize these bounds to Gaussian kernels defined on special infinite-dimensional compact subsets of the sequence space ℓ_2. More precisely, the considered domains are given by the image of the unit ℓ_∞-ball under some diagonal operator. In the third part we contribute some new insights to the compactness properties of diagonal operators from ℓ_p to ℓ_q for p ≠ q.