Menke points on the real line and their connection to classical orthogonal polynomials

AbstractWe investigate the properties of extremal point systems on the real line consisting of two interlaced sets of points solving a modified minimum energy problem. We show that these extremal points for the intervals [−1,1], [0,∞) and (−∞,∞), which are analogues of Menke points for a closed curve, are related to the zeros and extrema of classical orthogonal polynomials. Use of external fields in the form of suitable weight functions instead of constraints motivates the study of “weighted Menke points” on [0,∞) and (−∞,∞). We also discuss the asymptotic behavior of the Lebesgue constant for the Menke points on [−1,1]

Funktionentheorie

[no abstract available

Wigner quasi-probability distribution for the infinite square well: energy eigenstates and time-dependent wave packets

We calculate the Wigner quasi-probability distribution for position and momentum, P_W^(n)(x,p), for the energy eigenstates of the standard infinite well potential, using both x- and p-space stationary-state solutions, as well as visualizing the results. We then evaluate the time-dependent Wigner distribution, P_W(x,p;t), for Gaussian wave packet solutions of this system, illustrating both the short-term semi-classical time dependence, as well as longer-term revival and fractional revival behavior and the structure during the collapsed state. This tool provides an excellent way of demonstrating the patterns of highly correlated Schrodinger-cat-like `mini-packets' which appear at fractional multiples of the exact revival time.Comment: 45 pages, 16 embedded, low-resolution .eps figures (higher resolution, publication quality figures are available from the authors); submitted to American Journal of Physic

Training variational quantum circuits with CoVaR: covariance root finding with classical shadows

Exploiting near-term quantum computers and achieving practical value is a considerable and exciting challenge. Most prominent candidates as variational algorithms typically aim to find the ground state of a Hamiltonian by minimising a single classical (energy) surface which is sampled from by a quantum computer. Here we introduce a method we call CoVaR, an alternative means to exploit the power of variational circuits: We find eigenstates by finding joint roots of a polynomially growing number of properties of the quantum state as covariance functions between the Hamiltonian and an operator pool of our choice. The most remarkable feature of our CoVaR approach is that it allows us to fully exploit the extremely powerful classical shadow techniques, i.e., we simultaneously estimate a very large number $>10^4-10^7$ of covariances. We randomly select covariances and estimate analytical derivatives at each iteration applying a stochastic Levenberg-Marquardt step via a large but tractable linear system of equations that we solve with a classical computer. We prove that the cost in quantum resources per iteration is comparable to a standard gradient estimation, however, we observe in numerical simulations a very significant improvement by many orders of magnitude in convergence speed. CoVaR is directly analogous to stochastic gradient-based optimisations of paramount importance to classical machine learning while we also offload significant but tractable work onto the classical processor.Comment: 25 pages, 9 figure

Bounding the joint numerical range of Pauli strings by graph parameters

The interplay between the quantum state space and a specific set of measurements can be effectively captured by examining the set of jointly attainable expectation values. This set is commonly referred to as the (convex) joint numerical range. In this work, we explore geometric properties of this construct for measurements represented by tensor products of Pauli observables, also known as Pauli strings. The structure of pairwise commutation and anticommutation relations among a set of Pauli strings determines a graph $G$, sometimes also called the frustration graph. We investigate the connection between the parameters of this graph and the structure of minimal ellipsoids encompassing the joint numerical range. Such an outer approximation can be very practical since ellipsoids can be handled analytically even in high dimensions. We find counterexamples to a conjecture from [C. de Gois, K. Hansenne and O. G\"uhne, arXiv:2207.02197], and answer an open question in [M. B. Hastings and R. O'Donnell, Proc. STOC 2022, pp. 776-789], which implies a new graph parameter that we call $\beta(G)$. Besides, we develop this approach in different directions, such as comparison with graph-theoretic approaches in other fields, applications in quantum information theory, numerical methods, properties of the new graph parameter, etc. Our approach suggests many open questions that we discuss briefly at the end.Comment: 14+3 pages, 5 figure

Analysis of freeform optical systems based on the decomposition of the total wave aberration into Zernike surface contributions

The increasing use of freeform optical surfaces raises the demand for optical design tools developed for generalized systems. In the design process, surface-by-surface aberration contributions are of special interest. The expansion of the wave aberration function into the field- and pupil-dependent coefficients is an analytical method used for that purpose. An alternative numerical approach utilizing data from the trace of multiple ray sets is proposed. The optical system is divided into segments of the optical path measured along the chief ray. Each segment covers one surface and the distance to the subsequent surface. Surface contributions represent the change of the wavefront that occurs due to propagation through individual segments. Further, the surface contributions are divided with respect to their phenomenological origin into intrinsic induced and transfer components. Each component is determined from a separate set of rays. The proposed method does not place any constraints on the system geometry or the aperture shape. However, in this thesis only plane symmetric systems with near-circular apertures are studied. This enabled characterization of the obtained aberration components with Zernike fringe polynomials. The application of the proposed method in the design process of the freeform systems is demonstrated. The analysis of Zernike surface contributions provides valuable insights for selecting the starting system with the best potential for correcting aberrations with freeform surfaces. Further, it helps in determining the effective location of a freeform element in a system. Consequently, it is possible to design systems corrected for Zernike aberrations of order higher than the order of coefficients used for freeform sag contributions, described with the same Zernike polynomial set.Die zunehmende Verwendung von optischen Freiformflächen erhöht die Forderung nach optischen Designwerkzeugen die für allgemeine Systeme entwickelt wurden. Im Design-Prozess sind oberflächenbedingte Aberrationsbeiträge von besonderem Interesse. Die Erweiterung der Wellenaberrationsfunktion in feld- und pupillen-abhängige Koeffizienten ist eine zu diesem Zweck verwendete analytische Methode. Ein alternativer numerischer Ansatz, der Daten aus der Verlauf von mehreren Strahlenbündeln verwendet, wird vorgeschlagen. Das optische System ist in Segmente des optischen Weges unterteilt, die entlang des Hauptstrahls gemessen werden. Oberflächenbeiträge repräsentieren die Änderung der Wellenfront, die aufgrund der Propagation durch einzelne Segmente auftritt. Ferner sind die Oberflächenbeiträge hinsichtlich ihres phänomenologischen Ursprungs in intrinsische induzierte und transferierende Komponenten unterteilt. Jede Komponente wird aus einem separaten Strahlenbündel bestimmt. Die vorgeschlagene Methode stellt keine Beschränkungen für die Systemgeometrie oder die Apertur bereit. In dieser Arbeit werden jedoch nur ebene symmetrische Systeme mit nahezu kreisförmigen Aperturen untersucht. Dies ermöglichte eine Charakterisierung der erhaltenen Aberrationskomponenten mit Zernike-Randpolynomen. Die Anwendung der vorgeschlagenen Methode im Designprozess der Freiformsysteme wird demonstriert. Die Analyse der Zernike-Oberflächenbeiträge liefert wertvolle Erkenntnisse für die Auswahl des Startsystems mit dem besten Potenzial zur Korrektur von Aberrationen mit Freiformflächen. Außerdem hilft es beim Bestimmen der effektiven Position eines Freiformelements in einem System. Folglich ist es möglich, Systeme zu entwerfen, die für Zernike-Aberrationen höherer Ordnung korrigiert sind als die für die Freiform-Sag Beiträge verwendeten Koeffizienten, die mit demselben Zernike-Polynomsatz beschrieben sind