<%BANNER%>

The Hidden Subgroup Problem

Permanent Link: http://ufdc.ufl.edu/UFE0041526/00001

Material Information

Title: The Hidden Subgroup Problem
Physical Description: 1 online resource (73 p.)
Language: english
Creator: Debhaumik, Anales
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The topic of my research is the Hidden Subgroup Problem. The problem can be stated as follows: {(Hidden Subgroup Problem)} Let G be a finite group and $H$ a subgroup. Given a black-box function f : G -- > S which is constant on (left)-cosets gH of H and takes different values for different cosets, determine a set of generators for $H$. Efficient classical algorithms for the Hidden Subgroup Problem are not known. However, efficient quantum algorithms are known for this problem in some cases. One such algorithm implies Shor's famous efficient method for breaking the RSA cryptosystem. An efficient solution of this problem in all cases would have wide implications in the field of theoretical computer science. For example it would most likely solve some classical problems which are neither NP-complete nor are in P. A solution would also imply a solution for Graph Isomorphism problem which is a long standing problem in computer science. In the present thesis, we study some quantum algorithms for the Hidden Subgroup Problem. We discuss Quantum Fourier Tranform and its applications to the Hidden Subgroup Problem. We discuss Almost Abelian groups and show that there is a quantum algorithm that solves the Hidden Subgroup Problem for them. We also study the decision version of the problem. We compare two different formalizations of this concept. We show that these formalizations coincide in the case of abelian and Frobenius groups. We conclude with a family of groups where the formalizations may not coincide
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Anales Debhaumik.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Turull, Alexandre.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041526:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041526/00001

Material Information

Title: The Hidden Subgroup Problem
Physical Description: 1 online resource (73 p.)
Language: english
Creator: Debhaumik, Anales
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The topic of my research is the Hidden Subgroup Problem. The problem can be stated as follows: {(Hidden Subgroup Problem)} Let G be a finite group and $H$ a subgroup. Given a black-box function f : G -- > S which is constant on (left)-cosets gH of H and takes different values for different cosets, determine a set of generators for $H$. Efficient classical algorithms for the Hidden Subgroup Problem are not known. However, efficient quantum algorithms are known for this problem in some cases. One such algorithm implies Shor's famous efficient method for breaking the RSA cryptosystem. An efficient solution of this problem in all cases would have wide implications in the field of theoretical computer science. For example it would most likely solve some classical problems which are neither NP-complete nor are in P. A solution would also imply a solution for Graph Isomorphism problem which is a long standing problem in computer science. In the present thesis, we study some quantum algorithms for the Hidden Subgroup Problem. We discuss Quantum Fourier Tranform and its applications to the Hidden Subgroup Problem. We discuss Almost Abelian groups and show that there is a quantum algorithm that solves the Hidden Subgroup Problem for them. We also study the decision version of the problem. We compare two different formalizations of this concept. We show that these formalizations coincide in the case of abelian and Frobenius groups. We conclude with a family of groups where the formalizations may not coincide
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Anales Debhaumik.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Turull, Alexandre.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041526:00001


This item has the following downloads:


Full Text

PAGE 1

THEHIDDENSUBGROUPPROBLEM By ANALESDEBHAUMIK ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010

PAGE 2

c r 2010AnalesDebhaumik 2

PAGE 3

Idedicatethistoeveryonewhohelpedmeinmyresearch. 3

PAGE 4

ACKNOWLEDGMENTS FirstandforemostIwouldliketoexpressmysincerestgrati tudetomyadvisorDr AlexandreTurull.Withouthisguidanceandpersistenthelp thisthesiswouldnothave beenpossible. Iwouldalsoliketothanksincerelymycommitteemembersfor theirhelpand constantencouragement. FinallyIwouldliketothankmywifeManishaBrahmacharyfor herloveandconstant support. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 6 CHAPTER 1INTRODUCTION ................................... 8 1.1Preliminaries .................................. 8 1.2QuantumComputing .............................. 11 1.3QuantumFourierTransform .......................... 12 1.4HiddenSubgroupProblem .......................... 14 1.5Distinguishability ................................ 18 2HIDDENSUBGROUPSFORALMOSTABELIANGROUPS ........... 20 2.1Algorithmtondthehiddensubgroups .................... 20 2.2AnAlmostAbelianGroupofOrder 3 2 n ................... 22 3KEMPE-SHALEVDISTINGUISHABLITY ..................... 33 4ALGORITHMICDISTINGUISHABILITY ...................... 43 4.1Analgorithmfordistinguishability ....................... 43 4.2AbelianGroups ................................. 45 4.3FrobeniusGroups ............................... 46 5SOMECALCULATIONS ............................... 55 5.1Kempe-Shalevdistinguishability ........................ 55 5.2AlgorithmicDistinguishability ......................... 56 APPENDIX ATABLESFORCHAPTER4 ............................. 59 BTABLEFORCHAPTER5 .............................. 70 REFERENCES ....................................... 71 BIOGRAPHICALSKETCH ................................ 73 5

