633 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Randomized Joint Diagonalization of Symmetric Matrices

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    Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD) for performing this task. RJD applies a standard eigenvalue solver to random linear combinations of the matrices. Unlike existing optimization-based methods, RJD is simple to implement and leverages existing high-quality linear algebra software packages. Our main novel contribution is to prove robust recovery: Given a family that is ϵ\epsilon-near to a commuting family, RJD jointly diagonalizes this family, with high probability, up to an error of norm O(ϵ\epsilon). No other existing method is known to enjoy such a universal robust recovery guarantee. We also discuss how the algorithm can be further improved by deflation techniques and demonstrate its state-of-the-art performance by numerical experiments with synthetic and real-world data

    Mesoscopic Physics of Quantum Systems and Neural Networks

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    We study three different kinds of mesoscopic systems – in the intermediate region between macroscopic and microscopic scales consisting of many interacting constituents: We consider particle entanglement in one-dimensional chains of interacting fermions. By employing a field theoretical bosonization calculation, we obtain the one-particle entanglement entropy in the ground state and its time evolution after an interaction quantum quench which causes relaxation towards non-equilibrium steady states. By pushing the boundaries of the numerical exact diagonalization and density matrix renormalization group computations, we are able to accurately scale to the thermodynamic limit where we make contact to the analytic field theory model. This allows to fix an interaction cutoff required in the continuum bosonization calculation to account for the short range interaction of the lattice model, such that the bosonization result provides accurate predictions for the one-body reduced density matrix in the Luttinger liquid phase. Establishing a better understanding of how to control entanglement in mesoscopic systems is also crucial for building qubits for a quantum computer. We further study a popular scalable qubit architecture that is based on Majorana zero modes in topological superconductors. The two major challenges with realizing Majorana qubits currently lie in trivial pseudo-Majorana states that mimic signatures of the topological bound states and in strong disorder in the proposed topological hybrid systems that destroys the topological phase. We study coherent transport through interferometers with a Majorana wire embedded into one arm. By combining analytical and numerical considerations, we explain the occurrence of an amplitude maximum as a function of the Zeeman field at the onset of the topological phase – a signature unique to MZMs – which has recently been measured experimentally [Whiticar et al., Nature Communications, 11(1):3212, 2020]. By placing an array of gates in proximity to the nanowire, we made a fruitful connection to the field of Machine Learning by using the CMA-ES algorithm to tune the gate voltages in order to maximize the amplitude of coherent transmission. We find that the algorithm is capable of learning disorder profiles and even to restore Majorana modes that were fully destroyed by strong disorder by optimizing a feasible number of gates. Deep neural networks are another popular machine learning approach which not only has many direct applications to physical systems but which also behaves similarly to physical mesoscopic systems. In order to comprehend the effects of the complex dynamics from the training, we employ Random Matrix Theory (RMT) as a zero-information hypothesis: before training, the weights are randomly initialized and therefore are perfectly described by RMT. After training, we attribute deviations from these predictions to learned information in the weight matrices. Conducting a careful numerical analysis, we verify that the spectra of weight matrices consists of a random bulk and a few important large singular values and corresponding vectors that carry almost all learned information. By further adding label noise to the training data, we find that more singular values in intermediate parts of the spectrum contribute by fitting the randomly labeled images. Based on these observations, we propose a noise filtering algorithm that both removes the singular values storing the noise and reverts the level repulsion of the large singular values due to the random bulk

    Asymptotics, Geometry, and Soft Matter

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    This dissertation is concerned with two problems that lie at the interface of soft-matter physics, geometry, and asymptotic analysis, but otherwise have no bearing on one another. In the first problem, I consider the equilibrium thermal fluctuations of deformable mechanical frameworks. These frameworks have served as highly idealized representations of mechanical structures that underlie a plethora of soft, few-body systems at the submicron scale such as colloidal clusters and DNA origami. When the holonomic constraints in a framework cease to be linearly independent, singularities can appear in its configuration space, where it becomes energetically softer. Consequently, the framework\u27s free-energy landscape becomes dominated by the neighborhoods of points corresponding to these singularities. In the second problem, I study the localization of elastic waves in thin elastic structures with spatially varying curvature profiles, using a curved rod and a uniaxially-curved shell as concrete examples. Waves propagating on such structures have multiple components owing to the curvature-mediated coupling of the tangential and normal components of the displacement field. Here, using the semiclassical approximation, I show that these waves form localized, bound states around points where the absolute curvature of the structure has a minimum. Both these problems exemplify the subtle interplay between the mechanical properties of soft materials and their geometry, which further sets the stage for many interesting consequences

