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.

Thesis (MTech (Electrical Engineering))--Cape Peninsula University of Technology, 2018. / In this research, we undertake a numerical study of the interaction between laser beams and two as well as multi-level atoms. The main aim of this research is to obtain a deeper understanding of laser-atom interactions and non-linear processes such as optical four-wave mixing. This work will supplement experiments to be conducted by other members of the group, who are involved in generating entangled photons via four-wave mixing in cold rubidium atoms. We begin by performing a basic study of the interaction between laser beams and two-level atoms as an aid to gain knowledge of numerical techniques, as well as an understanding of the physics behind light-atom interactions. We make use of a semi-classical approach to describe the system where the atoms are treated quantum mechanically and the laser beams are treated classically. We study the interaction between atoms and laser beams using the density matrix operator and Maxwell's equations respectively. By solving the optical Bloch equations for two-level atoms we examine the atomic populations and coherences and present plots of the density matrix elements as a function of time. The e ects of various parameters such as laser intensity, detuning and laser modulation have been tested. The behaviour of the laser beam as it propagates through the atomic sample is also studied. This is determined by Maxwell's equation where the atomic polarization is estimated from the coherence terms of the density matrix elements. / French South African Institute of Technology National Research Foundation

An investigation of the electronic structure of some transition metal clusters comprising anti-ferromagnetically coupled, heterometallic arrays of eight metal ions that are wheel-shaped, is reported. The compounds were synthesized and provided by Dr. Grigore Timco of The University of Manchester and are formulated by their metal content as Cr7M, where M = a divalent 3d metal. Two families of wheels are the subject of this research, defined ‘green’ and ‘purple’ from their physical appearance. Within each family, the compounds are all isostructural. From simulation using a single Hamiltonian for Cr7M-purple compounds, where M = Zn, Mn, or Ni, it is shown that with only two exchange parameters, one JCr-Cr and one JCr-M, data from bulk magnetization, neutron scattering, Electron Paramagnetic Resonance (EPR) spectroscopy at multiple frequencies and specific heat measurements can be modelled and that there is transferability of parameters. Preliminary attempts to measure electron spin relaxation times for two of the purple wheels have shown values of T1 and T2 that are comparable with those of the more extensively studied green wheels and hence further studies in this area are warranted. Variable temperature Q- and W-band EPR spectra for a series of nine heterodimers comprising one green and one purple wheel, M=Zn, Mn or Ni in each case, are reported. For Cr7Zn-purple there is no magnetic exchange detected, whereas weak and quantifiable exchange is required to interpret the spectra from the other six dimers. EPR studies of three trimers of the form purple-green-purple are reported and the presence of magnetic exchange is identified by comparison with the spectra of the component single and double wheel compounds, although this is not quantified because of the numerical size of the simulations that are required. The process of comparing simulated to experimental spectra is a complex problem and one which is central to the work reported in this thesis. The problem of fitting has been investigated and two novel solutions, one based upon pixel mapping and the other based on wavelet transformation are proposed.

An adversary armed with a quantum computer has algorithms[66, 33, 34] at their disposal, which are capable of breaking our current methods of encryption. Even with the birth of post-quantum cryptography[52, 62, 61], some of best cryptanalytic algorithms are still quantum [45, 8]. This thesis contains several experiments on the efficacy of lattice reduction algorithms, BKZ and LLL. In particular, the difficulty of solving Learning With Errors is assessed by reducing the problem to an instance of the Unique Shortest Vector Problem. The results are used to predict the behavior these algorithms may have on actual cryptographic schemes with security based on hard lattice problems. Lattice reduction algorithms require several floating-point operations including multiplication. In this thesis, I consider the resource requirements of a quantum circuit designed to simulate floating-point multiplication with high precision. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

This project was motivated by the following question: what information do the properties of a random graph contain about the performance of a quantum search acting on it? To investigate this problem, we define a notion of search time to quantify the behavior of a quantum search, and find strong evidence of a relation between its distribution and the model of random graph on which the search was performed. Surprisingly, we also find strong evidence that the return time of a classical random walk initialized at the marked vertex is closely related to its search time, and that the distribution of degrees over the graph vertices may play a significant role in this relation. / Mathematics

