Many important tasks in finance often rely on complex and time-consuming computations. The rapid development of quantum technology has raised the question of whether quantum computing can be used to solve these tasks more efficiently than classical computing. This thesis studies the potential use of quantum computing in finance by solving differently-sized problem instances of the mean-variance portfolio selection model using commercially available quantum resources. The experiments employ gate-based quantum computers and quantum annealing, the two main technologies for realizing a quantum computer. To solve the mean-variance optimization problem on gate-based quantum computers, the model was formulated as a quadratic unconstrained binary optimization (QUBO) problem, which was then used as input to quantum resources available on the largest quantum computing as a service (QCaaS) platforms, IBM Quantum Lab, Microsoft Azure Quantum and Amazon Braket. To solve the problem using quantum annealing, a hybrid quantum-classical solver available on the service D-Wave Leap was employed, which takes as input the mean-variance model’s constrained quadratic form. The problem instances were also solved classically on the model’s QUBO form, where the results acted as benchmarks for the performances of the quantum resources. The results were evaluated based on three performance metrics: time-to-solve, solution quality, and cost-to-solve. The findings indicate that gate-based quantum computers are not yet mature enough to consistently find optimal solutions, with the computation times being long and costly as well. Moreover, the use of gate-based quantum computers was not trouble-free, with the majority of quantum computers failing to even complete the jobs. Quantum annealing, on the other hand, demonstrated greater maturity, with the hybrid solver being capable of fast and accurate optimization, even for very large problem instances. The results from using the hybrid solver justify further research into quantum annealing, to better understand the capabilities and limitations of the technology. The results also indicate that quantum annealing has reached a level of maturity where it has the potential to make a significant impact on financial institutions, creating value that cannot be obtained by using classical computing.

Quantum computing has generated a lot of interested in the past two decades. Research into powerful models of quantum computation has yielded important and elegant results like an efficient algorithm for factoring and a quadratic speed-up for unordered search. At the same time, given the current difficulty in the physical implementation of these general purpose models, considerable effort has also been made in estimating the power of weaker models of quantum computation: models that have a small quantum component. The first part of this thesis is an investigation into the power of interference in quantum computing. While interference in probability amplitudes is a central feature even in powerful models, it is the only extra resource available to quantum finite automata. Of particular interest is interference in automata that have both classical and quantum states (2QCFA) as proposed by Ambainis and Watrous, since it inquires into the power of a classical deterministic finite automaton when augmented with a quantum component of constant size. Our contributions in this part are as follows: • To abstract out the phenomenon of interference in quantum computing, we propose a model called the 2-way Optical Interference Automata (2OIA). The model consists of a 2DFA augmented with a simple optical arrangement. We show different ways of harnessing the power of interference in the form of algorithms on this model to recognize some non-trivial languages. We then go on to show a language recognizable by a Turing machine using O(n2) space but by no 2OIA. • A natural classical model for comparison with 2QCFA is the weighted automaton, since it has the potential to capture interference in sum of path weights. Using the Cortes-Mohri definition of language recognition, we give an efficient simulation of 2QCFAwith algebraic amplitudes by weighted automata over the complex semi ring. • We introduce quantum non-determinism to the Measure-Once 1-way Quantum Finite Automata of Moore and Crutchfield and Kondacs and Watrous and show that even then, the model can recognize only regular languages with bounded error. • We propose a group theoretic generalization of counter automata that allows a notion of counter reversal complexity. To obtain this generalization, we combine concepts from classical counter automata theory with results in 2QCFA. We examine specific instances of this generalization and compare their ii iii powers. We also show an instance recognizing a language that is not recognized by conventional 2-way counter automata. Finally, we show a strict hierarchy among the 1-way versions of the instances Discussed. The second part of the thesis deals with Quantum Adiabatic Optimization algorithms. A common trick for designing faster quantum adiabatic algorithms is to apply the adiabatic condition locally at every instant. However it is often difficult to determine the instantaneous gap between the lowest two eigen values, which is an essential ingredient in the adiabatic condition. We present a simple linear algebraic technique for obtaining a lower bound on the instantaneous gap even in such a situation. As an illustration, we investigate the adiabatic unordered search of van Dam et al. and Roland and Cerf when the non-zero entries of the diagonal final Hamiltonian are perturbed by a polynomial (in logN, where N is the length of the unordered list) amount. We use our technique to derive a bound on the running time of a local adiabatic schedule in terms of the minimum gap between the lowest two eigenvalues.

