A reliable large-scale quantum computer, if built, can solve many real-life problems exponentially faster than the existing digital devices. The biggest obstacle to building one is that they are extremely sensitive and error-prone regardless of the selection of physical implementation. Both data storage and data manipulation require careful implementation and precise control due to its quantum mechanical nature. For the development of a practical and scalable computer, it is essential to identify possible quantum errors and reduce them throughout every layer of the hierarchy of quantum computation. In this dissertation, we present our investigation into new methods to reduce errors in quantum computers from three different directions: quantum memory, quantum control, and quantum error correcting codes. For quantum memory, we pursue the potential of the quantum equivalent of a magnetic hard drive using two-body-interaction structures in fractal dimensions. With regard to quantum control, we show that it is possible to arbitrarily reduce error when manipulating multiple quantum bits using a technique popular in nuclear magnetic resonance. Finally, we introduce an efficient tool to study quantum error correcting codes and present analysis of the codes' performance on model quantum architectures.

Open quantum systems refer to systems that are affected by interaction with the environment. The effects of these unwanted interactions, called \emph{quantum noise}, are studied using dynamical maps. We study the geometry of these maps in this work. We review the canonical representations of dynamical maps such as reduced dynamics, $\mathcal{A}$ and $\mathcal{B}$ forms and operator sum representation. We develop a framework for simplifying the action of dynamical maps in terms of their action on the coherence vector associated with the density matrix. We use the framework to describe the geometry of depolarization, dephasing and dissipation in the domain of complete positivity. We give a geometric picture of how two-, three- and four-level systems are affected by these common forms of quantum noises. We show useful similarities between two- and four-level depolarizing maps and give a generalization for $n$-qubits. We also derive important results that restrict dephasing and dissipation. / text

This thesis is concerned with various aspects of ion traps and their use as a quantum simulation and computation device. In its first part we investigate various sources of noise and decoherence in ion traps. As quantum information is very fragile, a detailed knowledge of noise and decoherence sources in a quantum computation device is essential. In the special case of an ion trap quantum computer we investigate the effects of intensity and phase noise in the laser, which is used to perform the gate operations. We then look at other sources of noise which are present without a laser being switched on. These are fluctuations in the trapping frequency caused by noise in the electric potentials applied to the trap and fluctuating electrical fields which will cause heating of the centre-of-mass vibrational state of the ions in the trap. For the case of fluctuating electrical fields we estimate the effect on a quantum gate operation. We then propose a scheme for performing quantum gates without having the ions cooled down to their motional ground state. The second part deals with various aspects of the use of ion traps as a device for quantum computation. We start with the use of ionic qubits as a measurement device for the centre-of-mass vibrational mode and investigate in detail the effect these measurements will have on the vibrational mode. If one wants to use quantum computation devices as systems to simulate quantum mechanics, it is of interest to know how to simulate say a k-level system with N qubits. We investigate the easiest case of this wider problem and look at how to simulate a three-level system (a so called trit) with two qubits in an ion trap quantum computer. We show how to get and measure a SU (3) geometric phase with this toy model. Finally we investigate how to simulate collective angular momentum models with a string of qubits in an ion trap. We assume that the ionic qubits are coupled to a thermal reservoir and derive a master equation for this case. We investigate the semiclassical limit of this master equation and, in the case for two qubits in the trap, determine the entanglement of the steady state. We also outline a way to find the steady state for the master equation using coherence vectors.

This thesis considers the use of quantum mechanics for information security related tasks. Two secure quantum bit commitment protocols are introduced and the security of the protocols against attackers is discussed. The use of quantum entanglement breaking channels for making a protocol secure is considered and some security bounds are given. Entanglement measurement in multipartite systems and a universal entanglement measure are also introduced and discussed. / Graduate

In this thesis, we discuss several phenomena exhibiting `limiting behaviour' in physics. This includes the duality principle, delegated quantum computation, and super-resolution. The duality principle places a limit on the coexistence of wave and particle behaviours. We develop a framework that explains apparent violations of this principle while staying within the scope of quantum mechanics. In addition, we relate the duality principle to the sub-fidelity and weak-values. We also show that the maximum recoverable coherence of a qubit has a sharp transition from 0 to 1 when we have access to half of the environment to which the qubit is correlated. Delegated quantum computation consists of a computational weak client who wishes to delegate a complex quantum computation to a powerful quantum server. We develop a new protocol for delegated quantum computation requiring less quantum power than its predecessor. Finally, we develop and test a new theory for eigenmode super-resolution.