PAGE 6

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy THEHIDDENSUBGROUPPROBLEM By AnalesDebhaumik May2010 Chair:AlexandreTurullMajor:Mathematics Thetopicofmyresearchisthe HiddenSubgroupProblem .Theproblemcanbe statedasfollows: Problem. (HiddenSubgroupProblem)Let G beanitegroupand H asubgroup.Given ablack-boxfunction f : G S whichisconstanton(left)-cosets gH of H andtakes differentvaluesfordifferentcosets,determineasetofge neratorsfor H EfcientclassicalalgorithmsfortheHiddenSubgroupProb lemarenotknown. However,efcientquantumalgorithmsareknownforthispro bleminsomecases. OnesuchalgorithmimpliesShor'sfamousefcientmethodfo rbreakingtheRSA cryptosystem. Anefcientsolutionofthisprobleminallcaseswouldhavew ideimplicationsin theeldoftheoreticalcomputerscience.Forexampleitwou ldmostlikelysolvesome classicalproblemswhichareneither NP completenorarein P Asolutionwouldalso implyasolutionforGraphIsomorphismproblemwhichisalon gstandingproblemin computerscience. Inthepresentthesis,westudysomequantumalgorithmsfort he HiddenSubgroupProblem .WediscussQuantumFourierTranformanditsapplicationst othe HiddenSubgroupProblem.WediscussAlmostAbeliangroupsa ndshowthatthere isaquantumalgorithmthatsolvestheHiddenSubgroupProbl emforthem.Wealso studythedecisionversionoftheproblem.Wecomparetwodif ferentformalizationsof 6

PAGE 7

thisconcept.Weshowthattheseformalizationscoincidein thecaseofabelianand Frobeniusgroups.Weconcludewithafamilyofgroupswheret heformalizationsmay notcoincide. 7

PAGE 8

CHAPTER1 INTRODUCTION 1.1Preliminaries Aquantumcomputerisan,atpresenttheoretical,machinewh ichperforms computationsonthebasisofthelawsofquantummechanics.S uchadevicecan preparequantumstatesandperformmeasurementsonthem.As withclassical computers,numericaldataformtheinputandtheoutputfort hequantumcomputer.The processingofthisdatainaquantumcomputermayinvolveact ionsonthemwhichare notclassical.Asaresult,theoutputofaquantumcomputeri snotfullydeterminedbyits input.Givenaparticularinput,variousoutputsmayoccurw ithdifferentprobabilities.In thissense,quantumcomputersbehavedifferentlythanclas sicalones. Thoughquantumcomputingasaeldofresearchisstillinits infancysignicant amountofworkhasalreadybeendoneinthiseld.In1985Deut schgaveaquantum algorithmforaverysimpleproblemandshowedthatitworked betterthanaclassical one.Inhisproblemwearegivenablackboxwhichcomputesasi mplefunction.The boxtakestwobitsasinputandoutputstwobits.Itimplement soneofthefourfollowing functionsfrom F 22 F 22 : f 1 ( x y )=( x y ) f 2 ( x y )=( x y +1) f 3 ( x y )=( x x + y ) f 4 ( x y )=( x x + y +1) Thegoalistodeterminewhetherthefunctionimplementedby theboxisintheset f f 1 f 2 g or f f 3 f 4 g Onaclassicalcomputeritwilltake2queriestondtheanswe rto thequestion.Butonaquantumcomputeritwilltakejust1que rytondtheanswer.In 1992DeutschandJozsa( 7 )gavetherstnon-trivialquantumalgorithm.Theirproble m isthegeneralizationoftheaboveproblem.Inthisproblemw earegivenafunction f : F n2 F 2 Thegoalistondouthowmanyqueriestotheboxareneededint he 8

