Most quantum algorithms that are efficient as opposed to their equivalent classical algorithms are solving variants of the Hidden Subgroup Problem (HSP), therefore HSP is a central problem in the field of quantum computing. In this thesis, we offer some interesting results about the subgroup and coset structure of certain groups, including the dihedral group. We describe classical algorithms to solve the HSP over various abelian groups and the dihedral group. We also discuss some existing quantum algorithms to solve the HSP and give our own novel algorithms and ideas to approach the HSP for the dihedral groups.

This thesis investigates the impact noise has on Grover’s algorithm when being used to search for multiple items in a database. The main metric being looked at is the probability of the algorithm successfully finding a correct item. The Qiskit framework was used to implement and evaluate the algorithm’s performance in noise-free and noisy environments. Results from the experiments show significant findings. In noiseless tests, the algorithm performs effectively and as expected. However, with the introduction of a noise model, the algorithm’s performance declines noticeably. The probability of it finding a marked item was close to the probability of randomly selecting the same item from the database. This was the case regardless of how many items were marked or the database size. These unexpected outcomes illustrate the disabling effect of noise on Grover’s algorithm. Limitations of the study include noise completely disrupting the algorithm, challenges in accurately modelling quantum noise, and the use of relatively small databases. Further research is needed to explore noise mitigation strategies and assess the algorithm’s robustness in larger-scale scenarios. This research strengthens our understanding of noise’s impact on Grover’s algorithm, showcasing the challenges and limitations of its implementation. It highlights the importance of properly managing noise in quantum computing to fully utilize its potential in efficiently solving complex problems. / Denna avhandling undersöker effekten av brus på Grover’s algoritm för att söka efter flera markerade element i en databas. Huvudfokuset var att undersöka sannolikheten att algoritmen korrekt skulle hitta ett av flera markerade element i en databas. Qiskit-ramverket användes för att utvärdera algoritmens prestanda i brusfria och brusiga miljöer. Resultaten från experimenten var betydelsefulla. I brusfria tester presterar algoritmen effektivt och som förväntat. Men, med införandet av brus minskar algoritmens prestanda avsevärt. Sannolikheten för att algoritmen hittar ett markerat element liknar sannolikheten för att slumpmässigt välja ut samma element från databasen. Detta var fallet oavsett hur många element som var markerade och databasens storlek. Dessa oväntade resultat illustrerar brusets söndrande effekt på Grover’s algoritm. Begränsningar i studien inkluderar att bruset helt får algoritmen att sluta fungera, utmaningar med att noggrant modellera kvantbrus och användningen av relativt små databaser. Vidare forskning behövs för att undersöka strategier för att mitigera brus och bedöma algoritmens robusthet i storskaliga scenarier. Denna forskning stärker vår förståelse för brusets påverkan på Grover’s algoritm och betonar utmaningar och begränsningar vid dess implementering. Den betonar vikten av att hantera brus inom kvantdatorer för att kunna utnyttja deras potential för effektiv lösning av komplexa problem.

Previous work was successful at delineating reaction pathways for the photoactivated synthesis of an amine, [CztBu(PyriPr)(NH2−PyriPr)], by double intramolecular C−H activation and functionalization via irradiating a metal(II) azido complex, [CztBu(PyriPr)2NiN3. The present work seeks to expand upon earlier research, and to substitute the metal with iron or cobalt, and to expand the study to photocatalyzed intermolecular C−H activation and functionalization of organic substrates. Density functional theory (DFT) – B3LYP/6-31+G(d') and APFD/Def2TZVP – and time-dependent density functional theory (TDDFT) were used to propose a detailed pathway comprised of intermediates of low, intermediate, or high spin multiplicity and photo-generated excited states for the reaction of the azido complex, [CztBu(PyriPr)2MN3] to form the amine complex [CztBu(PyriPr)M(NH2−PyriPr)], M = Co, Ni or Fe, and the intermediates along the reaction pathway. For applications on quantum computing, the photophysical properties of photoactive d8 nickel(II) complexes are modeled. Such systems take advantage of a two-level system pathway between ground to excited state electronic transitions and could be useful for the discovery of successful candidates for a room temperature qubit, the analogue of a classical computational bit. A modified organometallic model, inspired by a nitrogen vacancy selective intersystem crossing model in diamond, was developed to take advantage of the formation of excited states. Tanabe-Sugano diagrams predict areas where these excited states may relax via phosphorescent emission. Under Zeeman splitting, these transitions create the conditions required for a two-level system needed to design a functional organometallic qubit.

