[Mathematical symbols can be only approximated here. For the correct display see the Abstract in the PDF files linked below] This work has demonstrated that hyperfine decoherence times sufficiently long for QIP and quantum optics applications are achievable in rare earth ion centres. Prior to this work there were several QIP proposals using rare earth hyperfine states for long term coherent storage of optical interactions [1, 2, 3]. The very long T_1 (~weeks [4]) observed for rare-earth hyperfine transitions appears promising but hyperfine T_2s were only a few ms, comparable to rare earth optical transitions and therefore the usefulness of such proposals was doubtful. ¶ This work demonstrated an increase in hyperfine T_2 by a factor of 7 × 10^4 compared to the previously reported hyperfine T_2 for Pr^[3+]:Y_2SiO_5 through the application of static and dynamic magnetic field techniques. This increase in T_2 makes previous QIP proposals useful and provides the first solid state optically active Lamda system with very long hyperfine T_2 for quantum optics applications. ¶ The first technique employed the conventional wisdom of applying a small static magnetic field to minimise the superhyperfine interaction [5, 6, 7], as studied in chapter 4. This resulted in hyperfine transition T_2 an order of magnitude larger than the T_2 of optical transitions, ranging fro 5 to 10 ms. The increase in T_2 was not sufficient and consequently other approaches were required. ¶ Development of the critical point technique during this work was crucial to achieving further gains in T_2. The critical point technique is the application of a static magnetic field such that the Zeeman shift of the hyperfine transition of interest has no first order component, thereby nulling decohering magnetic interactions to first order. This technique also represents a global minimum for back action of the Y spin bath due to a change in the Pr spin state, allowing the assumption that the Pr ion is surrounded by a thermal bath. The critical point technique resulted in a dramatic increase of the hyperfine transition T_2 from ~10 ms to 860 ms. ¶ Satisfied that the optimal static magnetic field configuration for increasing T_2 had been achieved, dynamic magnetic field techniques, driving either the system of interest or spin bath were investigated. These techniques are broadly classed as Dynamic Decoherence Control (DDC) in the QIP community. The first DDC technique investigated was driving the Pr ion using a CPMG or Bang Bang decoupling pulse sequence. This significantly extended T_2 from 0.86 s to 70 s. This decoupling strategy has been extensively discussed for correcting phase errors in quantum computers [8, 9, 10, 11, 12, 13, 14, 15], with this work being the first application to solid state systems. ¶ Magic Angle Line Narrowing was used to investigate driving the spin bath to increase T_2. This experiment resulted in T_2 increasing from 0.84 s to 1.12 s. Both dynamic techniques introduce a periodic condition on when QIP operation can be performed without the qubits participating in the operation accumulating phase errors relative to the qubits not involved in the operation. ¶ Without using the critical point technique Dynamic Decoherence Control techniques such as the Bang Bang decoupling sequence and MALN are not useful due to the sensitivity of the Pr ion to magnetic field fluctuations. Critical point and DDC techniques are mutually beneficial since the critical point is most effective at removing high frequency perturbations while DDC techniques remove the low frequency perturbations. A further benefit of using the critical point technique is it allows changing the coupling to the spin bath without changing the spin bath dynamics. This was useful for discerning whether the limits are inherent to the DDC technique or are due to experimental limitations. ¶ Solid state systems exhibiting long T_2 are typically very specialised systems, such as 29Si dopants in an isotopically pure 28Si and therefore spin free host lattice [16]. These systems rely on on the purity of their environment to achieve long T_2. Despite possessing a long T_2, the spin system remain inherently sensitive to magnetic field fluctuations. In contrast, this work has demonstrated that decoherence times, sufficiently long to rival any solid state system [16], are achievable when the spin of interest is surrounded by a concentrated spin bath. Using the critical point technique results in a hyperfine state that is inherently insensitive to small magnetic field perturbations and therefore more robust for QIP applications.

