The main topic of this thesis is linear optics and the implementation of quantum operations (measurements, quantum channels, and unitary rotations) on optical systems. In the opening chapter the basic notions needed to understand the rest of the thesis will be explained. These notions include defining a quantum state, measurement, quantum channel and the linear optics tool set. The work in this thesis takes both fundamental and practical approaches to studying linear optical networks. For instance in the first chapter a proof is provided that shows that any unitary on a single mode Fock state can be realised with linear optics. The proof is constructive, however the approach to realising the unitary is not suitable for experimental implementation because it requires complicated ancilla states. As in the KLM proposal the procedure works only stochastically however by allowing the size of the ancilla to grow the probability of failure can be made arbitrarly small. Furthermore we investigate the realisation of arbitrary channels in a specific encoding that we call a $d$-rail encoding. The only ancilla state that we allow is a vacuum ancilliary state and further restrictions were considered (e.g. photon counting). A proof is provided that using these resources only random unitaries can be applied deterministically using linear optics. An expression for the optimal probability of success for realising more general channels with these resources is also discussed. As a final topic we also investigate the realisation of a quantum non-demolition measurement onto the dual rail qubit space. The investigation is a blend of both fundamental and practical approaches. To begin we employ a modified KLM-like procedure and show that the scheme can be realised perfectly but stochastically. The probability that the proper measurement is made can be made arbitrarly close to one using a suitably large ancilla state. In addition we consider an existing scheme \cite{gisin10a} which uses practical sources (two single photon sources) to perform the measurement. The scheme does not realise the true measurement but instead has a free parameter in it which is the transmittiviy of a beamsplitter. The measurement will project onto a space that has a vacuum component. By adjusting the transmittivity of this beamsplitter the vacuum component can be made arbitrarly small but only at the expense of the probability of success of the procedure. In this thesis a modification that can be made to eliminate the vacuum component without changing the sources is introduced. The modification is surprisingly simple and only involves the addition of a single beamsplitter. In the proposal for the original amplifier it was used in simulations for DIQKD that included device imperfections. To show the improvement of our modification these DIQKD simulations are reproduced using the modified amplifier and its results are compared to the results of the original amplifier.

This thesis describes the back action of microwave-photon detection via a Josephson photomultiplier (JPM), a superconducting qubit coupled strongly to a high-quality mi- crowave cavity, and the applications of these devices. The back action operator depends qualitatively on the duration of the measurement interval, resembling the regular photon annihilation operator at short interaction times and approaching a variant of the photon subtraction operator at long times. The optimal operating conditions of the JPM differ from those considered optimal for processing and storing of quantum information, in that a short T2 of the JPM suppresses the cavity dephasing incurred during measurement. Un- derstanding this back action opens the possibility to perform multiple JPM measurements on the same state, hence performing efficient state tomography. In addition, this the- sis describes the creation of non-classical states of microwave radiation via single photon detection using JPMs. When operated in the low T2 regime, the back action of a JPM resembles the photon subtraction operator. Using the non-linearity of this back action, it is possible to create non-classical states of microwave radiation, including squeezed vacuum and odd Schro ̈dinger cat states, starting from a coherent state.

The main topic of this thesis is linear optics and the implementation of quantum operations (measurements, quantum channels, and unitary rotations) on optical systems. In the opening chapter the basic notions needed to understand the rest of the thesis will be explained. These notions include defining a quantum state, measurement, quantum channel and the linear optics tool set. The work in this thesis takes both fundamental and practical approaches to studying linear optical networks. For instance in the first chapter a proof is provided that shows that any unitary on a single mode Fock state can be realised with linear optics. The proof is constructive, however the approach to realising the unitary is not suitable for experimental implementation because it requires complicated ancilla states. As in the KLM proposal the procedure works only stochastically however by allowing the size of the ancilla to grow the probability of failure can be made arbitrarly small. Furthermore we investigate the realisation of arbitrary channels in a specific encoding that we call a $d$-rail encoding. The only ancilla state that we allow is a vacuum ancilliary state and further restrictions were considered (e.g. photon counting). A proof is provided that using these resources only random unitaries can be applied deterministically using linear optics. An expression for the optimal probability of success for realising more general channels with these resources is also discussed. As a final topic we also investigate the realisation of a quantum non-demolition measurement onto the dual rail qubit space. The investigation is a blend of both fundamental and practical approaches. To begin we employ a modified KLM-like procedure and show that the scheme can be realised perfectly but stochastically. The probability that the proper measurement is made can be made arbitrarly close to one using a suitably large ancilla state. In addition we consider an existing scheme \cite{gisin10a} which uses practical sources (two single photon sources) to perform the measurement. The scheme does not realise the true measurement but instead has a free parameter in it which is the transmittiviy of a beamsplitter. The measurement will project onto a space that has a vacuum component. By adjusting the transmittivity of this beamsplitter the vacuum component can be made arbitrarly small but only at the expense of the probability of success of the procedure. In this thesis a modification that can be made to eliminate the vacuum component without changing the sources is introduced. The modification is surprisingly simple and only involves the addition of a single beamsplitter. In the proposal for the original amplifier it was used in simulations for DIQKD that included device imperfections. To show the improvement of our modification these DIQKD simulations are reproduced using the modified amplifier and its results are compared to the results of the original amplifier.