PAGE 9

worstcasetodeterminewhetherthefunctionisaconstantor balanced(0ononehalf ofthedomainstatesand1ontheotherhalf).Fortheclassica lcasetheproblemhas exponentialquerycomplexity.Butforthequantumcaseaswa sshownbyDeutschand Jozsaonlyasinglequeryisneeded.MotivatedbythisBerste inandVazirani( 3 )gavea quantumalgorithmwhichworkedsignicantlyfaster(super -polynomialspeedup)than thebestclassicalalgorithm.TheBersteinVaziraniproble mhasanon-recursiveanda recursiveversion.Inthenon-recursiveversionwearegive nafunction f : F n2 F 2 Thefunctionis f s ( x )= x s forsomeunknown s where x s denotesthedotproduct. Thegoalistond s Intheclassicalcasethequerycomplexityisatleast n whereasfor thequantumcaseonlyasinglequeryisneeded.Therecursive versionisalittlemore complexthanthenon-recursiveone.Simon( 24 )thengaveaquantumalgorithmwhich performedexponentiallyfasterthanitsclassicalcounter part.InSimon'sproblemweare givenaninteger m 1 andafunction f : F m2 R where R isaniteset.Wealso knowthatthereexistsanonzeroelement s 2 F m2 suchthatforall x y 2 F m2 f ( x )= f ( y ) ifandonlyif x = y or x = y + s Thegoalistond s AllthesepavedthewayforShor's celebratedpaperonfactoringanddiscretelog( 23 ).Thereisnoknownefcientclassical algorithmforfactoringalargenumber n ThesecurityandrobustnessofRSApublic-key cryptosystemisbasedonthisfact.Shorshowedinhispapert hatgivenaquantum computerfactoringofanumber n canbedoneinpolynomialtime. Simon'sandShor'salgorithmslaidtheframeworkforaveryi mportantproblemof quantumcomputationknownastheHiddenSubgroupProblem.T heproblemcanbe statedasfollows. Problem. (HiddenSubgroupProblem)Let G beanitegroupand H asubgroup.Given ablack-boxfunction f : G S whichisconstanton(left)-cosets gH of H andtakes differentvaluesfordifferentcosets,determineasetofge neratorsfor H TheHiddenSubgroupProblemisattheheartofShor'sandSimo n'sproblems. InShor'sproblemgiven n let a beanyintegersuchgcd( a n )=1.Let r betheorderof 9

PAGE 10

a 2 Z n Thegoalistond r .Inthisproblem G is Z and H is r Z Forpracticalpurposes wecanworkwith Z m where m =2 l > n Thefunction f isgivenby f ( x )= a x + n Z Simon'sproblemreducestoaspecialcaseoftheHiddenSubgr oupProblemifwetake G tobe F m2 and H = f 0, s g EfcientquantumalgorithmsforTheHiddenSubgroupProble mareknownfor abeliangroupsthankstotheeffortsofShor,Simon,Deutsch etc( 7 ).Thenon-abelian casestillposesachallenge.Somepositiveresultshavebee nobtainedforDihedral groupbyM.EttingerandP.Hoyer( 8 ),G.Kuperberg( 19 ),Regev( 21 ),DaveBacon, AndrewM.ChildsandWimvanDam( 1 ).Efcientquantumalgorithmshavebeen obtainedforsemidirectproductoftheforms Z p r oZ p byYoshifumiInuiandFrancois LeGall( 15 ),for Z N oZ p where N = p r 1 1 p r 2 2 ... p r n n andprime p doesnotdivide p j 1 by DongPyoChi,JeongSanKimandSoojoonLee( 5 ),forsomemetacyclicgroupsand allgroupsoftheform Z rp oZ p foranyprime p andaxed r byDaveBacon,AndrewM. ChildsandWimvanDam( 1 ),forWreathproductoftheform Z n2 o Z 2 byM.RottelerandT. Beth( 4 ),forsomesolvablegroupsbyK.Friedl,G.Ivanyos,F.Magni ez,M.Santha,and P.Sen( 10 )andforgroupswithsmallcommutatorsbyG.Ivanyos,F.Magn iez,andM. Santha( 17 ).Unfortunatelyauniedsolutionhasstillnotbeenfound. Anefcientsolutionofthisproblemwouldhavewideimplica tionsintheeldof theoreticalcomputerscience.Forexampleitwouldmostlik elysolvesomeclassical problemswhichareneither NP completenorarein P Asolutionwouldalsoimplya solutionforGraphIsomorphismproblemwhichisalongstand ingproblemincomputer science. InthispaperwestudydistinguishabilityoftheHiddenSubg roupProblemfor Frobeniusgroupsoftheforms Z g ( ) ( ) oZ Z g ( i ) oZ m ( i ) oZ n ( i ) Z g ( i ) o ( SL (2,5) ( Z m ( i ) o Z n ( i ) ). Forgroupsoftheform Z g ( ) ( ) oZ wegiveanecessaryandsufcientconditionfor thedistinguishabilityofthesubgroupsgiventhat g ( ) > 1 c forsome c > 0 andforall sufcientlylarge Wealsogiveanecessarysufcientconditionforthedisting uishability 10