Quantum computation has attracted a great deal of attention from the scientific community in recent years. By using the quantum mechanical phenomena of superposition and entanglement, a quantum computer can solve certain problems much faster than classical computers. Several quantum algorithms have been developed to demonstrate this quantum speedup. Two important examples are Shor’s algorithm for the factorization problem, and Grover’s algorithm for the search problem. Significant efforts are on to build a large scale quantum computer for implementing these quantum algorithms. This thesis deals with Grover’s search algorithm, and presents its several generalizations that perform better in specific contexts. While writing the thesis, we have assumed the familiarity of readers with the basics of quantum mechanics and computer science. For a general introduction to the subject of quantum computation, see [1]. In Chapter 1, we formally define the search problem as well as present Grover’s search algorithm [2]. This algorithm, or more generally the quantum amplitude amplification algorithm [3, 4], drives a quantum system from a prepared initial state (s) to a desired target state (t). It uses O(α-1 = | (t−|s)| -1) iterations of the operator g = IsIt on |s), where { IsIt} are selective phase inversions selective phase inversions of the corresponding states. That is a quadratic speedup over the simple scheme of O(α−2) preparations of |s) and subsequent projective measurements. Several generalizations of Grover’s algorithm exist. In Chapter 2, we study further generalizations of Grover’s algorithm. We analyse the iteration of the search operator S = DsI t on |s) where Ds is a more general transformation than Is, and I t is a selective phase rotation of |t) by angle . We find sufficient conditions for S to produce a successful quantum search algorithm. In Chapter 3, we demonstrate that our general framework encapsulates several previous generalizations of Grover’s algorithm. For example, the phase-matching condition for the search operator requires the angles and and to be almost equal for a successful quantum search. In Kato’s algorithm, the search operator is where Ks consists of only single-qubit gates, which are easier to implement physically than multi-qubit gates. The spatial search algorithms consider the search operator where is a spatially local operator and provides implementation advantages over Is. The analysis of Chapter 2 provides a simpler understanding of all these special cases. In Chapter 4, we present schemes to improve our general quantum search algorithm, by controlling the operators through an ancilla qubit. For the case of two dimensional spatial search problem, these schemes yield an algorithm with time complexity . Earlier algorithms solved this problem in time steps, and it was an open question to design a faster algorithm. The schemes can also be used to find, for a given unitary operator, an eigenstate corresponding to a specified eigenvalue. In Chapter 5, we extend the analysis of Chapter 2 to general adiabatic quantum search. It starts with the ground state |s) of an initial Hamiltonian Hs and evolves adiabatically to the target state |t) that is the ground state of the final Hamiltonian The evolution uses a time dependent Hamiltonian HT that varies linearly with time . We show that the minimum excitation gap of HT is proportional to α. Also, the ground state of HT changes significantly only within a very narrow interval of width around the transition point, where the excitation gap has its minimum. This feature can be used to reach the target state (t) using adiabatic evolution for time In Chapter 6, we present a robust quantum search algorithm that iterates the operator on |s) to successfully reach |t), whereas Grover’s algorithm fails if as per the phase-matching condition. The robust algorithm also works when is generalized to multiple target states. Moreover, the algorithm provides a new search Hamiltonian that is robust against certain systematic perturbations. In Chapter 7, we look beyond the widely studied scenario of iterative quantum search algorithms, and present a recursive quantum search algorithm that succeeds with transformations {Vs,Vt} sufficiently close to {Is,It.} Grover’s algorithm generally fails if while the recursive algorithm is nearly optimal as long as , improving the error tolerance of the transformations. The algorithms of Chapters 6-7 have applications in quantum error-correction, when systematic errors affect the transformations The algorithms are robust as long as the errors are small, reproducible and reversible. This type of errors arise often from imperfections in apparatus setup, and so the algorithms increase the flexibility in physical implementation of quantum search. In Chapter 8, we present a fixed-point quantum search algorithm. Its state evolution monotonically converges towards |t), unlike Grover’s algorithm where the evolution passes through |t) under iterations of the operator . In q steps, our algorithm monotonically reduces the failure probability, i.e. the probability of not getting |t), from . That is asymptotically optimal for monotonic convergence. Though the fixed-point algorithm is of not much use for , it is useful when and each oracle query is highly expensive. In Chapter 9, we conclude the thesis and present an overall outlook.

Random walks describe diffusion processes, where movement at every time step is restricted only to neighbouring locations. Classical random walks are constructed using the non-relativistic Laplacian evolution operator and a coin toss instruction. In quantum theory, an alternative is to use the relativistic Dirac operator. That necessarily introduces an internal degree of freedom (chirality), which may be identified with the coin. The resultant walk spreads quadratically faster than the classical one, and can be applied to a variety of graph theoretical problems. We study in detail the problem of spatial search, i.e. finding a marked site on a hypercubic lattice in d-dimensions. For d=1, the scaling behaviour of classical and quantum spatial search is the same due to the restriction on movement. On the other hand, the restriction on movement hardly matters for d ≥ 3, and scaling behaviour close to Grover’s optimal algorithm(which has no restriction on movement) can be achieved. d=2 is the borderline critical dimension, where infrared divergence in propagation leads to logarithmic slow down that can be minimised using clever chirality flips. In support of these analytic expectations, we present numerical simulation results for d=2 to d=9, using a lattice implementation of the Dirac operator inspired by staggered fermions. We optimise the parameters of the algorithm, and the simulation results demonstrate that the number of binary oracle calls required for d= 2 and d ≥ 3 spatial search problems are O(√NlogN) and O(√N) respectively. Moreover, with increasing d, the results approach the optimal behaviour of Grover’s algorithm(corresponding to mean field theory or d → ∞ limit). In particular, the d = 3 scaling behaviour is only about 25% higher than the optimal value.