The development of integrated quantum photonics is integral to many areas of quantum information science, in particular linear optical quantum computing. In this context, a diversity of physical systems is being explored and thus versatility and adaptability are important prerequisites for any candidate platform. Silicon oxynitride is a promising material because its refractive index can be varied over a wide range. This dissertation describes the development of silicon oxynitride waveguides for applications in the field of integrated quantum photonics. The project consisted of three stages: design, characterisation, and application. First, the parameter space was studied through simulations. The structures were optimised to achieve low-loss devices with a small footprint at a wavelength of 900 nm. Buried channel waveguides with a cross-section of 1.6 ?m x 1.6 ?m and a core (cladding) refractive index of 1.545 (1.505) were chosen. Second, following their fabrication with plasma-enhanced chemical vapour deposition, electron beam lithography, and reactive ion etching, the waveguides were characterised. The refractive index was shown to be tunable from the silica to the silicon nitride regime. Optimised tapers significantly improved the coupling efficiency. The minimum bend radius was measured to be less than 2 mm. Propagation losses as low as 1.45 dB cm-1 were achieved. Directional couplers with coupling ratios ranging from 0 to 1 were realised. Third, building blocks for linear optical quantum computing were demonstrated. Reconfigurable quantum circuits consisting of Mach-Zehnder interferometers with near perfect visibilities were fabricated along with a four-port switch. The potential of quantum speedup was illustrated by carrying out the Deutsch-Jozsa algorithm with a fidelity of 100 % using on-demand single photons from a quantum dot. This dissertation presents the first implementation of tunable Mach-Zehnder interferometers, which act on single photons, based on silicon oxynitride waveguides. Furthermore, for the first time silicon oxynitride photonic quantum circuits were operated with on-demand single photons. Accordingly, this work has created a platform for the development of integrated quantum photonics.

