This thesis deals with the question of what resources are necessary and sufficient for quantum computational speedup. In particular, we study what resources are required to promote fault tolerant stabilizer computation to universal quantum computation. In this context we discover a remarkable connection between the possibility of quantum computational speedup and negativity in the discrete Wigner function, which is a particular distinguished quasi-probability representation for quantum theory. This connection allows us to establish a number of important results related to magic state computation, an important model for fault tolerant quantum computation using stabilizer operations supplemented by the ability to prepare noisy non-stabilizer ancilla states. In particular, we resolve in the negative the open problem of whether every non-stabilizer resource suffices to promote computation with stabilizer operations to universal quantum computation. Moreover, by casting magic state computation as resource theory we are able to quantify how useful ancilla resource states are for quantum computation, which allows us to give bounds on the required resources. In this context we discover that the sum of the negative entries of the discrete Wigner representation of a state is a measure of its usefulness for quantum computation. This gives a precise, quantitative meaning to the negativity of a quasi-probability representation, thereby resolving the 80 year debate as to whether this quantity is a meaningful indicator of quantum behaviour. We believe that the techniques we develop here will be widely applicable in quantum theory, particularly in the context of resource theories.

A continuous time quantum walk consists of a particle moving among the vertices of a graph G. Its movement is governed by the structure of the graph. More formally, the adjacency matrix A is the Hamiltonian that determines the movement of our particle. Quantum walks have found a number of algorithmic applications, including unstructured search, element distinctness and Boolean formula evaluation. We will examine the properties of periodicity and state transfer. In particular, we will prove a result of the author along with Godsil, Kirkland and Severini, which states that pretty good state transfer occurs in a path of length n if and only if the n+1 is a power of two, a prime, or twice a prime. We will then examine the property of strong cospectrality, a necessary condition for pretty good state transfer from u to v. We will then consider quantum walks involving more than one particle. In addition to moving around the graph, these particles interact when they encounter one another. Varying the nature of the interaction term gives rise to a range of different behaviours. We will introduce two graph invariants, one using a continuous-time multi-particle quantum walk, and the other using a discrete-time quantum walk. Using cellular algebras, we will prove several results which characterize the strength of these two graph invariants. Let A be an association scheme of n × n matrices. Then, any element of A can act on the space of n × n matrices by left multiplication, right multiplication, and Schur multiplication. The set containing these three linear mappings for all elements of A generates an algebra. This is an example of a Jaeger algebra. Although these algebras were initially developed by Francois Jaeger in the context of spin models and knot invariants, they prove to be useful in describing multi-particle walks as well. We will focus on triply-regular association schemes, proving several new results regarding the representation of their Jaeger algebras. As an example, we present the simple modules of a Jaeger algebra for the 4-cube.

Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.<p>This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithms known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.

Being a puzzling feature of quantum mechanics, entanglement caused many debates since the infancy days of quantum theory. But it is the last two decades that it has started to be seen as a resource for physical tasks which are not possible or extremely infeasible to be done classically. Popular examples are quantum cryptography - secure communication based on laws of physics - and quantum computation - an exponential speedup for factoring large integers. On the other hand, with current technological restrictions it seems to be difficult to preserve specific entangled states and to distribute them among distant parties. Therefore a precise measurement of quantum entanglement is necessary. In this thesis, common bipartite and multipartite entanglement measures in the literature are reviewed. Mathematical definitions, proofs of satisfaction of basic axioms and significant properties for each are given as far as possible. For Tangle and Geometric Measure of Entanglement, which is a multipartite measure, results of numerical calculations for some specific states are shown.

The main subject of this thesis is the investigation of the transformations of pure multipartite entangled states having Schmidt rank 2 by using only local operations assisted with classical communications (LOCC). A new parameterization is used for describing the entangled state of p particles distributed to p distant, spatially separated persons. Product, bipartite and truly multipartite states are identified in this new parametrization. Moreover, alternative parameterizations of local operations carried out by each party are provided. For the case of a deterministic transformation to a truly multipartite final state, one can find an analytic expression that determines whether such a transformation is possible. In this case, a chain of measurements by each party for carrying out the transformation is found. It can also be seen that, under deterministic LOCC transformations, there are some quantities that remain invariant. For the purpose of applying the results of this thesis in the context of the quantum information and computation, brief reviews of the entanglement purification, measurement based quantum computation and quantum codes are given.

In this thesis, We have calculated the optimized geometries, electronic structures and spin distributions of metal and non-metal elements Li, Na, N and P doped C60 fullerene dimers and trimers with different spin multiplicities using hybrid density functional theory (DFT) at the B3LYP/6-31G level of theory. Natural population analysis and Mulliken population analysis show that non-metal elements (N, P) inside the C60 fullerene dimers and trimers are well isolated and preserve their electronic structures while charge transfer processes occur between metal elements(Li, Na) and C60 structures. Energy calculations showed that both doped and undoped linear C60 structures are energetically lower than triangular C60 structures. Calculated spin density distributions make non-metal doped C60 structures advantageous over metal doped C60 cages as spin cluster qubits.

