Gauss ve kuaterniyon tam sayılarından kuantum kod elde etme

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Bu tez dört bölümden oluşmaktadır. Birinci bölümde cebir ve kodlama teorisinin temel tanım ve teoremleri, ikinci bölümde kısa bir literatür taraması, kuantum hesaplama ve kuantum bilgi hakkında temel tanım ve teoremler verilmektedir. Yine bu bölümde ikili olan ve ikili olmayan hata düzeltebilen kuantum kodlar açıklanmaktadır. Üçüncü bölümde Mannheim metriğine göre Gauss tamsayıları üzerindeki klasik kodlar yardımı ile Calderbank-Shor-Steane (kısaca CSS) kodları oluşturulmaktadır. Ayrıca bu bölümde Gauss tam sayıları için iyi hata bazları da tanımlanmaktadır. Dördüncü bölümde Lipschitz sayıları üzerindeki klasik kodlar yardımı ile CSS kodların nasıl inşa edileceği açıklanmakta ve bu sayılar için iyi hata bazları tanımlanmaktadır.This thesis consist of four chapters. In the first chapter, some notations and some basic definitions and theorems of abstract algebra are given. In the second chapter, the fundamental elements needed to perform quantum computation and quantum information are described and many elementary operations which may be used to develop more sophisticated applications of quantum computation and quantum information are presented. Moreover, binary and nonbinary quantum error-correcting codes are explained. In the third chapter, the CSS codes are constructed from codes over Gaussian integers with respect to the Mannheim metric. Moreover, the set of the nice error bases over Gaussian integers is introduced. In the fourth chapter, the CSS codes are constructed via codes over quaternion integers with respect to the Lipschitz metric. Moreover, the set of the nice error bases over quaternion integers is introduced

Quantal-Classical Duality and the Semiclassical Trace Formula

We consider Hamiltonian systems which can be described both classically and quantum mechanically. Trace formulas establish links between the energy spectra of the quantum description and the spectrum of actions of periodic orbits in the classical description. This duality is investigated in the present paper. The duality holds for chaotic as well as for integrable systems. For billiards the quantal spectrum (eigenvalues of the Helmholtz equation) and the classical spectrum (lengths of periodic orbits) are two manifestations of the billiard's boundary. The trace formula expresses this link as a Fourier transform relation between the corresponding spectral densities. It follows that the two-point statistics are also simply related. The universal correlations of the quantal spectrum are well known, consequently one can deduce the classical universal correlations. An explicit expression for the scale of the classical correlations is derived and interpreted. This allows a further extension of the formalism to the case of complex billiard systems, and in particular to the most interesting case of diffusive system. The concept of classical correlations allows a better understanding of the so-called diagonal approximation and its breakdown. It also paves the way towards a semiclassical theory that is capable of global description of spectral statistics beyond the breaktime. An illustrative application is the derivation of the disorder-limited breaktime in case of a disordered chain, thus obtaining a semiclassical theory for localization. A numerical study of classical correlations in the case of the 3D Sinai billiard is presented. We gain a direct understanding of specific statistical properties of the classical spectrum, as well as their semiclassical manifestation in the quantal spectrum.Comment: 42 pages, 17 figure

Multimedia Applications of the Wavelet Transform

This dissertation investigates novel applications of the wavelet transform in the analysis and compression of audio, still images, and video. Most recently, some surveys have been published on the restoration of noisy audio signals. Based on these, we have developed a wavelet-based denoising program for audio signals that allows flexible parameter settings. The multiscale property of the wavelet transform can successfully be exploited for the detection of semantic structures in images: A comparison of the coefficients allows the extraction of a predominant structure. This idea forms the basis of our semiautomatic edge detection algorithm. Empirical evaluations and the resulting recommendations follow. In the context of the teleteaching project Virtual University of the Upper Rhine Valley (VIROR), many lectures were transmitted between remote locations. We thus encountered the problem of scalability of a video stream for different access bandwidths in the Internet. A substantial contribution of this dissertation is the introduction of the wavelet transform into hierarchical video coding and the recommendation of parameter settings based on empirical surveys. Furthermore, a prototype implementation proves the principal feasibility of a wavelet-based, nearly arbitrarily scalable application. Mathematical transformations constitute a commonly underestimated problem for students in their first semesters of study. Motivated by the VIROR project, we spent a considerable amount of time and effort on the exploration of approaches to enhance mathematical topics with multimedia; both the technical design and the didactic integration into the curriculum are discussed. In a large field trial on "traditional teaching versus multimedia-enhanced teaching", the objective knowledge gained by the students was measured. This allows us to objectively rate positive the efficiency of our teaching modules

QUANTUM CODES FROM CODES OVER GAUSSIAN INTEGERS WITH RESPECT TO THE MANNHEIM METRIC

In this paper, some nonbinary quantum codes using classical codes over Gaussian integers are obtained. Also, some of our quantum codes are better than or comparable with those known before, (for instance [[8, 2, 5]](4+i))

Building models of the Universe with hydrodynamic simulations

Hydrodynamic simulations have become irreplaceable in modern cosmology for exploring complex systems and making predictions to steer future observations. In Chapter 1, we begin with a philosophical discussion on the role of simulations in science. We argue that simulations can bridge the gap between empirical and fundamental knowledge. The validation of simulations stresses the importance of achieving a balance between trustworthiness and scepticism. Next, Chapter 2 introduces the formation of structures and comparisons between synthetic and observational data. Chapter 3 describes the production pipeline of zoom-in simulations used to model individual objects and novel methods to mitigate known shortcomings. Then, we assessed the weak scaling of the SWIFT code and found it to be one of the hydrodynamic codes with the highest parallel efficiency. In Chapter 4, we study the rotational kinetic Sunyaev-Zeldovich (rkSZ) effect for high-mass galaxy clusters from the MACSIS simulations. We find a maximum signal greater than 100 $\mu$K, 30 times stronger than early predictions from self-similar models, opening prospects for future detection. In Chapter 5, we address a tension between the distribution of entropy measured from observations and predicted by simulations of groups and clusters of galaxies. We find that most recent hydrodynamic simulations systematically over-predict the entropy profiles by up to one order of magnitude, leading to profiles that are shallower and higher than the power-law-like entropy profiles that have been observed. We discuss the dependence on different hydrodynamic and sub-grid parameters using variations of the EAGLE model. Chapter 6 explores the evolution of the profiles as a function of cosmic time. We report power-law-like entropy profiles at high redshift for both objects. However, at late times, an entropy plateau develops and alters the shape of the profile.Comment: PhD thesis, University of Manchester. Also available at https://research.manchester.ac.uk/en/studentTheses/building-models-of-the-universe-with-hydrodynamic-simulation