Theory of Curie temperature enhancement in electron-doped EuO

We present a comparative, theoretical study of the doping dependence of the critical temperature $T_C$ of the ferromagnetic insulator-metal transition in Gd-doped and O-deficient EuO, respectively. The strong $T_C$ enhancement in Eu$_{1-x}$Gd$_x$O is due to Kondo-like spin fluctuations on the Gd sites, which are absent in EuO$_{1-x}$. Moreover, we find that the $T_C$ saturation in Eu$_{1-x}$Gd$_x$O for large $x$ is due to a reduced activation of dopant electrons into the conduction band, in agreement with experiments, rather than antiferromagnetic long-range contributions of the RKKY interaction. The results shed light on possibilities for further increasing $T_C$.Comment: Published version; 3 references adde

Ferromagnetic Semiconductor-Metal Transition in Heterostructures of Electron Doped Europium Monoxide

In the present work, we develop and solve a self-consistent theory for the description of the simultaneous ferromagnetic semiconductor-metal transition in electron doped Europium monoxide. We investigate two different types of electron doping, Gadolinium impurities and Oxygen vacancies. Besides the conduction band occupation, we can identify low lying spin fluctuations on magnetic impurities as the driving force behind the doping induced enhancement of the Curie temperature. Moreover, we predict the signatures of these magnetic impurities in the spectra of scanning tunneling microscope experiments. By extending the theory to allow for inhomogeneities in one spatial direction, we are able to investigate thin films and heterostructures of Gadolinium doped Europium monoxide. Here, we are able to reproduce the experimentally observed decrease of the Curie temperature with the film thickness. This behavior is attributed to missing coupling partners of the localized 4f moments as well as to an electron depletion at the surface which leads to a reduction of the number of itinerant electrons. By investigating the influence of a metallic substrate onto the phase transition in Gadolinium doped Europium monoxide, we find that the Curie temperature can be increased up to 20%. However, as we show, the underlying mechanism of metal-interface induced charge carrier accumulation is inextricably connected to a suppression of the semiconductor-metal transition

TiGL - An Open Source Computational Geometry Library for Parametric Aircraft Design

This paper introduces the software TiGL: TiGL is an open source high-fidelity geometry modeler that is used in the conceptual and preliminary aircraft and helicopter design phase. It creates full three-dimensional models of aircraft from their parametric CPACS description. Due to its parametric nature, it is typically used for aircraft design analysis and optimization. First, we present the use-case and architecture of TiGL. Then, we discuss it's geometry module, which is used to generate the B-spline based surfaces of the aircraft. The backbone of TiGL is its surface generator for curve network interpolation, based on Gordon surfaces. One major part of this paper explains the mathematical foundation of Gordon surfaces on B-splines and how we achieve the required curve network compatibility. Finally, TiGL's aircraft component module is introduced, which is used to create the external and internal parts of aircraft, such as wings, flaps, fuselages, engines or structural elements

Diagrammatic Analysis for Parameterized Quantum Circuits

Diagrammatic representations of quantum algorithms and circuits offer novel approaches to their design and analysis. In this work, we describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits, in particular for computing observable expectation values as functions of or for fixed parameters, which are important algorithmic quantities in a variety of applications ranging from combinatorial optimization to quantum chemistry. We provide several new ZX-diagram rewrite rules and generalizations for this setting. In particular, we give formal rules for dealing with linear combinations of ZX-diagrams, where the relative complex-valued scale factors of each diagram must be kept track of, in contrast to most previously studied single-diagram realizations where these coefficients can be effectively ignored. This allows us to directly import a number useful relations from the operator analysis to ZX-calculus setting, including causal cone and quantum gate commutation rules. We demonstrate that the diagrammatic approach offers useful insights into algorithm structure and performance by considering several ans\"atze from the literature including realizations of hardware-efficient ans\"atze and QAOA. We find that by employing a diagrammatic representation, calculations across different ans\"atze can become more intuitive and potentially easier approach systematically than by alternative means. Finally, we outline how diagrammatic approaches may aid in the design and study of new and more effective quantum circuit ans\"atze

Embedding of Complete Graphs in Broken Chimera Graphs

In order to solve real world combinatorial optimization problems with a D-Wave quantum annealer it is necessary to embed the problem at hand into the D-Wave hardware graph, namely Chimera or Pegasus. Most hard real world problems exhibit a strong connectivity. For the worst case scenario of a complete graph, there exists an efficient solution for the embedding into the ideal Chimera graph. However, since real machines almost always have broken qubits it is necessary to find an embedding into the broken hardware graph. We present a new approach to the problem of embedding complete graphs into broken Chimera graphs. This problem can be formulated as an optimization problem, more precisely as a matching problem with additional linear constraints. Although being NP-hard in general it is fixed parameter tractable in the number of inaccessible vertices in the Chimera graph. We tested our exact approach on various instances of broken hardware graphs, both related to real hardware as well as randomly generated. For fixed runtime, we were able to embed larger complete graphs compared to previous, heuristic approaches. As an extension, we developed a fast heuristic algorithm which enables us to solve even larger instances. We compared the performance of our heuristic and exact approaches.Comment: 26 pages, 9 figures, 2 table

