Here I report on our studies of the electronic properties of elliptical quantum dots in a perpendicular external magnetic field, and in the presence of the Rashba spin-orbit interaction. Our work indicates that the Fock-Darwin spectra, corresponding to the non-interacting electrons in an elliptical quantum dot display a strong signature of the Rashba spin-orbit coupling even in a low magnetic field, as the anisotropy of the quantum dot is increased. An explanation of this pronounced effect with respect to the anisotropy is presented. The strong spin-orbit coupling effect manifests itself prominently in the corresponding dipole-allowed optical transitions and hence is susceptible to direct experimental observation.

Coin-flipping is the cryptographic task of generating a random coin-flip between two mistrustful parties. Kitaev discovered that the security of quantum coin-flipping protocols can be analyzed using semidefinite programming. This lead to his result that one party can force a desired coin-flip outcome with probability at least 1/???2. We give sufficient background in quantum computing and semidefinite programming to understand Kitaev's semidefinite programming formulation for coin-flipping cheating strategies. These ideas are specialized to a specific class of protocols singled out by Nayak and Shor. We also use semidefinite programming to solve for the maximum cheating probability of a particular protocol which has the best known security. Furthermore, we present a family of protocols where one party has a greater probability of forcing an outcome of 0 than an outcome of 1. We also discuss a computer search to find specific protocols which minimize the maximum cheating probability.

The efficiency of compiling high-level quantum algorithms into instruction sets native to quantum computers defines the moment in the future when we will be able to solve interesting and important problems on quantum computers. In my work I focus on the new methods for compiling single qubit operations that appear in many quantum algorithms into single qubit operations natively supported by several popular architectures. In addition, I study several questions related to synthesis and optimization of multiqubit operations. When studying the single qubit case, I consider two native instruction sets. The first one is Clifford+T; it is supported by conventional quantum computers implementing fault tolerance protocols based on concatenated and surface codes, and by topological quantum computers based on Ising anyons. The second instruction set is the one supported by topological quantum computers based on Fibonacci anyons. I show that in both cases one can use the number theoretic structure of the problem and methods of computational algebraic number theory to achieve improvements over the previous state of the art by factors ranging from 10 to 1000 for instances of the problem interesting in practice. This order of improvement might make certain interesting quantum computations possible several years earlier. The work related to multiqubit operations is on exact synthesis and optimization of Clifford+T and Clifford circuits. I show an exact synthesis algorithm for unitaries generated by Clifford+T circuits requiring exponentially less number of gates than previous state of the art. For Clifford circuits two directions are studied: the algorithm for finding optimal circuits acting on a small number of qubits and heuristics for larger circuits optimization. The techniques developed allows one to reduce the size of encoding and decoding circuits for quantum error correcting codes by 40-50\% and also finds their applications in randomized benchmarking protocols.

Of the many available computational approaches, density functional theory is the most widely used in studying actinide complexes. This is generally because it incorporates electron correlation effects and is computationally inexpensive for modestly sized compounds. The first chapter of this thesis is an introductory chapter in which some basic concepts of electronic structure theory are discussed. The rest of this thesis is a compilation of several studies of the structural and electronic properties of a range of actinide compounds using predominantly density functional theory. The performances of the basis set/relativistic components as well as the density functional component of theoretical calculations were examined in Chapters 2 and 3 respectively. In Chapters 4, 5, 6 and 7, the electronic structures and properties of actinide species in the environment were explored. The speciation of actinyl aquo-hydroxo species at increasing pH values were studied in Chapter 4. In Chapter 5, the structural and electronic properties of uranyl peroxo complexes with other environmentally important ligands were studied. The adsorption of uranyl complexes to geochemical surfaces was studied in Chapter 6. In addition, the mechanistic pathways to the reduction of these complexes on surfaces and alcohols were examined. In chapter 7, the complexes formed by the uranyl moiety with the aquo and fluoride ligands were studied in gas and aqueous phases. The interactions of uranyl pentafluoride with a protein were examined using a hybrid QM/MM approach. Overall these studies (Chapters 4, 5, 6 and 7) provided valuable insights into the speciation and reduction of actinide species in the environment. In Chapter 8, the properties of novel pentavalent uranium complexes were studied using density functional theory. These complexes have promising roles in the retardation of uranium, via U(VI)-U(IV) reduction, in the environments of nuclear storage repositories. In Chapter 9, the existence of cation-cation interactions in an hexavalent bis-uranyl hydroxo complex was examined using density functional theory and wavefunction methods. In Chapter 10, a summary of the works compiled in this thesis is presented. Future directions for work on the chemistry of actinide complexes were also included in this chapter.