PAGE 11

ofthesubgroupsofFrobeniusgroupsofthetype Z g ( i ) oZ m ( i ) oZ n ( i ) giventhat g ( i ) c m ( i ) n ( i ) forsome c > 0 andsufcientlylarge i Basedontheconditionthat g ( i ) c m ( i ) n ( i ) forsome c > 0 andsufcientlylarge i wegiveanecessaryandsufcient conditionforthedistinguishabilityofthesubgroupsof Z g ( i ) o ( SL (2,5) ( Z m ( i ) oZ n ( i ) ). In( 11 )M.Grigni,L.J.Schulman,M.Vazirani,U.Vaziranigaveane fcientalgorithmfor ”almostabeliangroups.”Theyalsogiveanoutlineofhowthe algorithmworksfor”almost abeliangroups” Z 3 oZ m where m isapowerof 2 .Insection 7 wedescribethealgorithm forthesegroupsandgiveanestimateofthenumberofstepsre quiredtoobtainthe hiddensubgroupswithprobabilitybiggerthan 1 2 1.2QuantumComputing Thetheoryofquantumcomputingwaslaunchedinthebeginnin gof1980.The famousphysicistFeynmaninanarticle( 9 )proposedthatitisnotpossibletosimulate aquantumsystemonaclassicalcomputerwithoutexponentia lslowdown.Healso suggestedthataquantumcomputercouldbeawaytoavoidsuch slowdown.In1985 Deutsch( 6 )rstgaveadescriptionofauniversalquantumcomputerwhi chwasrened laterbysomeothercomputerscientists.Thoughaquantumco mputerisstilloutofreach alotoftheoreticalprogresshasbeenmadeeversince. Tounderstandaquantumcomputerwersthavetoknowafewthi ngsabout quantummechanicsandquantumsystem.Inquantummechanics astateinan n -levelsystemisaunitvectorinan n dimensionalcomplexvectorspace H n withan orthonormalbasischosenas j x 1 i j x 2 i ,..., j x n i .Thechoiceofsuchabasisisarbitrary. Anystateinaquantumsystemcanthusbewrittenas p 1 j x 1 i + p 2 j x 2 i +...+ p n j x n i where the p i 2 C arecalledtheamplitudes( 13 )withtherequirementthat P ni =1 j p i j 2 =1 The amplitudessignifythattheprobabilitythataproperty x i isobservedofthesystemis j p i j 2 Thelinearcombinationsofthebasisstatesarecalled superpositionofstates Thebasicunitofinformationforaclassicalcomputerisabi t.Similarlyaquantum computeroperatesonquantumbitsknownas qubits Itisbasicallya 2 -levelquantum 11

PAGE 12

systemandcanbeidentiedwitha 2 -dimensionalHilbertspacewithbasis j 0 i j 1 i .A generalstateofaquantumsystemisthusequalto c 1 j 0 i + c 2 j 1 i wherethe c i arecomplex numbersand j c 1 j 2 + j c 2 j 2 =1. Ifwehavetwosystemswithbasisstates j x 1 i j x 2 i ,..., j x n i and j y 1 i j y 2 i ,..., j y m i then thecompoundsystemwillhavebasisstatesas( j x i i j y i i ).Thusasystemoftwoqubits isa 4 -dimensionalHilbertspacewithanorthonormalbasis fj 0 ij 0 i j 0 ij 1 i j 1 ij 0 i j 1 ij 1 ig Aquantumregisteroflength m isanorderedsystemof m qubitsassociatedwiththe 2 m dimensionalspace H 2 N H 2 .... N H 2 Anoperationon m qubitscanberepresentedbyaunitarymap U : H H where H isa 2 m -dimensionalHilbertspacerepresentingthequantumsyste m.Itcanalsobe representedbyaunitarymatrix.Asaverysimpleexamplewec anconsiderthenot operationonasinglequbit.Theoperationtakes j 0 i to j 1 i and j 1 i to j 0 i Thematrixthat denesthisoperationis 0B@ 0110 1CA Theprocessofgettinginformationoutofaquantumstateisk nownasmeasurement. Thereareseveralwaystothinkaboutmeasurement.Thesimpl estoneismeasurement inthecomputationalbasis.Supposeweareinaquantumsyste mwith n -dimensional basisvectorsand j i = P n 1 0 p j j j i isaquantumstate.Measurementinthecomputational basiswillreturnthestate j j i withprobability j p j j 2 andafterthemeasurementtheoutput statebecomes j 0 i = j j i Thusthegivenstatecollapsestotheonereturnedbythe measurementandallotherinformationaboutthestateisdes troyed. 1.3QuantumFourierTransform Onereasonwhyaquantumcomputercanworkmoreefcientlyth anaclassical one,asdemonstratedbytheperformancesoftheefcientqua ntumalgorithms,isthat quantumfouriertransformcanbeimplementedonit.Itisthe keyingredientforsome veryinterestingquantumalgorithms.Italsoaccountsfort heefciencyoftheperiod 12

