Title page for ETD etd-06222006-133421


Type of Document Dissertation
Author Becnel, Jeremy James
Author's Email Address becnel@gmail.com
URN etd-06222006-133421
Title Extension of Shor's Period-Finding Algorithm to Infinite Dimensional Hilbert Spaces
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Ambar Sengupta Committee Chair
Hui-Hsiung Kuo Committee Member
Jorge Morales Committee Member
Lawrence Smolinsky Committee Member
Padmanabhan Sundar Committee Member
Mette Gaarde Dean's Representative
Keywords
  • white noise distribution theory
Date of Defense 2006-03-20
Availability unrestricted
Abstract
Over the last decade quantum computing has become a very popular field in various disciplines, such as physics, engineering, and mathematics. Most of the attraction stemmed from the famous Shor period--finding algorithm, which leads to an efficient algorithm for factoring positive integers. Many adaptations and generalizations of this algorithm have been developed through the years, some of which have not been ripened with full mathematical rigor. In this dissertation we use concepts from white noise analysis to rigorously develop a Shor algorithm adapted to find a hidden subspace of a function with domain a real Hilbert space. After reviewing the framework of quantum mechanics, we demonstrate how these principles can be used to develop algorithms which operate on a quantum computing device. We present a self-contained account of white noise analysis, including the main relevant results. Inspired by a generalized function in the algorithm, we develop a new distribution, the delta function for a subspace of an infinite dimensional Hilbert space. We then use this distribution to rigorously prove one of the main identities needed for the algorithm. Finally we provide a rigorous formulation of the hidden subspace algorithm in infinite dimensions.
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