This thesis considers a real massive free quantum scalar field propagating with arbitrary coupling to $n$-dimensional anti-de Sitter space. Analytical expressions are found for the field modes and Feynman Green function. The condition for the equivalence of rotational vacuum states is also established. The rotational and thermal anti-commutator functions are then derived. A method is developed for computing the Hadamard renormalised vacuum and thermal expectation values of the quadratic field fluctuations and the stress-energy tensor. Results are obtained for $n=2$ to $n=11$, satisfying Wald's axioms and exhibiting the trace anomaly.

Quantum computation is a new computational paradigm which can provide fundamentally faster computation than in the classical regime. This is dependent on finding efficient quantum algorithms for problems of practical interest. One of the most successful tools in developing new quantum algorithms is the quantum walk. In this thesis, we explore two applications of the discrete time quantum walk. In addition, we introduce an experimental scheme for generating cluster states, a universal resource for quantum computation. We give an explicit construction which provides a link between the circuit model of quantum computation, and a graph structure on which the discrete time quantum walk traverses, performing the same computation. We implement a universal gate set, proving the discrete time quantum walk is universal for quantum computation, thus confirming any quantum algorithm can be recast as a quantum walk algorithm. In addition, we study factors affecting the efficiency of the quantum walk search algorithm. Although there is a strong dependence on the spatial dimension of the structure being searched, we find secondary dependencies on other factors including the connectivity and disorder (symmetry). Fairly intuitively, as the connectivity increases, the efficiency of the algorithm increases, as the walker can coalesce on the marked state with higher probability in a quicker time. In addition, we find as disorder in the system increases, the algorithm can maintain the quantum speed up for a certain level of disorder before gradually reverting to the classical run time. Finally, we give an abstract scheme for generating cluster states. We see a linear scaling, better than many schemes, as doubling the size of the generating grid in our scheme produces a cluster state which is double the depth. Our scheme is able to create other interesting topologies of entangled states, including the unit cell for topological error correcting schemes.

In this thesis, we study the non-Abelian anyons that emerge as vortices in Ki-taev's honeycomb spin lattice model. By generalizing the solution of the model, we explicity demonstrate the non-Abelian fusion rules and the braid statistics that charaterize the anyons. This is based on showing the presence of vortices leads to zero modes in the spectrum. These can acquire finite energy due to short range vortex-vortex interactions. By studying the spectral evolution as a function of the vortex seperation, we unambigously identify the zero modes with the fusion degrees of freedom of non-Abelian anyons. To calculate the non-Abelian statistics, we show how the vortex transport can be implemented through local manipulation of the couplings. This enables us to employ the eigenstates of the model to simulate a process where a vortex winds around another. The corresponding evolution of the degenerate ground state space is given by a Berry phase, which under suitable conditions coincides with the statistics. By considering a range of finite size systems, we find a physical regime where the Berry phase gives the predicted statistics of the anoyonic vortices with high fidelity. Finally we study the full-vortex sector of the model and find that is supports a previously undiscovered topological phase. This new phase emerges from the phase with non-Abelian anyons due to their interactions. To study the transitions between the different topological phases appearing in the model, we consider Fermi surface, whose topology captures the long-range properties. Each phase is found to be characterized by a distinct number of Fermi points, with the number depending on distinct global Hamiltonian symmetries. To study how the Fermi surfaces evolve into each other at phase transitions, we consider the low-energy field theory that is described by Dirac fermions. We show that phase transition driving perturbations translate to a coupling to chiral gauge fields, that always lead to Fermi point transport. By studying this transport, we obtain analytically the extended phase space of the model and its properties.

