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Application of SDP to product rules and quantum query complexity

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TitleInfo
Title
Application of SDP to product rules and quantum query complexity
Name (type = personal)
NamePart (type = family)
Mittal
NamePart (type = given)
Rajat
DisplayForm
Rajat Mittal
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Szegedy
NamePart (type = given)
Mario
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Mario Szegedy
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Advisory Committee
Role
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chair
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NamePart (type = family)
Steiger
NamePart (type = given)
William
DisplayForm
William Steiger
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Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Saks
NamePart (type = given)
Michael
DisplayForm
Michael Saks
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Roetteler
NamePart (type = given)
Martin
DisplayForm
Martin Roetteler
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
outside member
Name (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (type = corporate)
NamePart
Graduate School - New Brunswick
Role
RoleTerm (authority = RULIB)
school
TypeOfResource
Text
Genre (authority = marcgt)
theses
OriginInfo
DateCreated (qualifier = exact)
2011
DateOther (qualifier = exact); (type = degree)
2011-05
Place
PlaceTerm (type = code)
xx
Language
LanguageTerm (authority = ISO639-2b); (type = code)
eng
Abstract (type = abstract)
In recent years, semidefinite programming has played a vital role in shaping complexity theory and quantum computing. There have been numerous applications ranging from estimating quantum values, over approximating combinatorial quantities, to proving various bounds. This work extends the use of semidefinite programs (SDPs) to proving product rules and to characterizing quantum query complexity. In the first application, we provide a general framework to establishing product rules for quantities that can be expressed (or approximated) using SDPs. We use duality theory to give product rules, which bound the value of the ``product'' of two problems in terms of their value. Some previous results have implicitly used the properties of SDPs to give such product rules. Here we give sufficient and necessary conditions under which these approaches work, thereby enabling us to capture these previous results under our unified framework. We also include a discussion about alternate definitions of what a ``product'' means and how they fit into our approach. The second application provides an SDP characterization of quantum query complexity, which is one of the ways in which complexity of a function can be measured. It is known that quantum query complexity can be lower bounded by the so-called ``adversary method'' which is expressible as a semidefinite program. Recently, Ben Reichardt showed that the adversary method leads to a tight lower bound for boolean functions by converting the solution of this SDP (of adversary method) into an algorithm. We show that a related SDP, called ``witness size'' in this thesis, provides a tight bound on the quantum query complexity of non boolean functions (total as well as partial). This witness size SDP is also used to give composition results for quantum query complexity. We also show that the witness size is bounded by a constant multiple of the adversary bound. Finally, we briefly explore whether other convex programming paradigms can be useful in complexity theory. One of them is copositive programming. We show that one of the recent result about parallel repetition of unique games, by Barak et.al., can be interpreted as an application of copositive programming.
Subject (authority = RUETD)
Topic
Computer Science
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
Identifier
ETD_3199
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
x, 86 p. : ill.
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = vita)
Includes vita
Note (type = statement of responsibility)
by Rajat Mittal
Subject (authority = ETD-LCSH)
Topic
Computer programming
Subject (authority = ETD-LCSH)
Topic
Quantum theory
Subject (authority = ETD-LCSH)
Topic
Querying (Computer science)
Identifier (type = hdl)
http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000061359
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Location
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NjNbRU
Identifier (type = doi)
doi:10.7282/T3B27TMJ
Genre (authority = ExL-Esploro)
ETD doctoral
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Rights

RightsDeclaration (ID = rulibRdec0006)
The author owns the copyright to this work.
RightsHolder (type = personal)
Name
FamilyName
Mittal
GivenName
Rajat
Role
Copyright Holder
RightsEvent
Type
Permission or license
DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2011-03-31 12:12:32
AssociatedEntity
Name
Rajat Mittal
Role
Copyright holder
Affiliation
Rutgers University. Graduate School - New Brunswick
AssociatedObject
Type
License
Name
Author Agreement License
Detail
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.
Copyright
Status
Copyright protected
Availability
Status
Open
Reason
Permission or license
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