A Measurement-based Algorithm for Graph Colouring

We present a novel algorithmic approach to find a proper vertex colouring of a graph with d colours, if it exists. We associate a d-dimensional quantum system with each vertex and the initial state is a mixture of all possible colourings, from which we obtain a random proper colouring of the graph by measurements. The non-deterministic nature of the quantum measurement is tackled by a reset operation, which can revert the effect of unwanted projections. As in the classical case, we find that the runtime scales exponentially with the number of vertices. However, we provide numerical evidence that the average runtime for random graphs scales polynomially in the number of edges

Global parameterization of trimmed NURBS based CAD geometries for mesh deformation

In this talk we want to present a novel global parameterization scheme for points on a CAD geometry. This algorithm can then be used to compute mesh deformations for changes of the underlying geometry. The creation of structured meshes is a time-consuming trial and error process, which is not suitable for e.g. automatic optimization. Particularly gradient based optimization often performs only small changes of the design variables, which should result in only slightly different meshes. Therefore, methods are required that deform an initial mesh based on the change of the initial geometry. Here, we present a projection method that computes a bijective mapping between a point in space and its global parameterization with respect to the trimmed NURBS based CAD geometry. After a geometry change, the parameterized points can be back-projected into 3D space which eventually yields the deformed mesh. Providing support for trimmed NURBS geometries is particularly challenging, as their surface parameters u;v mighty be valid only in a non-rectangular trimming region. This region however varies on geometry changes, which would lead to a loss of mesh points, if this is not properly handled. To overcome this issue, we reparametrize the trimming region such that the domain of some new parameters u0;v0 is rectangular. Our projection algorithm is separated into three different problems: first – finding the face a mesh point belongs; second – reparametrize the face to get a bijective mapping; third – project the point onto the reparametrized surface. The first and third problem are comparable simple and can be performed using standard CAD algorithms. For the reparameterization problem, we provide a method that converts the 2d trimming domain of the NURBS into a series of two-dimensional untrimmed patches. This is done by first subdividing the original NURBS face into multiple faces. Then, we identify or create four boundary curves for each of these sub-faces. The four boundary curves are finally used to create a reparameterization patch e.g. using the Coons method. The projection of a point leads only to a unique solution, if the reparameterization patch is invertible. We check invertibility of the patch, by separating the patch into rational Bezier spline surfaces and check that their Jacobian determinant is larger than 0. This strategy allows a large range of different face types, including faces with holes and faces with more or less than four boundary curves. The back-projection method is analogous, and also requires the creation of the reparameterization surfaces. This algorithm is implemented in a C++ based library, which utilizes the CAD functionality of the Open CASCADE framework. The library is designed to be used on computing clusters by providing functions for the serialization and deserialization of the geometry. This enables the parallelized projection and back-projection of large computation meshes with millions of points to reduce the computational runtime. Since this algorithm only works for small geometry changes - i.e. for geometries with the same topology - we also added functions to compare and store the topology of two CAD objects. Our method is currently used within a DLR-internal project to enable a large scale gradient based optimization of an aircraft. In our work flow, the software TiGL converts the parametric aircraft description into a CAD representation. The initial structured mesh is created with a commercial mesh generator. In the subsequent iterations of the optimization, the mesh is deformed using the presented method. Details to the robustness of this algorithm and its computational performance will be presented at the talk

Towards Finding an Optimal Flight Gate Assignment on a Digital Quantum Computer

We investigate the performance of the variational quantum eigensolver (VQE) for the optimal flight gate assignment problem. This problem is a combinatorial optimization problem that aims at finding an optimal assignment of flights to the gates of an airport, in order to minimize the passenger travel time. To study the problem, we adopt a qubit-efficient binary encoding with a cyclic mapping, which is suitable for a digital quantum computer. Using this encoding in conjunction with the Conditional Value at Risk (CVaR) as an aggregation function, we systematically explore the performance of the approach by classically simulating the CVaR-VQE. Our results indicate that the method allows for finding a good solution with high probability, and the method significantly outperforms the naive VQE approach. We examine the role of entanglement for the performance, and find that ans\"atze with entangling gates allow for better results than pure product states. Studying the problem for various sizes, our numerical data show that the scaling of the number of cost function calls for obtaining a good solution is not exponential for the regimes we investigate in this work

Quantum Annealing Applied to De-Conflicting Optimal Trajectories for Air Traffic Management

We present the mapping of a class of simplified air traffic management (ATM) problems (strategic conflict resolution) to quadratic unconstrained boolean optimization (QUBO) problems. The mapping is performed through an original representation of the conflict-resolution problem in terms of a conflict graph, where nodes of the graph represent flights and edges represent a potential conflict between flights. The representation allows a natural decomposition of a real world instance related to wind-optimal trajectories over the Atlantic ocean into smaller subproblems, that can be discretized and are amenable to be programmed in quantum annealers. In the study, we tested the new programming techniques and we benchmark the hardness of the instances using both classical solvers and the D-Wave 2X and D-Wave 2000Q quantum chip. The preliminary results show that for reasonable modeling choices the most challenging subproblems which are programmable in the current devices are solved to optimality with 99% of probability within a second of annealing time.Comment: Paper accepted for publication on: IEEE Transactions on Intelligent Transportation System