Anyons are quasiparticles that may be realized in two dimensional systems. They come in two types, the simpler Abelian anyons and the more complex non-Abelian anyons. Both of these have been considered as a means for quantum computation, but non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models. First the means to perform quantum computation with Abelian anyon models is considered. These models, like many non-Abelian models, cannot realize universal quantum computation by braiding alone. Non-topological operations must be used in addition, whose complexity depends on the physical means by which the anyons are realized. Here we consider anyons based on spin lattice models, with single spin measurements playing the role of non-topological operations. The computational power achieved by various kinds of measurement is explored and the requirements for universality are determined. The possibility to simulate non-Abelian anyons using Abelian ones is then considered. Finally, a non-Abelian quantum memory is dissected in order to determine the means by which it provides fault-tolerant storage of information. This understanding is then employed to build equivalent quantum memories with Abelian anyon models. The methodology provides with the means to demonstrate that Abelian models have the capability to simulate non-Abelian anyons, and to realize the same computational power and fault-tolerance as non Abelian models. Apart from the intellectual interest in relating topological models with each other, and of understanding the properties of non-Abelian anyons in terms of the simpler Abelian ones, these results can also be applied in the lab. The simpler structure of Abelian anyons means that their physical realization is more straightforward. The demonstration of non-Abelian properties with Abelian models therefore allows features of non-Abelian anyons to be realized with present and near future technology. Based on this possibility, proposals are made here for proof of principle experiments.

No description available.

The motion of fluids, such as air or water, is central to many engineering systems of significant economic and environmental importance. Examples range from air/fuel mixing in combustion engines to turbulence induced noise and fatigue on aircraft. Recent advances in novel sensor/actuator technologies have raised the intriguing prospect of actively sensing and manipulating the motion of the fluid within these systems, making them ripe for feedback control, provided a suitable control model exists. Unfortunately, the models for many of these systems are described by nonlinear, partial differential-algebraic equations for which few, if any, controller synthesis techniques exist. In stark contrast, the majority of established control theory assumes plant models of finite (and typically small) state dimension, expressed as a linear system of ordinary differential equations. Therefore, this thesis explores the problem of how to apply the mainstream tools of control theory to the class of systems described by partial differential-algebraic equations, that are either linear, or for which a linear approximation is valid. The problems of control system design for infinite-dimensional and algebraically constrained systems are treated separately in this thesis. With respect to the former, a new method is presented that enables the computation of a bound on the n-gap between a discretisation of a spatially distributed plant, and the plant itself, by exploiting the convergence rate of the v-gap metric between low-order models of successively finer spatial resolution. This bound informs the design, on loworder models, of H[infinity] loop-shaping controllers that are guaranteed to robustly stabilise the actual plant. An example is presented on a one-dimensional heat equation. Controller/estimator synthesis is then discussed for finite-dimensional systems containing algebraic, as well as differential equations. In the case of fluid flows, algebraic constraints typically arise from incompressibility and the application of boundary conditions. A numerical algorithm is presented, suitable for the semi-discrete linearised Navier-Stokes equations, that decouples the differential and algebraic parts of the system, enabling application of standard control theory without the need for velocity-vorticity type methods. This algorithm is demonstrated firstly on a simple electrical circuit, and secondly on the highly non-trivial problem of flow-field estimation in the transient growth region of a flat-plate boundary layer, using only wall shear measurements. These separate strands are woven together in the penultimate chapter, where a transient energy controller is designed for a channel-flow system, using wall mounted sensors and actuators.

The Very High Temperature Reactor (VHTR) is designed to push the boundaries and capabilities of existing High Temperature Gas Reactor technology to higher levels, challenging the desired inherent and passive safety features. To ascertain the viability of the design requires a detailed understanding of the complex multiphysics within the reactor core and the associated energy removal system. Due to the scale of the calculation computational numerical models are utilised. During a transient the greatest challenge to inherent and passive safety design features will occur. To understand the core dynamics during these off normal conditions requires the use and development of coupled radiation transport thermal hydraulic codes. In this thesis the coupled radiation transport computational multiphase fluid dynamic FETCH model is applied to a generic block type VHTR. The purpose of this research is twofold. First to analyse the suitability of the FETCH model to be capable of capturing the physics inherent within the generic VHTR of interest. Secondly to analyse the suitability of the generic VHTR to operate within certain key safety constraints of interest. A necessary component of this research was to provide evidence to support the reliability and credibility of model solutions through the use of a continuous verification and validation automated framework. Also this PhD thesis includes the development and analysis of a Sub Grid Scale finite element methodology applied in the context of the multigroup neutron diffusion equations. The method was found to be superior to standard Continuous Galerkin finite element methods but suffered from stability issues associated with low, or zero, absorption coefficient terms.