    Electron Momentum Distributions from Strong-Field-Induced Ionization of Atoms and Molecules

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    High-intensity femtosecond laser pulses in the visible or infrared range can induce electron emission. This single-ionization process may be interpreted as a sequence of (nonadiabatic) tunnel ionization and subsequent acceleration of the electron by the external oscillating field in the presence of the electrostatic force between electron and parent ion. Based on the analysis of photoelectron momentum distributions from the numerical solution of the time-dependent Schrödinger equation, this thesis theoretically studies a variety of phenomena taking place in atoms as well as in molecules in strong fields. The underlying physical mechanisms are revealed by simplified models which take the nonperturbative character of the ionization process into account. The simulation results for several settings are directly compared to measurements, offering the possibility to benchmark state-of-the-art theory and experiment against each other. One example of this is an investigation of the nonadiabatic strong-field ionization of atomic hydrogen in an attoclock setting. More generally, the deflection of the photoelectrons is analyzed in different attoclock configurations to explore the initial conditions of electrons at the tunnel exit—the position where the electron appears after tunneling. When a molecule is ionized, its orbital structure influences the liberated electron wave packet. The orbital imprint on the momentum-space phase of the wave packet, which encodes spatial information, is demonstrated and an interferometric approach to access these phases is evaluated. A characterization of the freed wave packet is crucial as it influences subsequent processes. Such secondary processes are induced when the electron is driven back to the parent ion and scatters off. Similar to focusing of light by a lens, the Coulomb attraction forces scattered electron wave packets through focal points, causing a shift of their phase. Due to the interference of outgoing waves, these phases become visible in electron momentum distributions. For a faithful description, these focal-point effects must be included in a prefactor of the exponentiated action in semiclassical models. Furthermore, the control of electron scattering dynamics is demonstrated for low-energy electrons close to the continuum threshold by means of near-single-cycle terahertz pulses. The temporally-localized preparation of the electron wave packet by a femtosecond laser pulse at a well-defined time within the terahertz field enables a switching between different regimes of dynamics, ranging from recollision-free acceleration to extensive scattering phenomena. In contrast to most studies in the electric dipole approximation that consider only the temporal evolution of the external electric field, various beyond-dipole effects in strong-field ionization are explored in the present work. The microscopic mechanisms of nondipole modifications are thoroughly analyzed. There, the effects of the spatially-varying electric field and of the magnetic field as well as their fingerprints on the geometry of the momentum distributions are identified. Furthermore, the subcycle time resolution of the light-induced momentum transfer in an attoclock-like setup is explored theoretically. Electron recollisions entirely change the observed nondipole effects and render the observations sensitive to the electronic target structure. The high-order above-threshold ionization caused by large-angle scattering is investigated both for exemplary atoms and for diatomic molecules through examination of nondipole shifts of the lateral momentum distribution. The phases of the electron wave packets are also altered by beyond-dipole effects. It is shown that this results in a displacement of ring-link structures known as above-threshold ionization rings that are caused by intercycle interference. In addition, the holographic structures arising from the subcycle interference of scattered and nonscattered electrons are modified

    Symbol Detection in 5G and Beyond Networks

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    Beyond 5G networks are expected to provide excellent quality of service in terms of delay and reliability for users, where they could travel with high mobility (e.g., 500 km/h) and achieve better spectral efficiency. To support these demands, advanced wireless architectures have been proposed, i.e., orthogonal time frequency space (OTFS) modulation and multiple-input multiple-output (MIMO), which are used to handle high mobility communications and increase the network’s spectral efficiency, respectively. Symbol detection in these advanced wireless architectures is essential to satisfy reliability requirements. On the one hand, the optimal maximum likelihood symbol detector is prohibitively complex as its complexity is non-deterministic polynomial-time (NP)-hard. On the other hand, conventional low-complexity symbol detectors pose a significant performance loss compared to the optimal detector. Thus they cannot be used to satisfy high-reliability requirements. One solution to this problem is to develop a low-complexity algorithm that can achieve near-optimal performance in a particular scenario (e.g., M-MIMO). Nevertheless, there are some cases where we cannot design low-complexity algorithms. To alleviate this issue, deep learning networks can be integrated into an existing algorithm and trained using a dataset obtained by simulating a corresponding scenario. In this thesis, we design symbol detectors for advanced wireless architectures (i.e., MIMO and OTFS) to support an excellent quality of service in terms of delay and reliability and better spectral efficiency beyond 5G networks

    Hands-on Science. Celebrating Science and Science Education

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    The book herein aims to contribute to the improvement of Science Education in our schools and to an effective implementation of a sound widespread scientific literacy at all levels of society

    50 Years of quantum chromodynamics – Introduction and Review

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