Abstract Iris The power of quantum computing lies in its ability to perform certain calculations and solve complex problems exponentially faster than classical computers. This potential has profound implications for a wide range of fields, including particle physics. This thesis lays a fundamental foundation for understanding quantum computing. Particular emphasis is placed on the intricate process of quantum gate decomposition, an elementary lynchpin that underpins the development of quantum algorithms and plays a crucial role in this research. In particular, this concerns the implementation of quantum algorithms designed to simulate the dynamic evolution of multi-particle quantum systems - so-called Hamiltonian simulations. The concept of quantum gate decomposition is introduced and linked to quantum circuit optimisation. The decomposition of quantum gates plays a crucial role in fault-tolerant quantum computing in the sense that an optimal implementation of a quantum gate is essential to efficiently perform a quantum simulation, especially for near-term quantum computers. Part of this thesis aims to propose a new explicit tensorial notation of quantum computing. Two notations are commonly used in the literature. The first is the Dirac notation and the other standard formalism is based on the so-called computational basis. The main disadvantage of the latter is the exponential growth of vector and matrix dimensions and the fact that it hides some relevant quantum properties of the operations by increasing the apparent number of independent variables. A third possible notation is introduced here, which describes qubit states as tensors and quantum gates as multilinear or quasi-multilinear maps. Some advantages for the detection of separable and entangled systems and for measurement techniques are also shown. Finally, this thesis demonstrates the advantage of quantum computing in the description of multi-particle quantum systems by proposing a quantum algorithm to simulate collective neutrino oscillations. Collective flavour oscillations of neutrinos due to forward neutrino-neutrino scattering provide an intriguing many-body system for time evolution simulations on a quantum computer. These phenomena are of particular interest in extreme astrophysical settings such as core-collapse supernovae, neutron star mergers and the early universe. A detailed description of the physical phenomena and environments in which collective flavor oscillations occur is first reported, and the derivation of the Hamiltonian governing the evolution of flavor oscillations is detailed. The aim is to reproduce this evolution using a quantum algorithm. To manage the computational complexity, we use the Trotter approximation of the time evolution operator, which mitigates the exponential growth of circuit complexity. The quantum algorithm was designed to work on a trapped-ion based testbed (the theory of which is presented in detail). After machine-aware optimisation, the quantum circuit implementing the algorithm was run on the real quantum machine 'Quantinuum', and the results are presented and discussed.

Quantum information technologies include secure quantum communications and ultra precise quantum sensing that are significantly more efficient than their classical counterparts. To enable such technologies, we need a scalable quantum platform in which qubits are con trollable. Color centers provide controllable optically-active spin qubits within the coherence time limit. Moreover, the nearby nuclear spins have long coherence times suitable for quantum memories. In this thesis, I present a theoretical understanding of and control protocols for various color centers. Using group theory, I explore the wave functions and laser pumping-induced dynamics of VSi color centers in silicon carbide. I also provide dynamical decoupling-based high-fidelity control of nuclear spins around the color center. I also present a control technique that combines holonomic control and dynamically corrected control to tolerate simultaneous errors from various sources. The work described here includes a theoretical understanding and control techniques of color center spin qubits and nuclear spin quantum memories, as well as a new platform-independent control formalism towards robust qubit control. / Doctor of Philosophy / Quantum information technologies promise to offer efficient computations of certain algorithms and secure communications beyond the reach of their classical counterparts. To achieve such technologies, we must find a suitable quantum platform to manipulate the quantum information units (qubits). Color centers host spin qubits that can enable such technologies. However, it is challenging due to our incomplete understanding of their physical properties and, more importantly, the controllability and scalability of such spin qubits. In this thesis, I present a theoretical understanding of and control protocols for various color centers. By using group theory that describes the symmetry of color centers, I give a phenomenological model of spin qubit dynamics under optical control of VSi color centers in silicon carbide. I also provide an improved technique for controlling nuclear spin qubits with higher precision. Moreover, I propose a new qubit control technique that combines two methods - holonomic control and dynamical corrected control - to provide further robust qubit control in the presence of multiple noise sources. The works in this thesis provide knowledge of color center spin qubits and concrete control methods towards quantum information technologies with color center spin qubits.

Micro– and subsequently nano–scale fabrication techniques have reshaped our world more drastically than almost any other development of the last half-century. Spurred by the invention of the transistor at Bell Labs in 1947, monolithic integrated circuits—or microchips in the colloquial lexicon—were developed in ’59, kickstarting the modern digital age as we know it. More recently, the maturation of classical computing technology and significant advancements in materials science have led to a boom of interest in and progress by the quantum sector on both computation and communication fronts. The explosive growth currently underway in the field of quantum information science (QIS) marks the dawning of a new age, which will undoubtedly transform our world in ways we have yet to imagine. This dissertation seeks to leverage advanced nanofabrication approaches, atomically thin materials, and state of the art microscopy techniques to develop room-temperature single photon sources for QIS applications. A basic overview of 2D materials is provided in Chapter 1. Particular emphasis is placed on the optical properties of tungsten diselenide (WSe2), which is followed by a brief discussion of quantum emitters in 2D and other material systems. Chapter 2 describes the scanning near-field optical microscopy (SNOM) technique we use to investigate the photoluminescence (PL) response of strained WSe₂ with resolution well below the classical diffraction limit. The third chapter is dedicated to the various fabrication methods explored and developed to produce the plasmonic substrates necessary for near-field optical studies. The first section focuses on the creation of extremely flat metallic surfaces, while the second deals with extremely sharp metallic stressors. These two platforms enable the investigations of nanobubbles—touched upon in Chapter 2—and nanowrinkles, which are the subject of discussion in Chapter 4. The strain confinement provided by these wrinkles leads to highly localized quantum dot-like states that exhibit excitation power saturation at room temperature. Together, these studies lay the groundwork for achieving high-temperature quantum emission in atomically thin semiconducting van der Waals materials.