This thesis is concerned with the development of an intermediate magnetic field "clock-qubit" in ⁴³Ca⁺ at 146G and techniques to manipulate this qubit using microwaves and lasers. While ⁴³Ca⁺ has previously been used as a qubit, its relatively complicated level structure - with a nuclear spin of 7/2 and low-lying D-states -- makes cooling it in the intermediate field an intimidating prospect. As a result, previous experiments have used small magnetic fields of a few gauss where coherence times are limited and off-resonant excitation is a significant source of experimental error. We demonstrate a simple scheme that allows ⁴³Ca⁺ to be cooled in the intermediate field without any additional experimental complexity compared with low fields. Using the clock-qubit, we achieve a coherence time of T^*<sub style='position:relative;left:-.5em;'>2</sub> = 50 (10)s - the longest demonstrated in any single qubit. We also demonstrate a combined state preparation and measurement error of 6.8(6)x 10^-4 - the lowest achieved for a hyperfine trapped ion qubit [NVG⁺13] - and single-qubit logic gates with average errors of 1.0(3) x 10^-6 - more than an order of magnitude better than the previous record [BWC⁺11]. These results represent the state-of-the-art in the field of single-qubit control. Moreover, we achieve them all in a single scalable room-temperature ion trap using experimentally robust techniques and without relying on the use of narrow-linewidth lasers, magnetic field screening or dynamical decoupling techniques. We also present work on a recent scheme [OWC⁺11] to drive two-qubit gates using microwaves. We have constructed an ion trap with integrated microwave circuitry to perform these gates. Using this trap, we have driven motional sideband transitions, demonstrating the spin-motion coupling that underlies the two-qubit gate. We present an analysis of likely sources of experimental error during a future two-qubit gate and the design and preliminary characterisation of apparatus to minimise the main error contributions. Using this apparatus, we hope to perform a two-qubit gate in the near future.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The goal of this thesis is to present evolutionary algorithms, and demonstrate their applicability in quantum computing. As an introduction to evolutionary algorithms, it is applied to the simple but still challenging (from a computational viewpoint) Travelling Salesman Problem (TSP). This example is used to illustrate the e ect of various parameters like selection method, and maximum population size on the accuracy and e ciency of the evolutionary algorithms. For the sample problem, the 48 continental state capitals of the USA, solutions are evolved and compared to the known optimal solution. From this investigation tournament selection was shown to be the most e ective selection method, and that a population of 200 individuals per generation gave the most e ective convergence rates. In the next part of the thesis, evolutionary algorithms are applied to the generation of optimal quantum circuits for the following cases: The identity transformation : Picked for its simplicity as a test of the correct implementation of the evolutionary algorithm. The results of this investigation showed that the solver program functions correctly and that evolutionary algorithms can indeed nd valid solutions for this kind of problem. The work by Ding et al. [16] on optimal circuits for the two-qubit entanglement gate, controlled-S gate as well as the three qubit entanglement gate are solved by means of EA and the results compared. In all cases similar circuits are produced in fewer generations than the application of Ding et al. [16]. The three qubit quantum Fourier transform gate was also attempted, but no convergence was attained. The quantum teleportation algorithm is also investigated. Firstly the nature of the transformation that leads to quantum teleportation is considered. Next an e ective circuit is sought using evolutionary algorithms. The best result is one gate longer than Brassard [11], and seven gates longer than Yabuki [61]. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om evolusionêre algoritmes te ondersoek en hulle toepaslikheid op kwantumkomputasie te demonstreer. As 'n inleiding tot evolusionêre algoritmes is die eenvoudige, maar steeds komputasioneel uitdagende handelsreisigerprobleem ondersoek. Die invloed van die keuse van 'n seleksie metode, sowel as die invloed van die maksimum aantal individue in 'n generasie op die akkuraatheid en e ektiwiteit van die algoritmes is ondersoek. As voorbeeld is die 48 kontinentale hoofstede van die state van die VSA gekies. Die oplossings wat met evolusionêre algoritmes verkry is, is met die bekende beste oplossings vergelyk. Die resultate van hierdie ondersoek was dat toernooi seleksie die mees e ektiewe seleksie metode is, en dat 200 individue per generasie die mees e ektiewe konvergensie tempo lewer. Evolusionêre algoritmes word vervolgens toegepas om optimale oplossings vir die volgende kwantumalgoritmes te genereer: Die identiteitstransformasie: Hierdie geval is gekies as 'n eenvoudige toepassing met 'n bekende oplossing. Die resultaat van hierdie toepassing van die program was dat dit korrek funksioneer, en vinnig by die korrekte oplossings uitkom. Vervolgens is daar ondersoek ingestel na vier van die gevalle wat in Ding et al. [16] bespreek word. Die spesi eke transformasies waarna gekyk is, is 'n optimale stroombaan vir twee kwabis verstrengeling, 'n beheerde-S hek, 'n drie kwabis verstrengelings hek, en 'n drie kwabis kwantum Fourier transform hek. In die eerste drie gevalle stem die oplossings ooreen met die van Ding et al. [16], en is die konvergensie tempo vinniger. Daar is geen oplossing vir die kwantum Fourier transform verkry nie. Laastens is daar na die kwantumteleportasiealgoritme gekyk. Die eerste stap was om te kyk na die transformasie wat in hierdie geval benodig word, en daarna is gepoog om 'n e ektiewe stroombaan te evolueer. Die beste resultaat was een hek langer as Brassard [11], en sewe hekke langer as Yabuki [61].

The purpose of this thesis is the theoretical study of interference and correlation effects in multimode and continuum mode quantum systems. We are concerned with interference effects in multiport devices which in a sense are generalised Mach-Zehnder interferometers. It is shown how these multimode devices can be employed for the study of negative result and interaction free measurements. Interference and coherence effects are also studied in relation to the radiation fields generated by atoms through the process of spontaneous emission. Besides first order interference, higher order coherence effects are investigated with the aid of Glauber's photodetection theory and it is found that detectors that lie in spacelike regions may display nonclassical correlations under certain conditions. It is well known that the vanishing of field commutators between regions that cannot be connected by subluminal signals reflects the locality of quantum field theory. But is it possible that these spacelike regions exhibit correlations that violate Bell type inequalities? This is the main question and principal concern of the thesis and the answer is affirmative, nonclassical correlations between spacelike regions are indeed possible. A scheme of four detectors that lie in spacelike points was also studied. In this case we do not consider the radiation field but a free scalar field in vacuum state. Nevertheless the virtual quanta of this field may induce nonclassical correlations if the intervals between the detectors are spacelike but small enough. The fundamental reason for this fact is the nonvanishing of the Feynman propagator outside the light cone. Since this propagator is decaying expotentially with the distance it is demonstrated that for large spacelike intervals field correlations obey classical inequalities. We should also note that different inertial observers will agree on the violation or not of these inequalities since the results are manifestly Lorentz invariant.

