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Quantum walks and ground state problems

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Title
Quantum walks and ground state problems
Name (ID = NAME001); (type = personal)
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Richter
NamePart (type = given)
Peter C. (Peter Courtland)
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Peter C. Richter
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author
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Szegedy
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Mario
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Advisory Committee
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Mario Szegedy
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chair
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Eric
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Advisory Committee
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Eric Allender
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Kilian
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Joe
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Advisory Committee
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Joe Kilian
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Kendon
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Viv
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Advisory Committee
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Viv Kendon
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outside member
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Rutgers University
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degree grantor
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Graduate School - New Brunswick
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school
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theses
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DateCreated (qualifier = exact)
2007
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2007
Language
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English
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electronic
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viii, 101 pages
Abstract
Since the appearance of Shor's factoring algorithm in 1994, the search for novel quantum computer algorithms has proved surprisingly difficult. Two design approaches that have yielded some progress are quantum walks and adiabatic computing. The former has been shown to speed up algorithms whose complexity is related to the classical hitting time of a symmetric Markov chain, and there is evidence that the latter speeds up simulated annealing algorithms for computing ground states of classical Hamiltonians.
In this thesis, we look into the possibility of obtaining a quantum speedup for the mixing time of a symmetric Markov chain. We prove that by subjecting a quantum walk to a small amount of decoherence (typically the adversary of a quantum computer), it can be forced to mix to the correct stationary distribution, often considerably faster than its classical counterpart. A more general theorem to this effect would imply quantum speedups for a variety of approximation algorithms for #P-complete problems.
We conclude with some observations on adiabatic computing -- a time-dependent generalization of the quantum walk framework -- and the problem of estimating the ground state energy of a quantum Hamiltonian with local spin interactions.
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references (p. 94-100).
Subject (ID = SUBJ1); (authority = RUETD)
Topic
Computer Science
Subject (ID = SUBJ2); (authority = ETD-LCSH)
Topic
Computer algorithms
Subject (ID = SUBJ3); (authority = ETD-LCSH)
Topic
Quantum field theory
Subject (ID = SUBJ4); (authority = ETD-LCSH)
Topic
Random walks (Mathematics)
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Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Identifier (type = hdl)
http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.16768
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ETD_521
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3W37WQ1
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
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Copyright protected
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Open
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Name
Peter Richter
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Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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