Quantum error correction codes were introduced as a means to protect quantum information from decoherance and operational errors. Based on their approach to error control, error correcting codes can be divided into two different classes: block codes and convolutional codes. There has been significant development towards finding quantum block codes, since they were first discovered in 1995. In contrast, quantum convolutional codes remained mainly uninvestigated. In this thesis, we develop the stabilizer formalism for quantum convolutional codes. We define distance properties of these codes and give a general method for constructing encoding circuits, given a set of generators of the stabilizer of a quantum convolutional stabilizer code, is shown. The resulting encoding circuit enables online encoding of the qubits, i.e., the encoder does not have to wait for the input transmission to end before starting the encoding process. We develop the quantum analogue of the Viterbi algorithm. The quantum Viterbi algorithm (QVA) is a maximum likehood error estimation algorithm, the complexity of which grows linearly with the number of encoded qubits. A variation of the quantum Viterbi algorithm, the Windowed QVA, is also discussed. Using Windowed QVA, we can estimate the most likely error without waiting for the entire received sequence.

Quantum error correction and fault-tolerance are at the heart of any scalable quantum computation architecture. Developing a set of tools that satisfy the requirements of fault- tolerant schemes is thus of prime importance for future quantum information processing implementations. The Clifford gate set has the desired fault-tolerant properties, preventing bad propagation of errors within encoded qubits, for many quantum error correcting codes, yet does not provide full universal quantum computation. Preparation of magic states can enable universal quantum computation in conjunction with Clifford operations, however preparing magic states experimentally will be imperfect due to implementation errors. Thankfully, there exists a scheme to distill pure magic states from prepared noisy magic states using only operations from the Clifford group and measurement in the Z-basis, such a scheme is called magic state distillation [1]. This work investigates the robustness of magic state distillation to faults in state preparation and the application of the Clifford gates in the protocol. We establish that the distillation scheme is robust to perturbations in the initial state preparation and characterize the set of states in the Bloch sphere that converge to the T-type magic state in different fidelity regimes. Additionally, we show that magic state distillation is robust to low levels of gate noise and that performing the distillation scheme using noisy Clifford gates is a more efficient than using encoded fault-tolerant gates due to the large overhead in fault-tolerant quantum computing architectures.

Quantum error correction codes were introduced as a means to protect quantum information from decoherance and operational errors. Based on their approach to error control, error correcting codes can be divided into two different classes: block codes and convolutional codes. There has been significant development towards finding quantum block codes, since they were first discovered in 1995. In contrast, quantum convolutional codes remained mainly uninvestigated. In this thesis, we develop the stabilizer formalism for quantum convolutional codes. We define distance properties of these codes and give a general method for constructing encoding circuits, given a set of generators of the stabilizer of a quantum convolutional stabilizer code, is shown. The resulting encoding circuit enables online encoding of the qubits, i.e., the encoder does not have to wait for the input transmission to end before starting the encoding process. We develop the quantum analogue of the Viterbi algorithm. The quantum Viterbi algorithm (QVA) is a maximum likehood error estimation algorithm, the complexity of which grows linearly with the number of encoded qubits. A variation of the quantum Viterbi algorithm, the Windowed QVA, is also discussed. Using Windowed QVA, we can estimate the most likely error without waiting for the entire received sequence.

Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. Quantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. This thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. Three techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.

Understanding the performance of realistic noisy encoded circuits is an important task for the development of large-scale practical quantum computers. Specifically, the development of proposals for quantum computation must be well informed by both the qualities of the low-level physical system of choice, and the properties of the high-level quantum error correction and fault-tolerance schemes. Gaining insight into how a particular computation will play out on a physical system is in general a difficult problem, as the classical simulation of arbitrary noisy quantum circuits is inefficient. Nevertheless, important classes of noisy circuits can be simulated efficiently. Such simulations have led to numerical estimates of threshold errors rates and resource estimates in topological codes subject to efficiently simulable error models. This thesis describes and analyzes a method that my collaborators and I have introduced for leveraging efficient simulation techniques to understand the performance of large quantum processors that are subject to errors lying outside of the efficient simulation algorithm's applicability. The idea is to approximate an arbitrary gate error with an error from the efficiently simulable set in a way that ``honestly'' represents the original error's ability to preserve or distort quantum information. After introducing and analyzing the individual gate approximation method, its utility as a means for estimating circuit performance is studied. In particular, the method is tested within the use-case for which it was originally conceived; understanding the performance of a hypothetical physical implementation of a quantum error-correction protocol. It is found that the method performs exactly as desired in all cases. That is, the circuits composed of the approximated error models honestly represent the circuits composed of the errors derived from the physical models.

Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. Quantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. This thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. Three techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.

Quantum error correction and fault-tolerance are at the heart of any scalable quantum computation architecture. Developing a set of tools that satisfy the requirements of fault- tolerant schemes is thus of prime importance for future quantum information processing implementations. The Clifford gate set has the desired fault-tolerant properties, preventing bad propagation of errors within encoded qubits, for many quantum error correcting codes, yet does not provide full universal quantum computation. Preparation of magic states can enable universal quantum computation in conjunction with Clifford operations, however preparing magic states experimentally will be imperfect due to implementation errors. Thankfully, there exists a scheme to distill pure magic states from prepared noisy magic states using only operations from the Clifford group and measurement in the Z-basis, such a scheme is called magic state distillation [1]. This work investigates the robustness of magic state distillation to faults in state preparation and the application of the Clifford gates in the protocol. We establish that the distillation scheme is robust to perturbations in the initial state preparation and characterize the set of states in the Bloch sphere that converge to the T-type magic state in different fidelity regimes. Additionally, we show that magic state distillation is robust to low levels of gate noise and that performing the distillation scheme using noisy Clifford gates is a more efficient than using encoded fault-tolerant gates due to the large overhead in fault-tolerant quantum computing architectures.

The development of a quantum computer presents one of the greatest challenges in science and engineering to date. The promise of more ecient computing based on entangled quantum states and the superposition principle has led to a worldwide explosion of interest in the elds of quantum information and computation. Decoherence is one of the main problems that gives rise to dierent errors in the quantum system. However, the discovery of quantum error correction and the establishment of the accuracy threshold theorem provide us comprehensive tools to build a quantum computer. This thesis contributes to this eort by investigating a particular class of quantum error correcting codes, called Decoherence free subsystems. The passive approach to error correction taken by these encodings provides an ecient means of protection for symmetrically coupled system-bath interactions. Here I will present methods for determining the subsystem-preserving evolutions for noiseless subsystem encodings and more importantly implementing a Universal quantum computing over three-quantum dots.

Fault tolerant quantum error correction is a procedure which satisfies the feature that if one of the gates in the procedure has failed then the failure causes at most one error in the output qubits of the encoded block. Quantum computer is based on the idea of two quantum state systems (Qubits). However, the majority of systems are constructed from higher than two- level subspace. Bad control and environmental interactions in these systems lead to leakage fault. Leakage errors are errors that couple the states inside a code subspace to the states outside a code subspace. One example for leakage fault is loss errors. Since the fault tolerant procedure may be unable to recognize the leakage fault because it was designed to deal with Pauli errors. In that case a single leakage fault might disrupt the fault tolerant technique. In this thesis we investigate the effect of ancilla losses on fault-tolerant quantum error correction in the [[7,1,3]] Steane code. We proved that both Shor and Steane methods are still fault tolerant if loss errors occur.

Quantum computing has enormous potential, but this can only be realised if quantum errors can be controlled sufficiently to allow quantum algorithms to be completed reliably. However, quantum-error-corrected logical quantum bits (qubits) which can be said to have achieved meaningful error suppression have not yet been demonstrated. This thesis reports research on several topics related to the challenge of designing fault-tolerant quantum computers. The first topic is a proposal for achieving large-scale error correction with the surface code in a silicon donor based quantum computing architecture. This proposal relaxes some of the stringent requirements in donor placement precision set by previous ideas from the single atom level to the order of 10 nm in some regimes. This is shown by means of numerical simulation of the surface code threshold. The second topic then follows, it is the development of a method for benchmarking and assessing the performance of small error correcting codes in few-qubit systems, introducing a metric called 'integrity' - closely linked to the trace distance -- and a proposal for experiments to demonstrate various stepping stones on the way to 'strictly superior' quantum error correction. Most quantum error correcting codes, including the surface code, do not allow for fault-tolerant universal computation without the addition of extra gadgets. One method of achieving universality is through a process of distilling and then consuming high quality 'magic states'. This process adds additional overhead to quantum computation over and above that incurred by the use of the base level quantum error correction. The latter parts of this thesis report an investigation into how many physical qubits are needed in a `magic state factory' within a surface code quantum computer and introduce a number of techniques to reduce the overhead of leading magic state techniques. It is found that universal quantum computing is achievable with ∼ 16 million qubits if error rates across a device are kept below 10^-4. In addition, the thesis introduces improved methods of achieving magic state distillation for unconventional magic states that allow for logical small angle rotations, and show that this can be more efficient than synthesising these operations from the gates provided by traditional magic states.