Universal quantum computation requires precision control of the dynamics of qubits. Frequently accurate quantum control is impeded by systematic drifts and other errors. Compensating composite pulse sequences are a resource efficient technique for quantum error reduction. This work describes compensating sequences for ion-trap quantum computers. We introduce a Lie-algebraic framework which unifies all known fully-compensating sequences and admits a novel geometric interpretation where sequences are treated as vector paths on a dynamical Lie algebra. Using these techniques, we construct new narrowband sequences with improved error correction and reduced time costs. We use these sequences to achieve laser addressing of single trapped 40Ca+ ions, even if neighboring ions experience significant field intensity. We also use broadband sequences to achieve robust control of 171Yb+ ions even with inhomogeneous microwave fields. Further, we generalize compensating sequences to correct certain multi-qubit interactions. We show that multi-qubit gates may be corrected to arbitrary accuracy if there exists either two non-commuting controls with correlated errors or one error-free control. A practical ion-trap quantum computer must be extendible to many trapped ions. One solution is to employ microfabricated surface-electrode traps, which are well-suited for scalable designs and integrated systems. We describe two novel surface-electrode traps, one with on-chip microwave waveguides for hyperfine 171Yb+ qubit manipulations, and a second trap with an integrated high numerical aperture spherical micromirror for enhanced fluorescence collection.

This thesis deals with the question of what resources are necessary and sufficient for quantum computational speedup. In particular, we study what resources are required to promote fault tolerant stabilizer computation to universal quantum computation. In this context we discover a remarkable connection between the possibility of quantum computational speedup and negativity in the discrete Wigner function, which is a particular distinguished quasi-probability representation for quantum theory. This connection allows us to establish a number of important results related to magic state computation, an important model for fault tolerant quantum computation using stabilizer operations supplemented by the ability to prepare noisy non-stabilizer ancilla states. In particular, we resolve in the negative the open problem of whether every non-stabilizer resource suffices to promote computation with stabilizer operations to universal quantum computation. Moreover, by casting magic state computation as resource theory we are able to quantify how useful ancilla resource states are for quantum computation, which allows us to give bounds on the required resources. In this context we discover that the sum of the negative entries of the discrete Wigner representation of a state is a measure of its usefulness for quantum computation. This gives a precise, quantitative meaning to the negativity of a quasi-probability representation, thereby resolving the 80 year debate as to whether this quantity is a meaningful indicator of quantum behaviour. We believe that the techniques we develop here will be widely applicable in quantum theory, particularly in the context of resource theories.

A physical system is said to be in topological phase if at low energies and long wavelengths the observable quantities are invariant under diffeomorphisms. Such physical systems are of great interest in condensed matter physics and computer science where they can be applied to form topological insulators and fault–tolerant quantum computers. Physical systems in topological phase may be rigorously studied through their algebraic manifestations, (pre)modular categories. A complete classification of these categories would lead to a taxonomy of the topological phases of matter. Beyond their ties to physical systems, premodular categories are of general mathematical interest as they govern the representation theories of quasi–Hopf algebras, lead to manifold and link invariants, and provide insights into the braid group. In the course of this work, we study the classification problem for (pre)modular categories with particular attention paid to their arithmetic properties. Central to our analysis is the question of rank finiteness for modular categories, also known as Wang’s Conjecture. In this work, we lay this problem to rest by exploiting certain arithmetic properties of modular categories. While the rank finiteness problem for premodular categories is still open, we provide new methods for approaching this problem. The arithmetic techniques suggested by the rank finiteness analysis are particularly pronounced in the (weakly) integral setting. There, we use Diophantine techniques to classify all weakly integral modular categories through rank 6 up to Grothendieck equivalence. In the case that the category is not only weakly integral, but actually integral, the analysis is further extended to produce a classification of integral modular categories up to Grothendieck equivalence through rank 7. It is observed that such classification can be extended provided some mild assumptions are made. For instance, if we further assume that the category is also odd–dimensional, then the classification up to Grothendieck equivalence is completed through rank 11. Moving beyond modular categories has historically been difficult. We suggest new methods for doing this inspired by our work on (weakly) integral modular categories and related problems in algebraic number theory. The allows us to produce a Grothendieck classification of rank 4 premodular categories thereby extending the previously known rank 3 classification.

I present in this thesis the design, implementation and measurement results of a narrowband quantum light source based on cavity-enhanced Parametric Down-Conversion (PDC). Spontaneous Parametric Down-Conversion (SPDC) is the workhorse in the field of optical quantum information and quantum computation, yet it is not suitable for applications where deterministic nonlinearities are required due to its low spectral brightness. By placing the nonlinear crystal inside a cavity, the spectrum of down-conversion is actively modified, such that all the non-resonant modes of down-conversion experience destructive interference, while the resonant mode sees constructive interference, resulting in great enhancement in spectral brightness. I design and construct such a cavity-enhanced down-conversion source with record high spectral brightness, making it possible to use cold atoms as the interaction medium to achieve large nonlinearity between photons. The frequency of the photons is tunable and their coherence time is measured to be on the order of 10 nanoseconds, matching the lifetime of the excited state of typical alkali atoms. I characterize extensively the output of the source by measuring the second-order correlation function, quantifying two-photon indistinguishability, performing quantum state tomography of entangled states, and showing different statistics of the source. The unprecedented long coherence time of the photon pairs has also made possible the encoding of quantum information in the time domain of the photons. I present a theoretical proposal of multi-dimensional quantum information with such long-coherence-time photons and analyze its performance with realistic parameter settings. I implement this proposal with the quantum light source I have built, and show for the first time that a qutrit can be encoded in the time domain of the single photons. I demonstrate the coherence is preserved for the qutrit state, thus ruling out any classical probabilistic explanation of the experimental data. Such an encoding scheme provides an easy access to multi-dimensional systems and can be used as a versatile platform for many quantum information and quantum computation tasks.