In the first part of this thesis we concern ourselves with the problem of generating pseudo-random circuits. These are a series of quantum gates chosen at random, with the overall effect of implementing unitary operations with statistical properties close to that of unitaries drawn at random with respect to the Haar measure. Such circuits have a growing number of applications in quantum-information processing, but all known algorithms require an external input of classical randomness. We suggest a scheme to implement random circuits in a weighted graph state. The input state is entangled with the weighted graph state and a random circuit is implemented by performing local measurements in one fixed basis only. A central idea in the analysis of this proposal is the average bipartite entanglement generated by the repeated application of such circuits on a large number of randomly chosen input product states. For a truly random circuit, this should agree with that obtained by applying unitaries at random chosen uniformly with respect to the Haar measure, values which can be calculated using Pages Conjecture. Part II is largely concerned with continuous variables (CV) systems. In particular, we are interested in two descriptions. That of the class of Gaussian states, and that of systems which can be adequately described through the use of Markovian master equations. In the case of the latter, there are a number of approaches one may take in order to derive a suitable equation, all of which require some sort of approximation. These approximations can be made based on a mixture of mathematical and physical grounds. However, unfortunately it is not always clear how justified we are in making a particular choice, especially when the test system we wish to describe includes its own internal interactions. In an attempt to clarify this situation, we derive Markovian master equations for single and interacting harmonic systems under different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding master equations with numerical simulations of the global systems evolution, we assess the robustness of the assumptions usually made in the process of deriving the reduced Markovian dynamics. This serves to clarify the general properties of other open quantum system scenarios subject to treatment within a Markovian approximation. Finally, we extend the notions of the smooth min- and smooth max-entropies to the continuous variable setting. Specifically, we have provided expressions to evaluate these measures on arbitrary Gaussian states. These expressions rely only on the symplectic eigenvalues of the corresponding covariance matrix. As an application, we have considered their use as a suitable measure for detecting thermalisation.

This thesis focuses on two important issues in turbulence theory of wall-bounded flows. One is the recent debate on the form of the mean velocity profile (is it a log-law or a power-law with very weak power exponent?) and on its scalings with Reynolds number. In particular, this study relates the mean flow profile of the turbulent channel flow with the underlying topological structure of the fluctuating velocity field through the concept of critical points, a dynamical systems concept that is a natural way to quantify the multiscale structure of turbulence. This connection gives a new phenomenological picture of wall-bounded turbulence in terms of the topology of the flow. This theory validated against existing data, indicates that the issue on the form of the mean velocity profile at the asymptotic limit of infinite Reynolds number could be resolved by understanding the scaling of turbulent kinetic energy with Reynolds number. The other major issue addressed here is on the fundamental mechanism(s) of viscoelastic turbulence that lead to the polymer-induced turbulent drag reduction phenomenon and its dynamical aspects. A great challenge in this problem is the computation of viscoelastic turbulent flows, since the understanding of polymer physics is restricted to mechanical models. An effective numerical method to solve the governing equation for polymers modelled as nonlinear springs, without using any artificial assumptions as usual, was implemented here for the first time on a three-dimensional channel flow geometry. The superiority of this algorithm is depicted on the results, which are much closer to experimental observations. This allowed a more detailed study of the polymer-turbulence dynamical interactions, which yields a clearer picture on a mechanism that is governed by the polymer-turbulence energy transfers.

Interpolation of discrete FIelds arises frequently in computational physics. This thesis focuses on the novel implementation and analysis of Galerkin projection, an interpolation technique with three principal advantages over its competitors: it is optimally accurate in the L2 norm, it is conservative, and it is well-defined in the case of spaces of discontinuous functions. While these desirable properties have been known for some time, the implementation of Galerkin projection is challenging; this thesis reports the first successful general implementation. A thorough review of the history, development and current frontiers of adaptive remeshing is given. Adaptive remeshing is the primary motivation for the development of Galerkin projection, as its use necessitates the interpolation of discrete fields. The Galerkin projection is discussed and the geometric concept necessary for its implementation, the supermesh, is introduced. The efficient local construction of the supermesh of two meshes by the intersection of the elements of the input meshes is then described. Next, the element-element association problem of identifying which elements from the input meshes intersect is analysed. With efficient algorithms for its construction in hand, applications of supermeshing other than Galerkin projections are discussed, focusing on the computation of diagnostics of simulations which employ adaptive remeshing. Examples demonstrating the effectiveness and efficiency of the presented algorithms are given throughout. The thesis closes with some conclusions and possibilities for future work.