Finite Element Method Applied to the One-dimensional Westervelt Equation

In this thesis we researched the applicability, properties and efficiency of the finite element method to solve the one-dimensional Westervelt equation, which describes nonlinear plane wave propagation. The goal was to investigate whether this lesser-known solution method has advantages or disadvantages compared to more commonly used solution techniques. We developed an understanding of nonlinear wave propagation by analyzing the Burgers equation, which we used to benchmark solutions. We used the commercial finite element software package COMSOL to calculate first solutions, where we found that numerical errors occur as the wave propagates through the shock wave formation distance. We examined the effect of several numerical parameters and concluded that reducing the element size decreases the overall error of the solution, both near the shock wave front and elsewhere. This also helps reduce numerical oscillations if present. Increasing the element order also improved the solution. The time stepping algorithm was found to have a strong connection to the element size. The maximum time step depends strongly on the minimum element size. Reducing physical parameters such as the amplitude of the source, or adding damping, were also researched but were shown to have little effect on reducing the numerical error around the shock wave front. The finite element method can solve inhomogeneous domains with relative ease compared to homogeneous domains, which may be an advantage over other methods. We then developed our own Matlab implementation of Galerkin's finite element method for the Westervelt equation to get more insight into the algorithms behind this method and get a better understanding of the effect of numerical parameters. We implemented two different time solvers and we concluded that our specific choice of backward differential formulas was producing more accurate results than more general build-in time solvers that come with COMSOL or Matlab. Furthermore we saw that the accuracy of the solution does not only depend on spatial numerical parameters, but also on the time solving parameters. Different time solving techniques can yield different degrees of accuracy and efficiency, and must therefore be chosen with care. We finally turned to adaptive finite element method techniques in order to improve overall accuracy and efficiency. We have shown that a simple form of adaptiveness can help improve the accuracy of the solution, but its efficiency depends on the implementation and the number of spatial dimensions in which the equation is solved. The finite element method provides different types of adaptiveness, such as local refinement/coarsening, node movement and local change of the order of the basis functions, which may be combined together. We showed the advantages and disadvantages of a node movement implementation based on the MMPDE-6 algorithm. We concluded that more research can be put in incorporating (combined types of) adaptiveness to solve the Westervelt equation.Bachelor Applied Physics and Applied MathematicsNumerical AnalysisElectrical Engineering, Mathematics and Computer Scienc

Efficient unitarity randomized benchmarking of few-qubit Clifford gates

Unitarity randomized benchmarking (URB) is an experimental procedure for estimating the coherence of implemented quantum gates independently of state preparation and measurement errors. These estimates of the coherence are measured by the unitarity. A central problem in this experiment is relating the number of data points to rigorous confidence intervals. In this work we provide a bound on the required number of data points for Clifford URB as a function of confidence and experimental parameters. This bound has favorable scaling in the regime of near-unitary noise and is asymptotically independent of the length of the gate sequences used. We also show that, in contrast to standard randomized benchmarking, a nontrivial number of data points is always required to overcome the randomness introduced by state preparation and measurement errors even in the limit of perfect gates. Our bound is sufficiently sharp to benchmark small-dimensional systems in realistic parameter regimes using a modest number of data points. For example, we show that the unitarity of single-qubit Clifford gates can be rigorously estimated using few hundred data points under the assumption of gate-independent noise. This is a reduction of orders of magnitude compared to previously known bounds.QuTechQID/Wehner GroupQuantum Information and SoftwareQuantum Internet Divisio

Witnessing entanglement in experiments with correlated noise

The purpose of an entanglement witness experiment is to certify the creation of an entangled state from a finite number of trials. The statistical confidence of such an experiment is typically expressed as the number of observed standard deviations of witness violations. This method implicitly assumes that the noise is well-behaved so that the central limit theorem applies. In this work, we propose two methods to analyze witness experiments where the states can be subject to arbitrarily correlated noise. Our first method is a rejection experiment, in which we certify the creation of entanglement by rejecting the hypothesis that the experiment can only produce separable states. We quantify the statistical confidence by a p-value, which can be interpreted as the likelihood that the observed data is consistent with the hypothesis that only separable states can be produced. Hence a small p-value implies large confidence in the witnessed entanglement. The method applies to general witness experiments and can also be used to witness genuine multipartite entanglement. Our second method is an estimation experiment, in which we estimate and construct confidence intervals for the average witness value. This confidence interval is statistically rigorous in the presence of correlated noise. The method applies to general estimation problems, including fidelity estimation. To account for systematic measurement and random setting generation errors, our model takes into account device imperfections and we show how this affects both methods of statistical analysis. Finally, we illustrate the use of our methods with detailed examples based on a simulation of NV centers.QID/Wehner GroupQID/Hanson LabQuTechQN/Hanson LabQuantum Internet DivisionQuantum Information and Softwar

Realization of a multinode quantum network of remote solid-state qubits

The distribution of entangled states across the nodes of a future quantum internet will unlock fundamentally new technologies. Here, we report on the realization of a three-node entanglement-based quantum network. We combine remote quantum nodes based on diamond communication qubits into a scalable phase-stabilized architecture, supplemented with a robust memory qubit and local quantum logic. In addition, we achieve real-time communication and feed-forward gate operations across the network. We demonstrate two quantum network protocols without postselection: the distribution of genuine multipartite entangled states across the three nodes and entanglement swapping through an intermediary node. Our work establishes a key platform for exploring, testing, and developing multinode quantum network protocols and a quantum network control stack.Accepted Author ManuscriptQuTechQID/Hanson LabGeneralBUS/Quantum DelftQID/Wehner GroupQuantum Internet DivisionQuantum Information and SoftwareQN/Hanson La

Distributed entanglement and teleportation on a quantum network

We report on the realization of a multi-node quantum network. Using the network, we have demonstrated three protocols; generation of a entangled state shared by all nodes, entanglement swapping and quantum teleportation between non-neighboring nodes.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.QID/Hanson LabQCD/Vandersypen LabALG/GeneralBUS/Quantum DelftQID/Wehner GroupQN/Borregaard groepQuantum Computer ScienceQN/Hanson La