Universität Stuttgart

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Open Access
Quantum machine learning for time series prediction
(2024) Fellner, Tobias
Time series prediction is an essential task in various fields, such as meteorology, finance and healthcare. Traditional approaches to time series prediction have primarily relied on regression and moving average methods, but recent advancements have seen a growing interest in applying machine learning techniques. With the rise of quantum computing, it is of interest to explore whether quantum machine learning can offer advantages over classical methods for time series forecasting. This thesis presents the first large-scale systematic benchmark comparing classical and quantum models for time series prediction. A variety of quantum models are evaluated against classical counterparts on different datasets. A novel quantum reservoir computing architecture is proposed, demonstrating promising results in handling nonlinear prediction tasks. The findings suggest that, for simpler time series prediction tasks, quantum models achieve accuracy comparable to classical methods. However, for more complex tasks, such as long-term forecasting, certain quantum models show improved performance. While current quantum machine learning models do not consistently outperform classical approaches, the results point to specific contexts where quantum methods may be beneficial.
Open Access
Quantum kernel methods and applications to differential equations
(2024) Flórez Ablan, Roberto
Quantum computers have the potential to surpass classical computers in specific tasks, promising advantages in many fields. Machine Learning (ML), a domain with significant societal impact, is a key area of interest for exploring the applications of quantum computing. Here, we investigate two research directions aimed at understanding how current quantum computers can be used to solve ML problems. First, we study Quantum Kernels (QKs). By calculating inner products between quantum states, QKs can be used to define similarity measures between points. QKs are a promising approach to Quantum Machine Learning (QML) but, in general, they have not been shown to outperform classical ML methods. A key reason for this is that QKs suffer from the exponential concentration problem. As the number of qubits increases, the kernel matrices become similar to the identity matrix, preventing generalization. One strategy to alleviate the exponential concentration problem is to rescale the data points that enter the quantum model. This technique is known as bandwidth tuning and has been shown to allow generalization in QKs. However, it has been numerically demonstrated that using this method results in QKs that cannot provide a quantum advantage over classical methods. In this thesis, we propose an explanation for this phenomenon. We show that due to the size of the rescaling factors, the QKs become similar to polynomial and RBF kernels, which are classically tractable. Second, we implemented a Differential Equation (DE) solver based on variational quantum methods. A Quantum Neural Network (QNN) or QK, is used to represent an ansatz for the solution of a DE. The DE information is included into a loss function, which is minimized using a classical optimizer. In the case of a QK, the optimized parameters are the coefficients of a linear combination of QKs evaluated at the data points. In the case of a QNN, the optimized parameters are the phases of the quantum gates. The QNN implementation was included into the open-source QML python library sQUlearn. A preliminary hyperparameter study was conducted for QKs. Based on our limited investigation, we conclude that QKs leveraging the fidelity between quantum states, known as Fidelity Quantum Kernels (FQKs), demonstrate superior performance compared to those employing a semi-classical approach, referred to as Projected Quantum Kernels (PQKs).
Open Access
MD simulations of 3D laser printing
(2024) Schmid, Jonas
Open Access
Wave functions and oscillator strengths in a two-band model for Rydberg excitons in cuprous oxide quantum wells
(2024) Kühner, Leon
Rydberg physics is the study of systems involving highly excited states of atoms or molecules, known as Rydberg states. In these states, one or more electrons are far from the nucleus, giving the atom exaggerated properties such as large size, long lifetimes, and strong interactions with external fields and nearby particles. These unique features make Rydberg systems a valuable tool for exploring a range of phenomena in atomic physics, quantum optics, and condensed matter physics. They are particularly important for applications in quantum technologies, such as quantum simulation and computation and sensing. Another candidate for Rydberg physics are excitons. When an electron is excited from the valence band to the conduction band the electron in the conduction band and the positively charged hole in the valance band can form hydrogen-like states. Excitons in cuprous oxide, though with relatively low principal quantum numbers, have already been detected in the 1950s by Gross and Hayashi. In 2014 it was possible to measure exciton states with a principal quantum number up to n=25, since then the exciton Rydberg physics has attracted large attention. These states have radii in the range of microns. Rydberg excitons show a large variety of phenomena which do not occur in atomic physics, for example the structure of the valence band leads to a breaking of the spherical symmetry, the spin-orbit coupling leads to the occurrence of a green and yellow exciton series, and central-cell corrections have effects on even parity states. Other effects occur when Rydberg excitons are confined in quantum wells. Such effects have been experimentally observed in GaAs. Thin layers in cuprous oxide have already been produced. Therefore, the observation of excitons in cuprous oxide quantum wells is expected soon. Excitons in quantum wells allow one to investigate the dimensional crossover from three-dimensional systems with weak confinement to two-dimensional systems with strong confinement. For this system the energy spectra have already been computed and effects like overlapping Rydberg series and resonances have been discussed. The theoretical calculations have so far been restricted to the computation of eigenenergies in a hydrogen-like model ignoring the impact of the valence band. The aim of this thesis is to study the effects of Rydberg excitons which rely on the wave functions. Such effects are the behavior of wave functions from weak to strong confinement and the quenching behavior in these regions that are visualized in this thesis. Numerically the wave functions are expanded in a B-spline basis. Also resonances above the first scattering threshold as well as bound states in the continuum above this threshold are visualized. Further, wave functions that undergo an avoided crossing are investigated. Another aspect is the influence of electrostatic effects for exciton states in quantum wells. These lead to the appearance of surface excitons, which can be seen in the visualization of these states. Oscillator strengths are investigated and rely on the behavior of the wave function. In our system the oscillator strengths are no longer translational invariant. Ultimately, this work provides a comprehensive exploration of Rydberg exciton wave functions, which could be instrumental in advancing the use of these systems in emerging quantum applications.

Universität Stuttgart

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