The potential impact of quantum computing has stimulated a worldwide effort to develop the necessary experimental and theoretical resources. In the race for the quantum computer, several candidate systems have emerged, but the ultimate system is still unclear. We study theoretically how to realize atomic Fock states both for fermionic and bosonic atoms, mainly in one-dimensional optical traps. We demonstrate a new approach of quantum computing based on ultracold fermionic atomic Fock states in optical traps. With the Pauli exclusion principle, producing fermionic atomic Fock states in optical traps is straightforward. We find that laser culling of fermionic atoms in optical traps can produce a scalable number of ultra-high fidelity qubits. We show how each qubit can be independently prepared, and how to perform the required entanglement operations and detect the qubit states with spatially resolved, single-atom detection with adiabatic trap-splitting and fluorescence imaging. On the other hand, bosonic atoms have a strong tendency to stay together. One must rely on strong repulsive interactions to produce bosonic atomic Fock states. To simulate the physical conditions of producing Fock states with ultracold bosonic atoms, we study a many-boson system with arbitrary interaction strength using the Bethe ansatz method. This approach provides a general framework, enabling the study of Fock state production over a wide range of realistic experimental parameters. / text

Single atoms are promising candidates for physically implementing quantum bits, the fundamental unit of quantum information. We have built an apparatus for cooling, trapping and imaging single rubidium atoms in microscopic optical tweezers. The traps are formed from a tightly focused off-resonant laser beam, which traps atoms using the optical dipole force. The traps have a diameter of ~1 &mu;m and a depth of ~1 mK. The novelty of our approach is the use a digital mirror device (DMD) to generate multiple independently movable tweezers from a single laser beam. The DMD consists of an array of micro-mirrors that can be switched on and off, thus acting as a binary amplitude modulator. We use the DMD to imprint a computer-generated hologram on the laser beam, which is converted in to the desired arrangement of traps in the focal plane of a lens. We have developed fast algorithms for calculating binary holograms suitable for the DMD. In addition, we use this method to measure and correct for errors in the phase of the wavefront caused by optical aberrations, which is necessary for producing diffraction-limited focal spots. Using this apparatus, we have trapped arrays of up to 20 atoms with arbitrary geometrical arrangements. We exploit light-assisted collisions between atoms to ensure there is at most one atom per trapping site. We measure the temperature of the atoms in the traps to be 12 &mu;K, and their lifetime to be 1.4 s. Finally, we demonstrate the ability to select individual atoms from an array and transport them over a distance of 14μm with laser cooling, and 5 &mu;m without.

<p>Atomic ions trapped in microfabricated surface traps can be utilized as a physical platform with which to build a quantum computer. They possess many of the desirable characteristics of such a device, including high fidelity state preparation and readout, universal logic gates, and long coherence times, and can be readily entangled with each other through photonic interconnects. The use of optical cavities integrated with trapped ion qubits as a photonic interface presents the possibility for order of magnitude improvements in performance in several key areas for their use in quantum computation. The first part of this thesis describes the design and fabrication of a novel surface trap for integration with an optical cavity. The trap is custom made on a highly reflective mirror surface and includes the capability of moving the ion trap location along all three trap axes with nanometer scale precision. The second part of this thesis demonstrates the suitability of small microcavities formed from laser ablated, fused silica substrates with radii of curvature in the 300-500 micron range for use with the mirror trap as part of an integrated ion trap cavity system. Quantum computing applications for such a system include dramatic improvements in the photon entanglement rate of up to 10 kHz, the qubit measurement time down to 1 microsecond, and the qubit measurement error rate down to the 1e-5 range. The final part of this thesis describes a performance simulator for exploring the physical resource requirements and performance demands to scale a quantum computer to sizes capable of implementing quantum algorithms beyond the limits of classical computation.</p> / Dissertation

Cavity quantum electrodynamics (cavity QED) with single atoms and single photons provides a promising route toward scalable quantum information processing (QIP) and computing. A strongly coupled atom-cavity system should act as a universal quantum interface, allowing the generation and storage of quantum information. This thesis describes the realisation of an atom-cavity system used for the production and manipulation of single photons. These photons are shown to exhibit strong sub-Poissonian statistics and indistinguishability, both prerequisites for their use in realistic quantum systems. Further, the ability to control the temporal shape and internal phase of the photons, as they are generated in the cavity, is demonstrated. This high degree of control presents a novel mechanism enabling the creation of arbitrary photonic quantum bits.