In this thesis, we develop a number of operational tests and tools for benchmarking the quantum nature of optical quantum communication devices. Using the laws of quantum physics, ideal quantum devices can fundamentally outperform their classical counterparts, or even achieve objectives which are classically impossible. Actual devices will not be ideal, but they may still be capable of facilitating quantum communication. Benchmarking tests, based on the presence of entanglement, can be used to verify whether or not imperfect quantum devices offer any advantage over their classical analogs. The general goal in this thesis is to provide strong benchmarking tools which simultaneously require minimal experimental resources but also offer a wide range of applicability. Another major component is the extension of existing qualitative benchmarks (`Is it quantum or classical?') to more quantitative forms (`How quantum is it?'). We provide a number of benchmarking results applicable to two main situations, namely discrete remote state preparation protocols and continuous-variable quantum device testing. The theoretical tools derived throughout this thesis are also applied to the tasks of certifying a remote state preparation experiment and a continuous-variable quantum memory.

Quantum optics provides powerful means to probe quantum mechanics. In this thesis, we study various aspects of quantum phenomena arising in quantum optical systems. Part I studies broadband quantum teleportation. After presenting three different methods of analyzing the standard teleportation protocol, we study the interplay between various bandwidths in determining the fidelity of a broadband quantum field teleportation. Explicit formulae for the degrees of first- and secondorder coherence for the teleportation of resonance fluorescence are derived for this purpose. Part II studies entanglement arising in cascaded open quantum (optical) systems. First, a detailed laser model is produced within quantum trajectory theory to study the total decoherence rate of a laser-driven qubit. Second, using this model, we address the issue of laser quantum state, viewed in connection with separability of the laser-driven-qubit system. Third, a measure of entanglement within quantum trajectory theory called ‘Contextual Entanglement’ is calculated for a few simple systems and compared with the ‘Entanglement of Formation’. Lastly, we introduce a method to quantify entanglement (based on the contextual entanglement) between a source and the field it emits, which we call the ‘Entanglement Spectrum’. It is applied to study the entanglement between a laser-driven qubit and the field the qubit scatters.

Quantum optics provides powerful means to probe quantum mechanics. In this thesis, we study various aspects of quantum phenomena arising in quantum optical systems. Part I studies broadband quantum teleportation. After presenting three different methods of analyzing the standard teleportation protocol, we study the interplay between various bandwidths in determining the fidelity of a broadband quantum field teleportation. Explicit formulae for the degrees of first- and secondorder coherence for the teleportation of resonance fluorescence are derived for this purpose. Part II studies entanglement arising in cascaded open quantum (optical) systems. First, a detailed laser model is produced within quantum trajectory theory to study the total decoherence rate of a laser-driven qubit. Second, using this model, we address the issue of laser quantum state, viewed in connection with separability of the laser-driven-qubit system. Third, a measure of entanglement within quantum trajectory theory called ‘Contextual Entanglement’ is calculated for a few simple systems and compared with the ‘Entanglement of Formation’. Lastly, we introduce a method to quantify entanglement (based on the contextual entanglement) between a source and the field it emits, which we call the ‘Entanglement Spectrum’. It is applied to study the entanglement between a laser-driven qubit and the field the qubit scatters.

Quantum optics provides powerful means to probe quantum mechanics. In this thesis, we study various aspects of quantum phenomena arising in quantum optical systems. Part I studies broadband quantum teleportation. After presenting three different methods of analyzing the standard teleportation protocol, we study the interplay between various bandwidths in determining the fidelity of a broadband quantum field teleportation. Explicit formulae for the degrees of first- and secondorder coherence for the teleportation of resonance fluorescence are derived for this purpose. Part II studies entanglement arising in cascaded open quantum (optical) systems. First, a detailed laser model is produced within quantum trajectory theory to study the total decoherence rate of a laser-driven qubit. Second, using this model, we address the issue of laser quantum state, viewed in connection with separability of the laser-driven-qubit system. Third, a measure of entanglement within quantum trajectory theory called ‘Contextual Entanglement’ is calculated for a few simple systems and compared with the ‘Entanglement of Formation’. Lastly, we introduce a method to quantify entanglement (based on the contextual entanglement) between a source and the field it emits, which we call the ‘Entanglement Spectrum’. It is applied to study the entanglement between a laser-driven qubit and the field the qubit scatters.

Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematical Physics, 1974.

Thesis (Ph.D.) -- University of Adelaide, Department of Mathematical Physics, 1976.