In 2003, Radford introduced a new method to construct simple modules for the Drinfel’d double of a graded Hopf algebra. Until then, simple modules for such algebras were usually constructed by taking quotients of Verma modules by maximal submodules. This new method gives a more explicit construction, in the sense that the simple modules are given as subspaces of the Hopf algebra and one can easily find spanning sets for them. I use this method to study the representations of two types of pointed Hopf algebras: restricted two-parameter quantum groups, and the Drinfel’d double of rank one pointed Hopf algebras of nilpotent type. The groups of group-like elements of these Hopf algebras are abelian; hence, they fall among those Hopf algebras classified by Andruskiewitsch and Schneider. I study, in particular, under what conditions a simple module can be factored as the tensor product of a one dimensional module with a module that is naturally a module for a special quotient. For restricted two-parameter quantum groups, given θ a primitive ℓth root of unity, the factorization of simple uθy,θz (sln)-modules is possible, if and only if gcd((y − z)n, ℓ) = 1. I construct simple modules using the computer algebra system Singular::Plural and present computational results and conjectures about bases and dimensions. For rank one pointed Hopf algebras, given the data D = (G, χ, a), the factorization of simple D(HD)-modules is possible if and only if |χ(a)| is odd and |χ(a)| = |a| = |χ|. Under this condition, the tensor product of two simple D(HD)-modules is completely reducible, if and only if the sum of their dimensions is less or equal than |χ(a)| + 1.

Quantum communication between two remote locations often involves remote parties sharing an entangled quantum state. At present, entanglement distribution is usually performed using photons transmitted through optical fibers. However, the absorption of light in the fiber limits the communication distances to less than 200 km, even for optimal photon telecom wavelengths. To increase this distance, the quantum repeater idea was proposed. In the quantum repeater architecture, one divides communication distance into segments of the order of the attenuation length of the photons and places quantum memory nodes at the intermediate locations. Since the photon loss between intermediate locations is low, it is possible then to establish entanglement between intermediate quantum memory nodes. Once entanglement between adjacent nodes is established, one can extend it over larger distances using entanglement swapping. The long coherence time of a quantum memory is a crucial requirement for the quantum repeater protocol. It is obvious that the coherence time of a quantum memory should be much longer that the time it takes for light to travel between remote locations. For a communication distance l = 1000 km, the corresponding time is t = l/c = 3.3 ms. One can show that for a simple repeater protocol and realistic success probabilities of entanglement generation, the required coherence time should be on the order of many seconds, while for the more complicated protocols that involve multiplexing and several quantum memory cells per intermediate node, the required coherence time is on the order of milliseconds. In this thesis, I describe a quantum memory based on an ensemble of rubidium atoms confined in a one-dimensional optical lattice. The use of the magnetically- insensitive clock transition and suppression of atomic motion allows us to increase coherence time of the quantum memory by two-orders of magnitude compared to previous work. I also propose a method for determining the Zeeman content of atomic samples. In addition, I demonstrate the observation of quantum evolution under continuous measurement. The long quantum memory lifetime demonstrated in this work opens the way for scalable processing of quantum information and long distance quantum communication.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Supervisor: William C. Schieve. Vita. Includes bibliographical references. Available also from UMI Company.

Proefschrift--Rijksuniversireit te Leiden. / Includes bibliographical references (p. 119-121).

Thesis (M. Sc.)--University of Queensland, 2002. / Includes bibliographical references.

Thesis (Ph. D.) -- California Institute of Technology, 2010. / Title from home page (viewed 03/08/2010). Advisor and committee chair names found in the thesis' metadata record in the digital repository. Includes bibliographical references.

Thesis (Ph. D.) -- California Institute of Technology, 2010. / Title from home page (viewed 02/23/2010). Advisor and committee chair names found in the Acknowledgments pages of the thesis. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

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.