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    QUANTUM RANDOM WALK ON FRACTALS

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    Genre
    Thesis/Dissertation
    Date
    2018
    Author
    Zhao, Kai
    Advisor
    Yang, Wei-shih, 1954-
    Committee member
    Futer, David
    Szyld, Daniel
    Shi, Justin Y.
    Department
    Mathematics
    Subject
    Mathematics
    Quantum Physics
    Fractals
    Quantum Random Walk
    Sierpinski Gasket
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/3938
    
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    DOI
    http://dx.doi.org/10.34944/dspace/3920
    Abstract
    Quantum walks are the quantum mechanical analogue of classical random walks. Discrete-time quantum walks have been introduced and studied mostly on the line Z or higher dimensional space Z d but rarely defined on graphs with fractal dimensions because the coin operator depends on the position and the Fourier transform on the fractals is not defined. Inspired by its nature of classical walks, different quantum walks will be defined by choosing different shift and coin operators. When the coin operator is uniform, the results of classical walks will be obtained upon measurement at each step. Moreover, with measurement at each step, our results reveal more information about the classical random walks. In this dissertation, two graphs with fractal dimensions will be considered. The first one is Sierpinski gasket, a degree-4 regular graph with Hausdorff di- mension of df = ln 3/ ln 2. The second is the Cantor graph derived like Cantor set, with Hausdorff dimension of df = ln 2/ ln 3. The definitions and amplitude functions of the quantum walks will be introduced. The main part of this dissertation is to derive a recursive formula to compute the amplitude Green function. The exiting probability will be computed and compared with the classical results. When the generation of graphs goes to infinity, the recursion of the walks will be investigated and the convergence rates will be obtained and compared with the classical counterparts.
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