Quantum computing is a form of computation that exploits quantum mechanical phenomena for information processing, with promising applications (among others) in optimization and machine learning. Indeed, quantum machine learning is currently one of the most popular directions of research in quantum computing, offering solutions with an at-least-theoretical advantage compared to the classical counterparts. Nevertheless, the quantum devices available in the current Noisy Intermediate-Scale Quantum (NISQ) era are limited in the number of qubits and significantly affected by noise. An interesting alternative to the current prototypes of general-purpose quantum devices is represented by quantum annealers, specific-purpose quantum machines implementing the heuristic search for solving optimization problems known as quantum annealing. However, despite the higher number of qubits, the current quantum annealers are characterised by very sparse topologies. These practical issues have led to the development of hybrid classical-quantum schemes, aiming at leveraging the strengths of both paradigms while circumventing some of the limitations of the available devices. In this thesis, several hybrid classical-quantum algorithms for optimization and machine learning are introduced and/or empirically assessed, as the empirical evaluation is a fundamental part of algorithmic research. The quantum computing models taken into account are both quantum annealing and circuit-based universal quantum computing. The results obtained have shown the effectiveness of most of the proposed approaches.

<p dir="ltr">Quantum information science and technology has been at the forefront of science and technology since MIT mathematician Peter Shor discovered a quantum algorithm to factor large numbers in 1994. Advancement in quantum theory also advances practical technological applications. Quantum technology can be applied both in civilian society and the military field from encryption, artificial intelligence, sensing, to communications. This multi-purpose applicability, therefore, has the potential to alter international security as scientifically advanced nation-states vie for quantum supremacy. This research examines the applications of quantum science and how these applications can potentially impact international security. Because nation-states fund and support quantum science research, sources of method will include academic journals and online resources as well as government reports. Practical applications of quantum technology, including quantum computing, quantum sensing, and quantum communication, will constitute the primary scope of this research.</p>

We explore the paradigm shift in quantum computing and quantum information science, emphasizing the synergy between hardware advancements and algorithm development. Only now have the recent advances in quantum computing hardware, despite a century of quantum mechanics, unveiled untapped potential, requiring innovative algorithms for full utilization. Project 1 addresses quantum applications in radiative reactions, overcoming challenges in many-fermion physics due to imaginary time evolution, stochastic methods like Monte Carlo simulations, and the associated sign problem. The methodology introduces the Electromagnetic Transition System and a general two-level system for computing radiative capture reactions. Project 2 utilizes Variational Quantum Eigensolver (VQE) to address the difficulties in adiabatic quantum computations, highlighting Singular Value Decomposition (SVD) in quantum computing. Results demonstrate an accurate ground state wavefunction match with only a 0.016% energy error. These projects advance quantum algorithm design, error mitigation, and SVD integration, showcasing quantum computing’s transformative potential in computational science.

Potential energy surfaces are fundamental to materials simulation, providing acomprehensive landscape of the energy variations as a function of atomic positions within a system. These surfaces are crucial for understanding the behavior of materials at the atomic and molecular levels. By mapping out the potential energy surface, researchers can predict stable configurations, transition states, and thermodynamic and kinetic observables, essential for the design of novel materials. However, constructing accurate potential energy surfaces presents significant challenges due to the complexity constructing a potential with nearly the same accuracy of quantum mechanical calculations, the transferability to explore the high-dimensional space of atomic configurations, and the efficiency to be evaluated without the need for extensive computational resources. In this work, group theoretical irreducible derivativemethods are used to construct accurate Taylor series expansions of the potential energy surface. A new condition number optimized basis is developed which guarantees no amplification of error at the smallest computational cost allowed by group theory. Using this highly efficient method, the physics of the metal-insulator transition in LuNiO₃ is investigated within both a purely harmonic model and simplified anharmonic model, yielding reasonable and good agreement with the experimental phase transition temperature, respectively. Then the irreducible derivatives approach is used to benchmark the phonon anharmonicity of the several popular machine learning potentials: Gaussian approximation potentials, artificial neural network potentials, and graph neural network potentials. The benchmark indicated that the graph neural network potentials had the ability to accurate reproduce quantum mechanical derivatives up to 5th-order. Finally, insights from the benchmark are used to train an accurate graph neural network potential for strongly correlated UO₂ which yielded excellent agreement with quantum mechanical calculations and experiment. These methodological advancements underscore the potential of combining grouptheoretical approaches with cutting-edge machine learning techniques to enhance the precision and efficiency of materials simulations. By achieving high accuracy in modeling complex phenomena such as phase transitions and phonon anharmonicity, this work paves the way for future studies to not only explore the design of novel materials with unprecedented properties, but to design new machine learning techniques where group theory is built from the ground up.