In this thesis, classical computation is first introduced. Finite quantum systems are considered with D-dimensional Hilbert space, and position x and momentum p taking values in Z(D) (the integers modulo D). An analytic rep resentation of finite quantum systems that use Theta function is presented and considered. The first novel part of this thesis is contribution to study reversible classical CNOT gates and their binary inputs and outputs with reversible cir cuits. Furthermore, a reversible classical Toffoli gates are considered, as well as implementation of a Boolean expression with classical CNOT and Toffoli gates. Reversible circuits with classical CNOT and Toffoli gates are also considered. The second novel part of this thesis the study of quantum computation in terms of CNOT and Toffoli gates. Analytic representations and their zeros are considered, while zeros of the inputs and outputs for quantum CNOT and Toffoli gates are studied. Also, approximate computation of their zeros on the output are calculated. Finally, some quantum circuits are discussed. i

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<P> Trapped ions are one of the best candidate systems to realize quantum computation. In our laboratory, we are trying to implement quantum computing and information processing: two hyperfine ground-states of magnesium-25 ions will serve as the two-level system to store quantum information. The ions are confined in a linear radio-frequency trap under ultra-high vacuum conditions and will be cooled down to their motional ground-states. By illuminating the ions with frequency-stabilized lasers we will be able to initialize, manipulate, and read out their internal electronic quantum states in a well-controlled way and with high fidelity. In addition, the ions can be made to interact with each other by coupling their internal electronic states to a collective vibrational mode of motion along the trap axis. In this thesis, the focus will be on the process of building a trapped-magnesium-ion quantum information processor. </p> / Thesis / Master of Science (MSc)

Measurement-based quantum computation is a form of quantum computing that operates on a prepared entangled graph state, typically a cluster state. In this dissertation, we will detail the creation of graph states across various physical platforms using different entangling gates. We will then benchmark the quality of graph states created with error-prone interactions through quantum wire teleportation experiments. By leveraging underlying symmetry, we will design graph states as measurement-based quantum error correction codes to protect against perturbations, such as ZZ crosstalk in quantum wire teleportation. Additionally, we will explore other measurement-based algorithms used for the quantum simulation of time evolution in fermionic systems, using the Kitaev model and the Hubbard model as examples. / Doctor of Philosophy / A quantum computer refers to a device that performs general computational functions relying on logic gates using units dominated by microscopic quantum properties. The fundamental difference between quantum computers and classical computers lies in the distinction be- tween the basic quantum unit, the qubit, and the classical computational unit, the bit. Both qubits and bits can exist in states 0 and 1. However, qubits possess two characteristics that classical computational units do not: superposition and entanglement. Superposition allows a qubit to exist in a combination of both states 0 and 1 simultaneously. Entanglement refers to the phenomenon where qubits interact and form an inseparable unified state. The effec- tive utilization of these unique properties enables quantum computers to exhibit capabilities far surpassing those of classical computers. Analogous to classical computers, qubits can be interconnected in a circuit-like manner sim- ilar to classical bits, forming an architecture known as circuit-based quantum computation (CBQC). However, given the unique properties of quantum systems, particularly entan- glement, a novel architecture called measurement-based quantum computing (MBQC) can also be designed. MBQC relies on pre-entangled graph states, usually cluster states, and only requires single-qubit measurements to implement quantum algorithms. The MBQC framework also includes a universal gate set, similar to other quantum computing architec- tures like CBQC. In this dissertation, we will introduce the creation of graph states and the implementation of measurement-based quantum algorithms.

<p dir="ltr">This thesis advances the methodologies of quantum state tomography (QST) to validate and optimize quantum processing on Noisy Intermediate-Scale Quantum (NISQ) devices, crucial for the transition to practical quantum systems. Inspired by recent advancements in the field, we propose a novel QST method based on the maximal entropy formalism, specifically addressing scenarios with incomplete measurement sets to provide a robust framework for state reconstruction. We extend this formalism to an informationally complete (IC) set of observables and introduce a variational approach for quantum state preparation, easily implementable on near-term quantum devices. Our developed maximal entropy-based QST protocol is applied to ultrafast molecular dynamics specifically for studying photoexcited ammonia molecule, enabling direct measurement and manipulation of electronic quantum coherences and exploring entanglement effects in molecular systems. Through this approach, we achieve a groundbreaking milestone by, for the first time, constructing the entanglement entropy of the electronic subsystem - an otherwise inaccessible metric. In doing so it also provides the first physical interpretation of the maximal entropy parameters in an experimental setting and highlights the potential for feedback between time-resolved quantum dynamics and quantum information science. Furthermore, building upon our advancements in state tomography, we propose a variational quantum algorithm for Hamiltonian learning that leverages the time dynamics of observables. Additionally, we reverse engineer the maximal entropy approach and demonstrate the use of entropy to refine the traditional geometric Brownian motion (GBM) method for better capturing real system complexities by addressing its log-normality restrictions, which opens new avenues for quantum sampling techniques. Through these contributions, this thesis showcases the Maximal Entropy formalism’s efficacy in QST and set the stage for future innovations and applications in cutting-edge quantum research.</p>