This thesis describes the back action of microwave-photon detection via a Josephson photomultiplier (JPM), a superconducting qubit coupled strongly to a high-quality mi- crowave cavity, and the applications of these devices. The back action operator depends qualitatively on the duration of the measurement interval, resembling the regular photon annihilation operator at short interaction times and approaching a variant of the photon subtraction operator at long times. The optimal operating conditions of the JPM differ from those considered optimal for processing and storing of quantum information, in that a short T2 of the JPM suppresses the cavity dephasing incurred during measurement. Un- derstanding this back action opens the possibility to perform multiple JPM measurements on the same state, hence performing efficient state tomography. In addition, this the- sis describes the creation of non-classical states of microwave radiation via single photon detection using JPMs. When operated in the low T2 regime, the back action of a JPM resembles the photon subtraction operator. Using the non-linearity of this back action, it is possible to create non-classical states of microwave radiation, including squeezed vacuum and odd Schro ̈dinger cat states, starting from a coherent state.

The goal of this work is to understand the application of the evolutionary programming approach to the problem of quantum circuit design. This problem is motivated by the following observations: <ul> <li>In order to keep up with the seemingly insatiable demand for computing power our computing devices will continue to shrink, all the way down to the atomic scale, at which point they become quantum mechanical systems. In fact, this event, known as Moore?s Horizon, is likely to occur in less than 25 years. </li> <li> The recent discovery of several quantum algorithms which can solve some interesting problems more efficiently than any known classical algorithm. </li> <li> While we are not yet certain that quantum computers will ever be practical to build, there do now exist the first few astonishing experimental devices capable of briefly manipulating small quantities of quantum information. The programming of these devices is already a nontrivial problem, and as these devices and their algorithms become more complicated this problem will quickly become a significant challenge. </li> </ul> The Evolutionary Programming (EP) approach to problem solving seeks to mimic the processes of evolutionary biology which have resulted in the awesome complexity of living systems, almost all of which are well beyond our current analysis and engineering capabilities. This approach is motivated by the highly successful application of Koza?s Genetic Programming (GP) approach to a variety of circuit design problems, and specifically the preliminary reports byWilliams and Gray and also Rubinstein who applied GP to quantum circuit design. Accompanying this work is software for evolutionary quantum circuit design which incorporates several advances over previous approaches, including: <ul> <li>A formal language for describing parallel quantum circuits out of an arbitary elementary gate set, including gates with one or more parameters. </li> <li> A fitness assessment procedure that measures both average case fidelity with a respect for global phase equivalences, and implementation cost. </li> <li> A Memetic Programming (MP) based reproductive strategy that uses a combination of global genetic and local memetic searches to effectively search through diverse circuit topologies and optimize the parameterized gates they contain. </li> </ul> Several benchmark experiments are performed on small problems which support the conclusion that Evolutionary Programming is a viable approach to quantum circuit design and that further experiments utilizing more computational resources and more problem insight can be expected to yield many new and interesting quantum circuits.

Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations.

This thesis explores the synthesis and characterisation of a range of molecular wheels containing unpaired electron spins. These molecular spin systems are of considerable interest, both for the insight they provide into the physics of such systems and for their potential as quantum bits ("qubits") in a quantum information processing device. In particular, this thesis explores using these wheels to meet criteria 1 and 5 of the DiVincenzo criteria. The synthesis of a novel homometallic and nonametallic ring of CrIII ions is introduced, along with extensive physical characterisation. Inelastic Neutron Scattering measurements suggest that the molecule has an almost degenerate S = 1/2 ground state with only 0.1 meV separation, making this ring a near perfect example of a Type I frustrated spin system. Chemical modification of the heterometallic {Cr7M} family of wheels with both hard and soft Lewis base functionality is also explored. Using a triphenylphosphine derivative, the coordination chemistry of a highly sterically hindered mono-substituted triphenylphosphine derivative with gold is explored, yielding new arrangements of the wheels. Changes in the electronic and steric properties of the system are studied by a combination of 31P NMR spectroscopy and DFT modelling, revealing dramatic changes in the phosphorus donor properties. The effect of this ligand substitution on the anisotropy tensor of CoII contained in a heterometallic {Cr7Co} ring is explored using variable temperature 1H NMR spectroscopy. Using a combination of the experimentally observed 1H NMR dipolar shifts and computational modelling, a significant change in the anisotropy tensor of the cobalt is found. Finally, as part of a g-engineering approach to qubit design the chemistry of the octametallic {Cr7Ni} ring functionalised with triphenylphosphine oxide is introduced. Initial efforts towards developing a hybrid {Cr7Ni}2Ln (Ln = Gd, Eu) qubit system, along with characterisation by EPR and luminescence spectroscopy, suggest that this may be a route to developing a qubit with the capacity for optical control of the communication.

