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.

This dissertation consists of three sections. In the first section, we discuss the generation of arbitrary two-qubit entangled states and present three generation methods. The first method is based on the interaction of an atom with classical and quantized cavity fields. The second method is based on the interaction of two coupled two-level atoms with a laser field. In the last method, we use two spin-1/2 systems which interact with a tuned radio frequency pulse. Using those methods we have generated two qubit arbitrary entangled states which is widely used in quantum computing and quantum information. In the second section, we discuss a possible experimental implementation of quantum walk which is based on the passage of an atom through a high-Q cavity. The chirality is determined by the atomic states and the displacement is characterized by the photon number inside the cavity. Our scheme makes quantum walk possible in a cavity QED system and the results could be widely used on quantum computer. In the last section, we investigate the properties of teleporting an arbitrary superposition of entangled Dicke states of any number of atoms (qubits) between two distant cavities. We also studied teleporting continuous variables of an optical field. Teleportation of Dicke states relies on adiabatic passage using multiatom dark states in each cavity and a conditional detection of photons leaking out of both cavities. In the continuous variables teleportation scheme we first reformulate the protocol of quantum teleportation of arbitrary input optical field states in the density matrix form, and established the relation between the P-function of the input and output states. We then present a condition involving squeeze parameter and detection efficiency under which the P-function of the output state becomes the Q function of the input state such that any nonclassical features in the input state will be eliminated in the teleported state. Based on the research in this section we have made it possible of arbitrary atomic Dicke states teleportation from one cavity to another, and this teleortation will play an essential role in quantum communication. Since quantum properties is so important in quantum communication, the condition we give in this section to distinguish classical and quantum teleportation is also important.

It is conjectured that quantum computers are able to solve certain problems more quickly than any deterministic or probabilistic computer. For instance, Shor's algorithm is able to factor large integers in polynomial time on a quantum computer. A quantum computer exploits the rules of quantum mechanics to speed up computations. However, it is a formidable task to build a quantum computer, since the quantum mechanical systems storing the information unavoidably interact with their environment. Therefore, one has to mitigate the resulting noise and decoherence effects to avoid computational errors. In this dissertation, I study various aspects of quantum error control codes - the key component of fault-tolerant quantum information processing. I present the fundamental theory and necessary background of quantum codes and construct many families of quantum block and convolutional codes over finite fields, in addition to families of subsystem codes. This dissertation is organized into three parts: Quantum Block Codes. After introducing the theory of quantum block codes, I establish conditions when BCH codes are self-orthogonal (or dual-containing) with respect to Euclidean and Hermitian inner products. In particular, I derive two families of nonbinary quantum BCH codes using the stabilizer formalism. I study duadic codes and establish the existence of families of degenerate quantum codes, as well as families of quantum codes derived from projective geometries. Subsystem Codes. Subsystem codes form a new class of quantum codes in which the underlying classical codes do not need to be self-orthogonal. I give an introduction to subsystem codes and present several methods for subsystem code constructions. I derive families of subsystem codes from classical BCH and RS codes and establish a family of optimal MDS subsystem codes. I establish propagation rules of subsystem codes and construct tables of upper and lower bounds on subsystem code parameters. Quantum Convolutional Codes. Quantum convolutional codes are particularly well-suited for communication applications. I develop the theory of quantum convolutional codes and give families of quantum convolutional codes based on RS codes. Furthermore, I establish a bound on the code parameters of quantum convolutional codes - the generalized Singleton bound. I develop a general framework for deriving convolutional codes from block codes and use it to derive families of non-catastrophic quantum convolutional codes from BCH codes. The dissertation concludes with a discussion of some open problems.

Quantum computing resources have been known to provide speed-ups in computational complexity in many algorithms. The impact of these resources in communication, however, has not attracted much attention. We investigate the impact of quantum entanglement on communication complexity. We provide a positive result, by presenting a class of multi-party communication problems wherein the presence of a suitable quantum entanglement lowers the classical communication complexity. We show that, in evaluating certains function whose parameters are distributed among various parties, the presence of prior entanglement can help in reducing the required communication. We also present an outline of realizing the required entanglement through optical photon quantum computing. We also suggest the possible impact of our results on network information flow problems, by showing an instance of a lower bound which can be broken by adding limited power to the communication model.

First-principles electronic structure theory explains properties of atoms, molecules and solids from underlying physical principles without input from empirical parameters. Time-dependent density functional theory (TDDFT) has emerged as arguably the most widely used first-principles method for describing the time-dependent quantum mechanics of many-electron systems. In this thesis, we will show how the fundamental principles of TDDFT can be extended and applied in two novel directions: The theory of open quantum systems (OQS) and quantum computation (QC). In the first part of this thesis, we prove theorems that establish the foundations of TDDFT for open quantum systems (OQS-TDDFT). OQS-TDDFT allows for a first principles description of non-equilibrium systems, in which the electronic degrees of freedom undergo relaxation and decoherence due to coupling with a thermal environment, such as a vibrational or photon bath. We then discuss properties of functionals in OQS-TDDFT and investigate how these differ from functionals in conventional TDDFT using an exactly solvable model system. Next, we formulate OQS-TDDFT in the linear-response regime, which gives access to environmentally broadened excitation spectra. Lastly, we present a hybrid approach in which TDDFT can be used to construct master equations from first-principles for describing energy transfer in condensed phase systems. In the second part of this thesis, we prove that the theorems of TDDFT can be extended to a class of qubit Hamiltonians that are universal for quantum computation. TDDFT applied to universal Hamiltonians implies that single-qubit expectation values can be used as the basic variables in quantum computation and information theory, rather than wavefunctions. This offers the possibility of simplifying computations by using the principles of TDDFT similar to how it is applied in electronic structure theory. Lastly, we discuss a related result; the computational complexity of TDDFT. / Physics

Josh Folk explains how the traditional rules of physics don't make sense at the quantum-mechanical level - and how those discrepancies can be turned into opportunities.

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.

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 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.

In this work, we mostly concentrate on the hidden shift problem for non-injective functions. It is worthwhile to know that the query complexity of the non-injective hidden shift problem is exponential in the worst case by the well known bounds on the unstructured search problem. Hence, we can make this problem more tractable by imposing additional constraints on the problem. Perhaps the first constraint that comes to mind is to address the average case problem. In this work, we show that the average case non-injective hidden shift problem can be reduced to the injective hidden shift problem by giving one such reduction. The reduction is based on a tool we developed called injectivization. The result is strong in the sense that the underlying group can be any finite group and that the non-injective functions for which we have defined the hidden shift problem can have range in an arbitrary finite set. Using this tool, we simplify the main result of a recent paper by about the hidden shift problem for Boolean-valued functions by reducing that problem to Simon's problem. They also posed an open question which is subject to personal interpretation. We answer the seemingly most general interpretation of the question. However, we use our own techniques in doing so (the authors ask if their techniques can be used for addressing that problem). Another constraint that one can consider is to have a promise on the structure of the functions. In this work we consider the hidden shift problem for c-almost generalized bent functions. A class of functions which we defined that includes the generalized bent functions. Then we turn our attention toward the generalized hidden shift problem which is easier than injective hidden shift problem and hence more tractable. We state some of our observations about this problem. Finally we show that the average classical query complexity of the non-injective hidden shift problem over groups of form (Z/mZ)^n when m is a constant is exponential, which also immediately implies that the classical average query complexity of the non-injective hidden shift problem is exponential. We also show that the worst-case classical query complexity of the generalized injective hidden shift problem over the same group is high, which implies that the classical query complexity of the hidden shift problem is high.