I describe an exact method for computing roots of a system of multivariate polynomials with rational coefficients, called the rational univariate reduction. This method enables performance of exact algebraic computation of coordinates of the roots of polynomials. In computational geometry, curves, surfaces and points are described as polynomials and their intersections. Thus, exact computation of the roots of polynomials allows the development and implementation of robust geometric algorithms. I describe applications in robust geometric modeling. In particular, I show a new method, called numerical perturbation scheme, that can be used successfully to detect and handle degenerate configurations appearing in boundary evaluation problems. I develop a derandomized version of the algorithm for computing the rational univariate reduction for a square system of multivariate polynomials and a new algorithm for a non-square system. I show how to perform exact computation over algebraic points obtained by the rational univariate reduction. I give a formal description of numerical perturbation scheme and its implementation.

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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.

Thesis, PlanB (M.S.)--University of Wisconsin--Stout, 2004. / Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1972. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

How can a client extract useful work from a server without trusting it to compute correctly? A modern motivation for this classic question is third party computing models in which customers outsource their computations to service providers (as in cloud computing). In principle, deep results in complexity theory and cryptography imply that it is possible to verify that an untrusted entity executed a computation correctly. For instance, the server can employ probabilistically checkable proofs (PCPs) in conjunction with cryptographic commitments to generate a succinct proof of correct execution, which the client can efficiently check. However, these theoretical solutions are impractical: they require thousands of CPU years to verifiably execute even simple computations. This dissertation describes the design, implementation, and experimental evaluation viiiof a system, called Pepper, that brings this theory into the realm of plausibility. Pepper incorporates a series of algorithmic improvements and systems engineering techniques to improve performance by over 20 orders of magnitude, relative to an implementation of the theory without our refinements. These include a new probabilistically checkable proof encoding with nearly optimal asymptotics, a concise representation for computations, a more efficient cryptographic commitment primitive, and a distributed implementation of the server with GPU acceleration to reduce latency. Additionally, Pepper extends the verification machinery to handle realistic applications of third party computing: those that interact with remote storage or state (e.g., MapReduce jobs, database queries). To do so, Pepper composes techniques from untrusted storage with the aforementioned technical machinery to verifiably offload both computations and state. Furthermore, to make it easy to use this technology, Pepper includes a compiler to automatically transform programs in a subset of C into executables that run verifiably. One of the chief limitations of Pepper is that verifiable execution is still orders of magnitude slower than an unverifiable native execution. Nonetheless, Pepper takes powerful results from complexity theory and verifiable computation a few steps closer to practicality / text

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