In this dissertation we discuss decoherence in charge qubits formed by multiple lateral quantum dots in the framework of the spin-boson model and the Born-Markov approximation. We consider the intrinsic decoherence caused by the coupling to bulk phonon modes and electromagnetic environmental fluctuations. In the case of decoherence caused by phonon coupling, two distinct quantum dot configurations are studied and proposed as setups that mitigate its nocive effects : (i) Three quantum dots in a ring geometry with one excess electron in total and (ii) arrays of quantum dots where the computational basis states form multipole charge configurations. For the three-dot qubit, we demonstrate the possibility of performing one- and two-qubit operations by solely tuning gate voltages. Compared to a previous proposal involving a linear three-dot spin qubit, the three-dot charge qubit allows for less overhead on two-qubit operations. For small interdot tunnel amplitudes, the three-dot qubits have Q factors much higher than those obtained for double-dot systems. The high-multipole dot configurations also show a substantial decrease in decoherence at low operation frequencies when compared to the double-dot qubit. We also discuss decoherence due to electromagnetic fluctuations in charge qubits formed by two lateral quantum dots. We use effective circuit models to evaluate correlations of voltage fluctuations in the qubit setup. These correlations allows us to estimate energy (T1) and phase (T2) relaxation times of the the qubit system. We also discuss the dependence the quality factor Q shows with respect to parameters of the setup, such as temperature and capacitive coupling between the electrodes.

Trapped-ion systems are a promising route toward the realization of both near-term and universal quantum computers. However, one of the pressing challenges is improving the fidelity of two-qubit entangling gates. These operations are often implemented by addressing individual ions with laser pulses using the M\o lmer-S\o rensen (MS) protocol. Amplitude modulation (AM) is a well-studied extension of this protocol, where the amplitude of the laser pulses is controlled as a function of time. We present an analytical study of AM using a Fourier series expansion so that the laser amplitude may be represented as a general continuous function. Varying the Fourier coefficients used to generate the pulse produces trade-offs between the laser power, gate time, and fidelity. We specifically study gate-timing errors, and we have shown that the sensitivity of the fidelity to these errors can be improved without a significant increase in the average laser power or the gate time. We plot atomic population vs. time for both the traditional MS protocol and the protocol with AM, highlighting the increased robustness of the AM gates. Our central result is that we improve the leading order dependence on gate timing errors from $\order{\Delta t^2}$ to $\order{\Delta t^6}$, and the protocol allows for arbitrarily high orders of scaling to be achieved in principle.

No description available.

