Optimizing Robust Quantum Gates in Open Quantum Systems

We are currently at the cusp of a revolution in quantum technology that relies not just on the passive use of quantum effects, but on their active control. At the forefront of this revolution is the implementation of a quantum computer. Encoding information in quantum states as “qubits” allows to use entanglement and quantum superposition to perform calculations that are infeasible on classical computers. The fundamental challenge in the realization of quantum computers is to avoid decoherence – the loss of quantum properties – due to unwanted interaction with the environment. This thesis addresses the problem of implementing entangling two-qubit quantum gates that are robust with respect to both decoherence and classical noise. It covers three aspects: the use of efficient numerical tools for the simulation and optimal control of open and closed quantum systems, the role of advanced optimization functionals in facilitating robustness, and the application of these techniques to two of the leading implementations of quantum computation, trapped atoms and superconducting circuits. After a review of the theoretical and numerical foundations, the central part of the thesis starts with the idea of using ensemble optimization to achieve robustness with respect to both classical fluctuations in the system parameters, and decoherence. For the example of a controlled phasegate implemented with trapped Rydberg atoms, this approach is demonstrated to yield a gate that is at least one order of magnitude more robust than the best known analytic scheme. Moreover this robustness is maintained even for gate durations significantly shorter than those obtained in the analytic scheme. Superconducting circuits are a particularly promising architecture for the implementation of a quantum computer. Their flexibility is demonstrated by performing optimizations for both diagonal and non-diagonal quantum gates. In order to achieve robustness with respect to decoherence, it is essential to implement quantum gates in the shortest possible amount of time. This may be facilitated by using an optimization functional that targets an arbitrary perfect entangler, based on a geometric theory of two-qubit gates. For the example of superconducting qubits, it is shown that this approach leads to significantly shorter gate durations, higher fidelities, and faster convergence than the optimization towards specific two-qubit gates. Performing optimization in Liouville space in order to properly take into account decoherence poses significant numerical challenges, as the dimension scales quadratically compared to Hilbert space. However, it can be shown that for a unitary target, the optimization only requires propagation of at most three states, instead of a full basis of Liouville space. Both for the example of trapped Rydberg atoms, and for superconducting qubits, the successful optimization of quantum gates is demonstrated, at a significantly reduced numerical cost than was previously thought possible. Together, the results of this thesis point towards a comprehensive framework for the optimization of robust quantum gates, paving the way for the future realization of quantum computers.

On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Theory for the Photoacoustic Response to X-Ray Absorption

A microscopic theory is presented for the photoacoustic effect induced in solids by x-ray absorption. The photoacoustic effect results from the thermalization of the excited Auger electrons and photoelectrons. We explain the dependence of the photoacoustic signal S on photon energy and the proportionality to the x-ray absorption coefficient in agreement with recent experiments on Cu. Results are presented for the dependence of S on photon energy, sample thickness, and the electronic structure of the absorbing solid.

Theory for Optical Absorption in Small Clusters: Dependence on Atomic Structure and Cluster Size

We present a theory which permits for the first time a detailed analysis of the dependence of the absorption spectrum on atomic structure and cluster size. Thus, we determine the development of the collective excitations in small clusters and show that their broadening depends sensitively on the tomic structure, in particular at the surface. Results for Hg_n^+ clusters show that the plasmon energy is close to its jellium value in the case of spherical-like structures, but is in general between w_p/ \wurzel{3} and w_p/ \wurzel{2} for compact clusters. A particular success of our theory is the identification of the excitations contributing to the absorption peaks.

Theory for the optical properties of small Hg_n clusters

The static and dynamical polarizabilities of the Hg-dimer are calculated by using a Hubbard Hamiltonian to describe the electronic structure. The Hamiltonian is diagonalized exactly within a subspace of second-quantized electronic states from which only multiply ionized atomic configurations have been excluded. With this approximation we can describe the most important electronic transitions including the effect of charge fluctuations. We analyze the polarizability as a function of the intraatomic Coulomb interaction which represents the repulsion between electrons. We obtain that this interaction results in strong electronic correlations in the excited states and increases the first excitation energy of the dimer by 0.8 eV in comparison to a calculation which neglects correlations, resulting in a better agreement with the experiment.

Theory for the change of bond character in divalent-metal clusters

To determine the size dependence of the bonding in divalent-metal clusters we use a many-electron Hamiltonian describing the interplay between van der Waals (vdW) and covalent interactions. Using a saddle-point slave-boson method and taking into account the size-dependent screening of charge fluctuations, we obtain for Hg_n a sharp transition from vdW to covalent bonding for increasing n. We show also, by solving the model Hamiltonian exactly, that for divalent metals vdW and covalent bonding coexist already in the dimers.

Photoionization of Kr near 4s threshold: II. Intermediate-coupling theory

The Kr 4s-electron photoionization cross section as a function of the exciting-photon energy in the range between 30 eV and 90 eV was calculated using the configuration interaction (CI) technique in intermediate coupling. In the calculations the 4p spin-orbital interaction and corrections due to higher orders of perturbation theory (the so-called Coulomb interaction correlational decrease) were considered. Energies of Kr II states were calculated and agree with spectroscopic data within less than 10 meV. For some of the Kr II states new assignments were suggested on the basis of the largest component among the calculated CI wavefunctions.

Photoionization of Kr near the 4s threshold: I. Experiment and LS coupling theory

Absolute cross sections for the transitions of the Kr atom into the 4s^1 and 4p^4nl states of the Kr^+ ion were measured in the 4s-electron threshold region by photon-induced fluorescence spectroscopy (PIFS). The cross sections for the transitions of the Kr atom into the 4s^1 and 4p^4nl states were also calculated, as well as the 4p^4nln'l' doubly excited states, in the frame of LS-coupling many-body technique. The cross sections of the doubly-excited atomic states were used to illustrate the pronounced contributions of the latter to the photoionization process, evident from the measurements. The comparison of theory and experiment led to conclusions about the origin of the main features observed in the experiment.

Application of inclusive probability theory to heavy ion-atom collisions

Using the independent particle model as our basis we present a scheme to reduce the complexity and computational effort to calculate inclusive probabilities in many-electron collision system. As an example we present an application to K - K charge transfer in collisions of 2.6 MeV Ne{^9+} on Ne. We are able to give impact parameter-dependent probabilities for many-particle states which could lead to KLL-Auger electrons after collision and we compare with experimental values.

Further investigations on doubly excited four-electron ions: beam-foil experiment and MCDF theory

In continuation of our previous work on the quintet transitions 1s2s2p^2 ^5 P-1s2s2p3d ^5 P^0, ^5 D^0, results on other n = 2 - n' = 3 quintet transitions for elements N, 0 and F are presented. Assignments have been established by comparison with Multi-Configuration Dirac-Fock calculations. High spectral resolution on beam-foil spectroscopy was essential for the identification of most of the lines. For some of the quintet lines decay curves were measured, and the lifetimes extracted were found to be in reasonable agreement with MCDF calculations.

Algorithmen für q-holonome Funktionen und q-hypergeometrische Reihen

Die q-Analysis ist eine spezielle Diskretisierung der Analysis auf einem Gitter, welches eine geometrische Folge darstellt, und findet insbesondere in der Quantenphysik eine breite Anwendung, ist aber auch in der Theorie der q-orthogonalen Polynome und speziellen Funktionen von großer Bedeutung. Die betrachteten mathematischen Objekte aus der q-Welt weisen meist eine recht komplizierte Struktur auf und es liegt daher nahe, sie mit Computeralgebrasystemen zu behandeln. In der vorliegenden Dissertation werden Algorithmen für q-holonome Funktionen und q-hypergeometrische Reihen vorgestellt. Alle Algorithmen sind in dem Maple-Package qFPS, welches integraler Bestandteil der Arbeit ist, implementiert. Nachdem in den ersten beiden Kapiteln Grundlagen geschaffen werden, werden im dritten Kapitel Algorithmen präsentiert, mit denen man zu einer q-holonomen Funktion q-holonome Rekursionsgleichungen durch Kenntnis derer q-Shifts aufstellen kann. Operationen mit q-holonomen Rekursionen werden ebenfalls behandelt. Im vierten Kapitel werden effiziente Methoden zur Bestimmung polynomialer, rationaler und q-hypergeometrischer Lösungen von q-holonomen Rekursionen beschrieben. Das fünfte Kapitel beschäftigt sich mit q-hypergeometrischen Potenzreihen bzgl. spezieller Polynombasen. Wir formulieren einen neuen Algorithmus, der zu einer q-holonomen Rekursionsgleichung einer q-hypergeometrischen Reihe mit nichttrivialem Entwicklungspunkt die entsprechende q-holonome Rekursionsgleichung für die Koeffizienten ermittelt. Ferner können wir einen neuen Algorithmus angeben, der umgekehrt zu einer q-holonomen Rekursionsgleichung für die Koeffizienten eine q-holonome Rekursionsgleichung der Reihe bestimmt und der nützlich ist, um q-holonome Rekursionen für bestimmte verallgemeinerte q-hypergeometrische Funktionen aufzustellen. Mit Formulierung des q-Taylorsatzes haben wir schließlich alle Zutaten zusammen, um das Hauptergebnis dieser Arbeit, das q-Analogon des FPS-Algorithmus zu erhalten. Wolfram Koepfs FPS-Algorithmus aus dem Jahre 1992 bestimmt zu einer gegebenen holonomen Funktion die entsprechende hypergeometrische Reihe. Wir erweitern den Algorithmus dahingehend, dass sogar Linearkombinationen q-hypergeometrischer Potenzreihen bestimmt werden können. ________________________________________________________________________________________________________________

Theory of laser-induced ultrafast structural changes in solids

The present thesis is a contribution to the study of laser-solid interaction. Despite the numerous applications resulting from the recent use of laser technology, there is still a lack of satisfactory answers to theoretical questions regarding the mechanism leading to the structural changes induced by femtosecond lasers in materials. We provide here theoretical approaches for the description of the structural response of different solids (cerium, samarium sulfide, bismuth and germanium) to femtosecond laser excitation. Particular interest is given to the description of the effects of the laser pulse on the electronic systems and changes of the potential energy surface for the ions. Although the general approach of laser-excited solids remains the same, the potential energy surface which drives the structural changes is calculated with different theoretical models for each material. This is due to the difference of the electronic properties of the studied systems. We use the Falicov model combined with an hydrodynamic method to study photoinduced phase changes in cerium. The local density approximation (LDA) together with the Hubbard-type Hamiltonian (LDA+U) in the framework of density functional theory (DFT) is used to describe the structural properties of samarium sulfide. We parametrize the time-dependent potential energy surface (calculated using DFT+ LDA) of bismuth on which we perform quantum dynamical simulations to study the experimentally observed amplitude collapse and revival of coherent $A_{1g}$ phonons. On the basis of a time-dependent potential energy surface calculated from a non-orthogonal tight binding Hamiltonian, we perform molecular dynamics simulation to analyze the time evolution (coherent phonons, ultrafast nonthermal melting) of germanium under laser excitation. The thermodynamic equilibrium properties of germanium are also reported. With the obtained results we are able to give many clarifications and interpretations of experimental results and also make predictions.