A simplified model of a double quantum dot is presented and analyzed, with applications to spin-qubit quantum computation. The ability to trap single electrons in semiconductor nanostructures has led to the proposal of quantum computers with spin-based qubits coupled by the exchange interaction. Current theory predicts an exchange interaction with a -1 power-law dependence on the detuning ϵ, the energy offset between the two dots. However, experiment has shown a -3/2 power-law dependence on ϵ. Using WKB analysis, this thesis explores one possible source of the modified dependence, namely an ϵ-dependent tunnel coupling between the two wells. WKB quantization is used to find expressions for the tunnel coupling of a one-dimensional double-well, and these results are compared to the exact, numerical solutions, as determined by the finite difference method and the transfer matrix method. Small ϵ-dependent corrections to the tunnel coupling are observed. In typical cases, WKB correctly predicts a constant tunnel coupling at leading-order. WKB also predicts small ϵ-dependent corrections for typical cases and strongly ϵ-dependent tunnel couplings for certain exceptional cases. However, numerical simulations suggest that WKB is not accurate enough to analyze the small corrections, and is not valid in the exceptional cases. Deviations from the conventional form of the low-energy Hamiltonian for a double-well are also observed and discussed.

A simplified model of a double quantum dot is presented and analyzed, with applications to spin-qubit quantum computation. The ability to trap single electrons in semiconductor nanostructures has led to the proposal of quantum computers with spin-based qubits coupled by the exchange interaction. Current theory predicts an exchange interaction with a -1 power-law dependence on the detuning ϵ, the energy offset between the two dots. However, experiment has shown a -3/2 power-law dependence on ϵ. Using WKB analysis, this thesis explores one possible source of the modified dependence, namely an ϵ-dependent tunnel coupling between the two wells. WKB quantization is used to find expressions for the tunnel coupling of a one-dimensional double-well, and these results are compared to the exact, numerical solutions, as determined by the finite difference method and the transfer matrix method. Small ϵ-dependent corrections to the tunnel coupling are observed. In typical cases, WKB correctly predicts a constant tunnel coupling at leading-order. WKB also predicts small ϵ-dependent corrections for typical cases and strongly ϵ-dependent tunnel couplings for certain exceptional cases. However, numerical simulations suggest that WKB is not accurate enough to analyze the small corrections, and is not valid in the exceptional cases. Deviations from the conventional form of the low-energy Hamiltonian for a double-well are also observed and discussed.

Cold, trapped atomic ions have enabled the investigation of fundamental physics and generated a rich field of applications. Foremost among these is quantum computation which has recently driven the development of the sophisticated, scalable surface-electrode trap. Despite the many advantages of surface-electrode traps, the typically smaller ion-electrode distance, d, in these traps results in an increased ion heating rate that is proportional to d^(-4) and a decreased trap well-depth that is proportional to d^(-2). These shortcomings can be simultaneously addressed by installing the trap into a cryogenic environment. With this in mind, a closed-cycle, cryogenic ion trapping apparatus that maintains excellent vacuum, is highly modular, has increased optical access, and uses a simple vibration isolation system has been developed. Single ions are trapped and used to characterize system properties such as the motion of the vibration isolation stage. In order to compare this system to a similar room temperature apparatus, the ion trapping lifetime and heating rate are determined. A single ion also serves as a sensitive electric field probe that is used to measure and compensate stray electric fields across the trap. Due to the long dark ion lifetimes in this system, it is well-suited to probing the stability of small, linear ion crystals. Linear ion crystals of arbitrary length are built in an automated fashion using transport waveforms and the scaling of dark lifetime with ion number for N <= 6 is investigated. These data are then used to consider the relevance of various loss channels.

Topological order is a new field of research involving exotic physics. Among other things it has been suggested as a means for realising fault-tolerant quantum computation. Topological degeneracy, i.e. the ground state degeneracy of a topologically ordered state, is one of the quantities that have been used to characterize such states. Topological order has also been suggested as a possible quantum information storage. We study two-dimensional lattice models defined on a closed manifold, specifically on a torus, and find that these systems exhibit topological degeneracy proportional to the genus of the manifold on which they are defined. We also find that the addition of lattice dislocations increases the ground state degeneracy, a behaviour that can be interpreted as artificially increasing the genus of the manifold. We derive the fusion and braiding rules of the model, which are then used to calculate the braiding properties of the dislocations themselves. These turn out to resemble non-abelian anyons, a property that is important for the possibility to achieve universal quantum computation. One can also emulate lattice dislocations synthetically, by adding an external field. This makes them more realistic for potential experimental realisations. / Topologisk ordning är ett nytt område inom fysik som bland annat verkar lovande som verktyg för förverkligandet av kvantdatorer. En av storheterna som karakteriserar topologiska tillstånd är det totala antalet degenererade grundtillstånd, den topologiska degenerationen. Topologisk ordning har också föreslagits som ett möjligt sätt att lagra kvantdata. Vi undersöker tvådimensionella gittermodeller definierade på en sluten mångfald, specifikt en torus, och finner att dessa system påvisar topologisk degeneration som är proportionerlig mot mångfaldens topologiska genus. När dislokationer introduceras i gittret finner vi att grundtillståndets degeneration ökar, något som kan ses som en artificiell ökning av mångfaldens genus. Vi härleder sammanslagningsregler och flätningsregler för modellen och använder sedan dessa för att räkna ut flätegenskaperna hos själva dislokationerna. Dessa visar sig likna icke-abelska anyoner, en egenskap som är viktiga för möjligheten att kunna utföra universella kvantberäkningar. Det går också att emulera dislokationer i gittret genom att lägga på ett yttre fält. Detta gör dem mer realistiska för eventuella experimentella realisationer.

<p>Strong coupling between light and matter has been demonstrated both in classical</p><p>cavity quantum electrodynamics (QED) systems and in more recent circuit-QED</p><p>experiments. This enables the generation of strong nonlinear photon-photon interactions</p><p>at the single-photon level, which is of great interest for the observation</p><p>of quantum nonlinear optical phenomena, the control of light quanta in quantum</p><p>information protocols such as quantum networking, as well as the study of</p><p>strongly correlated quantum many-body systems using light. Recently, strong</p><p>coupling has also been realized in a variety of one-dimensional (1D) waveguide-</p><p>QED experimental systems, which in turn makes them promising candidates for</p><p>quantum information processing. Compared to cavity-QED systems, there are</p><p>two new features in waveguide-QED: the existence of a continuum of states and</p><p>the restricted 1D phase space, which together bring in new physical effects, such</p><p>as the bound-state effects. This thesis consists of two parts: 1) understanding the</p><p>fundamental interaction between local quantum objects, such as two-level systems</p><p>and four-level systems, and photons confined in the waveguide; 2) exploring</p><p>its implications in quantum information processing, in particular photonic</p><p>quantum computation and quantum key distribution.</p><p>First, we demonstrate that by coupling a two-level system (TLS) or three/fourlevel</p><p>system to a 1D continuum, strongly-correlated photons can be generated</p><p>inside the waveguide. Photon-photon bound states, which decay exponentially as a function of the relative coordinates of photons, appear in multiphoton scattering</p><p>processes. As a result, photon bunching and antibunching can be observed</p><p>in the photon-photon correlation function, and nonclassical light source can be</p><p>generated on demand. In the case of an N-type four-level system, we show</p><p>that the effective photon-photon interaction mediated by the four-level system,</p><p>gives rise to a variety of nonlinear optical phenomena, including photon blockade,</p><p>photon-induced tunneling, and creation of single-photon states and photon</p><p>pairs with a high degree of spectral entanglement, all in the absence of a cavity.</p><p>However, to enable greater quantum networking potential using waveguide-</p><p>QED, it is important to study systems having more than just one TLS/qubit.</p><p>We develop a numerical Green function method to study cooperative effects in</p><p>a system of two qubits coupled to a 1D waveguide. Quantum beats emerge in</p><p>photon-photon correlations, and persist to much longer time scales because of</p><p>non-Markovian processes. In addition, this system can be used to generate a</p><p>high-degree of long-distance entanglement when one of the two qubits is driven</p><p>by an on-resonance laser, further paving the way toward waveguide-QED-based</p><p>quantum networks.</p><p>Furthermore, based on our study of light-matter interactions in waveguide-</p><p>QED, we investigate its implications in quantum information processing. First,</p><p>we study quantum key distribution using the sub-Possonian single photon source</p><p>obtained by scattering a coherent state off a two-level system. The rate for key</p><p>generation is found to be twice as large as for other sources. Second, we propose</p><p>a new scheme for scalable quantum computation using flying qubits--propagating</p><p>photons in a one-dimensional waveguide--interacting with matter qubits. Photonphoton</p><p>interactions are mediated by the coupling to a three- or four-level system,</p><p>based on which photon-photon -phase gates (Controlled-NOT) can be implemented for universal quantum computation. We show that high gate fidelity is</p><p>possible given recent dramatic experimental progress in superconducting circuits</p><p>and photonic-crystal waveguides. The proposed system can be an important</p><p>building block for future on-chip quantum networks.</p> / Dissertation

This thesis describes experimental work towards developing a trapped ion quantum information processor. An existing ion trap apparatus was capable of trapping and laser-cooling single ions or small ion strings of 40 Ca+, and had been used for studies of quantum jumps and natural lifetime measurements in Ca. This thesis describes improvements in this apparatus, which have allowed the stability and the flexibility of experimental control of the ions to be greatly increased. This enabled experiments to read out the spin state of a single trapped ion, and to load ions with isotope selectivity through photoionization. The optical systems were improved by installation of new lasers, optical reference cavities, and a system of acousto-optic modulators for laser intensity switching and frequency control. The photon counting for fluorescence detection was improved, and a new photon time-of-arrival correlation circuit developed. This has permitted rapid and more sensitive detection of micromotion, and hence cancellation of stray fields in the trap. A study of resonant circuits in the low RF, high voltage (10 MHz, 1 kV) regime was carried out with a view to developing a new RF supply for the Paul trap with reduced noise and increased power. A new supply based on a helical resonator was built and used to trap ions. This technique has reduced noise and will permit higher secular frequencies to be attained in the future. A magnetic field B in the ion trap is used to define a quantization axis, and in one series of experiments was required to be of order 100 G to provide a substantial Zeeman splitting. A set of magnetic field coils to control the size and direction of B is described. The design of these posed some problems owing to an unforseen issue with the vacuum chamber. In short, it is magnetizable and acts to first approximation like a magnetic shield. The field coils had to be sufficiently substantial to produce the desired field at the ion even in the presence of this shielding effect, and dark resonance (and other) spectra with Zeeman splitting were obtained to calibrate the field using the ion as a probe. Finally, the thesis describes the successful loading of the ion trap by laser photoionization from a weak atomic beam. This involved two new lasers at 423 nm and 389 nm. Saturated absorption spectroscopy of neutral calcium is first described, then transverse excitation of an atomic beam in our vacuum chamber is used to identify all the main isotopes of calcium and confirm their abundances in our source (a heated sample of natural calcium). Finally, photoionization is used to load the trap. This has three advantages over electron-impact ionization. By avoiding an electron gun, we avoid charging of insulating patches and subsequent electric field drift as they discharge; the flux in the atomic beam and hence calcium (and other) deposits on the electrodes can be greatly reduced; and most importantly, the photoionization is isotope selective. Evidence is presented which suggests that even with an non-enriched source, the rare isotope 43 Ca can be loaded with reasonable efficiency. This isotope is advantageous for quantum information experiments for several reasons, but chiefly because its ground state hyperfine structure can act as a stable qubit.

Orientador: Anna Diva Plasencia Lotufo / Resumo: A lógica de múltiplos valores, em especial a ternária, apresenta inúmeras vantagens sobre a lógica binária em circuitos reversíveis/quânticos. A realização de funções usando a lógica reversível ternária é conhecida por requerer um menor número de linhas em comparação com a lógica reversível binária convencional. Este aspecto tem motivado as pesquisas em abordagens de síntese. A grande maioria dos métodos existentes requerem entradas adicionais, denominadas de ancillary lines, durante o processo de síntese, o que é dispendioso para implementação em tecnologias quânticas, quando disponíveis. Neste trabalho, foram propostas diferentes metodologias e análises comparativas para o problema da síntese de circuitos reversíveis ternários sem a adição de ancillary lines. A metodologia de síntese proposta, denominada de MMD plus, foi aplicado nos modos backward e top-down como referência a todas as 362880 possíveis funções reversíveis ternárias de 2 variáveis. Além do processamento top-down originário do algoritmo MMD, um processamento bottom-up é implementado e sua eficiência comparativa é avaliada. Por definição, as funções reversíveis ternárias são permutações. Realiza-se a decomposição das permutações em ciclos disjuntos de ordem natural, em ciclos de permutação com 3 elementos, e em transposições, para obtenção dos circuitos reversíveis ternários. Uma métrica é introduzida para mensurar a complexidade e custo dos circuitos, com base nas portas reversíveis de múltiplos valores Muthu... (Resumo completo, clicar acesso eletrônico abaixo) / Doutor