Quantum information and quantum computing (QIQC) systems, relying on the phenomena of superposition and entanglement, offer the potential for vast improvements in certain computations. A practical QC realization requires maintaining the stored information for time-scales long enough to implement algorithms. One primary cause of information loss is decoherence, i.e., the loss of coherence between two energy levels in a quantum system. This work attributes decoherence to dissipation occurring as the system evolves and uses steepest-entropy-ascent quantum thermodynamics (SEAQT) to predict the evolution of system state. SEAQT asserts that, at any instant of time, the system state evolves such that the rate of system entropy change is maximized while conserving system energy. With this principle, the SEAQT equation of motion is applicable to systems in any state, near or far from stable equilibrium, making SEAQT particularly well suited for predicting the dissipation occurring as quantum algorithms are implemented. In the present research, the dynamics of qubits (quantum-bits) using the SEAQT framework are first examined during common quantum gates (combinations of which form algorithms). This is then extended to modeling a system of multiple qubits implementing Shor's algorithm on a nuclear-magnetic-resonance (NMR) QC. Additionally, the SEAQT framework is used to predict experimentally observed dissipation occurring in a two-qubit NMR QC undergoing a so called ``quenching'' process. In addition, several methods for perturbing the density or so-called ``state'' operator used by the SEAQT equation of motion subject to an arbitrary set of expectation value constraints are presented. These are then used as the basis for randomly generating states used in analyzing the dynamics of entangled, non-interacting systems within SEAQT. Finally, a reservoir interaction model is developed for general quantum systems where each system locally experiences a heat interaction with an external reservoir. This model is then used as the basis for developing a decoherence control scheme, which effectively transfers entropy out of the QIQC system as it is generated, thus, reducing the decoherence. Reservoir interactions are modeled for single qubits and the control scheme is employed in modeling an NMR QC and shown to eliminate nearly all of the noise caused by decoherence/dissipation. / Doctor of Philosophy / Quantum computers (QCs) have the potential to perform certain tasks much more efficiently than today0 s supercomputers. One primary challenge in realizing a practical QC is maintaining the stored information, the loss of which is known as decoherence. This work attributes decoherence to dissipation (a classical analogue being heat generated due to friction) occurring while an algorithm is run on the QC. Standard quantum modeling approaches assume that for any dissipation to occur, the QC must interact with its environment. However, in this work, steepest-entropy-ascent quantum thermodynamics (SEAQT) is used to model the evolution of the QC as it runs an algorithm. SEAQT, developed by Hatsopolous, Gyftopolous, Beretta, and others over the past 40 years, supplements the laws of quantum mechanics with those of thermodynamics and in contrast to the standard quantum approaches does not require the presence of an environment to account for the dissipation which occurs. This work first applies the SEAQT framework to modeling single qubits (quantum bits) to characterize the effect of dissipation on the information stored on the qubit. This is later extended to a nuclear-magnetic-resonance (NMR) QC of 7 qubits. Additionally, SEAQT is used to predict experimentally observed dissipation in a two-qubit NMR QC. Afterwards, several methods for constrained perturbations of a QC0 s state are presented. These methods are then used with SEAQT to analyze the effect of dissipation on the entanglement of two qubits. Finally, a model is derived within the SEAQT framework accounting for a qubit interacting with its environment, which is at a constant temperature. This model is then used to develop a method for limiting the decoherence and shown to significantly lowering the resulting error due to decoherence.

Implementing high-fidelity quantum control and suppressing the unwanted environmental noise has been one of the essential challenges in developing quantum information technologies. In the past, driving pulse sequences based on Dirac delta functions or square wave functions, such as Hahn spin echo or CPMG, have been developed to dynamically correcting the noise effects. However, implementing these ideal pulses with high fidelity is a challenging task in real experiments. In this thesis, we provide a new and simple method to explore the entire solution space of driving pulse shapes that suppress environmental noise in the evolution of the system. In this method, any single-qubit phase gate that is first-order robust against quasi-static transversal noise corresponds to a closed curve on a two-dimensional plane, and more general first-order robust single-qubit gates correspond to closed three-dimensional space curves. Second-order robust gates correspond to closed curves having the property that their projection onto any two-dimensional planes shall enclose a zero net area. The driving pulse shapes that implement the gates can be determined by the curvature, torsion, and the length of the curve. By utilizing the framework it is possible to obtain globally optimal solutions in pulse shaping in respect of experimental constraints by mapping them into geometrical optimization problems. One such problem we solved is to prove that the fastest possible single-qubit phase gates that are second-order noise-resistant shall be implemented using sign-flipping square functions. Since square waves are not experimentally feasible, we provide a method to smooth these pulses with minimal loss in gate speed while maintaining the robustness, based on the geometrical framework. This framework can also be useful in diagnosing the noise-cancellation properties of pulse shapes generated from numerical methods such as GRAPE. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation. / Doctor of Philosophy / Controlling a quantum system with high-fidelity is one of the main challenges in developing quantum information technologies, and it is an essential task to reduce the error caused by unwanted environmental noise. In this thesis, we developed a new geometrical formalism that enables us to explore all possible driving fields and provides a simple recipe to generate an infinite number of experimentally feasible driving pulse shapes for implementing quantum gates. We show that single-qubit operations that could suppress quasi-static noise to first-order correspond to closed three-dimensional space curves, and single-qubit gates that are second-order robust correspond to closed curves with zero enclosed net area. This simple geometrical framework can be utilized to obtain optimal solutions in quantum control problems, and can also be used as a method to diagnose driving pulse shapes generated from other means. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation.