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.
Files	Filename       Size       Approximate Download Time (Hours:Minutes:Seconds)     28.8 Modem   56K Modem   ISDN (64 Kb)   ISDN (128 Kb)   Higher-speed Access    Becnel_dis.pdf 588.31 Kb 00:02:43 00:01:24 00:01:13 00:00:36 00:00:03