Quantum cryptography offers a wonderful source for current and future research. The idea started in the early 1970s, and it continues to inspire work and development toward a popular goal, large-scale communication networks with strong security guarantees, based on quantum-mechanical properties. Quantum cryptography builds on the idea of exploiting physical properties to establish secure cryptographic operations. A particular quantum-based protocol has gathered interest in recent years for its use of mesoscopic coherent states. The AlphaEta protocol has been designed to exploit properties of coherent states of light to transmit data securely over an optical channel. AlphaEta aims to draw security from the uncertainty of any measurement of the transmitted coherent states due to intrinsic quantum noise. We propose a framework to combine this protocol with classical preprocessing, taking into account error-correction for the optical channel and establishing a strong provable security guarantee. Integrating a state-of-the-art solution for fast authenticated encryption is straightforward, but in this case the security analysis requires heuristic reasoning. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

The goal of this work is to understand the application of the evolutionary programming approach to the problem of quantum circuit design. This problem is motivated by the following observations: <ul> <li>In order to keep up with the seemingly insatiable demand for computing power our computing devices will continue to shrink, all the way down to the atomic scale, at which point they become quantum mechanical systems. In fact, this event, known as Moore?s Horizon, is likely to occur in less than 25 years. </li> <li> The recent discovery of several quantum algorithms which can solve some interesting problems more efficiently than any known classical algorithm. </li> <li> While we are not yet certain that quantum computers will ever be practical to build, there do now exist the first few astonishing experimental devices capable of briefly manipulating small quantities of quantum information. The programming of these devices is already a nontrivial problem, and as these devices and their algorithms become more complicated this problem will quickly become a significant challenge. </li> </ul> The Evolutionary Programming (EP) approach to problem solving seeks to mimic the processes of evolutionary biology which have resulted in the awesome complexity of living systems, almost all of which are well beyond our current analysis and engineering capabilities. This approach is motivated by the highly successful application of Koza?s Genetic Programming (GP) approach to a variety of circuit design problems, and specifically the preliminary reports byWilliams and Gray and also Rubinstein who applied GP to quantum circuit design. Accompanying this work is software for evolutionary quantum circuit design which incorporates several advances over previous approaches, including: <ul> <li>A formal language for describing parallel quantum circuits out of an arbitary elementary gate set, including gates with one or more parameters. </li> <li> A fitness assessment procedure that measures both average case fidelity with a respect for global phase equivalences, and implementation cost. </li> <li> A Memetic Programming (MP) based reproductive strategy that uses a combination of global genetic and local memetic searches to effectively search through diverse circuit topologies and optimize the parameterized gates they contain. </li> </ul> Several benchmark experiments are performed on small problems which support the conclusion that Evolutionary Programming is a viable approach to quantum circuit design and that further experiments utilizing more computational resources and more problem insight can be expected to yield many new and interesting quantum circuits.

ESR quantum computing presents faster means to perform gates on nuclear spins than the traditional NMR methods. This means ESR is a test-bed that can potentially be useful in ways that are not possible with NMR. The first step is to demonstrate universal control in the ESR system. This work focuses on spin systems with one electron spin and two nuclear spins. We try to demonstrate control over the nuclear spins using the electron as an actuator. In order to perform the experiments, a customized ESR spectrometer was built in the lab. The main advantage of the home-built system is the ability to send arbitrary pulses to the spins. This ability is the key to perform high fidelity controls on the spin system. A customized low temperature probe was designed and built to have three features necessary for the experiments. First, it is possible to orient the sample, thus to change the spin Hamiltonian of the system, in situ. Second, the combined system is able to perform ESR experiments at liquid nitrogen and liquid helium temperatures and rotate the sample while it is cold. Last, the pulse bandwidth of the microwave resonator, which directly affects the fidelity of the gates, is held constant with respect to the sample temperature. Simulations of the experiments have been carried out and the results are promising. Preliminary experiments have been performed, the final set of experiments, demonstrating full quantum control of a three-spin system, are underway at present.