PAGE 13

ndingalgorithmsliketheonesdevisedforvariousinstanc esoftheHiddenSubgroup Problem. Let G beanitegroupoforder n andlet R = f 1 2 ,..., g beacompletesetof inequivalentcomplexirreduciblerepresentationsof G withdegrees d i i =1,..., We choosetheserepresentationstobeunitary(( 16 ), Theorem 4.17 ).Let S time beacomplex vectorspaceofdimension j G j withanorthonormalbasis B time = fj g 1 i j g 2 i ,..., j g n ig wherethe g i aretheelementsofthegroup G Let S space beanothercomplexvector spacewithanorthonormalbasis B space = fj 1 ,1,1 i j 1 ,1,2 i ,.. j k i j i ,..., j d d ig formedbyconsideringthe ( i j ) th entryofthematrix ( g ) foreach 2 R and g 2 G Sincethereare d 2 matrixentriesinthematrix ( g ) foreach 2 R thereare P i =1 d 2 i = j G j numberofbasisvectors.Sothedimensionof S space is j G j whichisthesameas S time WearenowreadytodeneQuantumFourierTransform. Denition 1 QFTorquantumfouriertransformoveragroup G isthelinearmap Q G : S time S space suchthat Q G ( j g i )= 1 p j G j X i j p d ( g ) i j j i j i where 2 R hasdimension d ,and ( g ) i j isthe ( i j ) thentryofthematrix ( g ) for g 2 G and 1 i j d WewillnowshowthattheQFTisaunitarytransformation.Let g p g q 2 G .Then,sincethebasis S space isorthonormal,wehave h Q G ( j g p i ), Q G ( j g q i ) i = 1 j G j X i j d ( g p ) i j ( g q ) i j hj i j i j i j ii Since isunitary, ( g ) ij = ( g 1 ) ji andthepreviousexpressionbecomes 1 j G j X i j d ( g p ) i j ( g 1 q ) j i = 1 j G j X d ( g p g 1 q ) 13

PAGE 14

where ischaracteraffordedby Bythesecondorthogonalityrelation(( 16 ), Theorem 2.18 ) 1 j G j X d ( g p g 1 q )= 8>><>>: 1 for p = q 0 otherwise Sowehaveshownthat hj g p i j g q ii = h Q G ( j g p i ), Q G ( j g q i ) i HenceQFTisunitary. QuantumFourierTransformisanextremelypowerfultoolwhi chhasbeenused todeviseefcientquantumalgorithmsfortheHiddenSubgro upProblem.QFTrunsin polynomialtimeforabeliangroupsandthisfacthasbeenexp loitedtondanefcient generalalgorithmfortheabelianHSP.Extensiveresearchi sbeingdonetondefcient QFTinthecaseofnon-abeliangroups.Zalka( 25 )givesanefcientHSPalgorithmfor wreathproductgroupsoftheform G = Z n2 o Z 2 wherehecomputesQFTefcientlyforthe group.PeterHoyer( 14 )givesconstructionofQFTforQuaternionsandsomeMetacyc lic groups.Beth,Puschel,Rotteler,( 4 )showhowtodoQFTefcientlyonaclassofgroups -solvable2groupscontainingacyclicnormalsubgroupofin dex 2. Beals( 2 )shows howtocomputeQFTover S n intime r ( poly ( n )). ChristopherMoore,DanielRockmore, AlexanderRussell( 20 )giveefcientQFT-thatiscircuitsofpoly( log( j G j ) )size-forgroups liketheCliffordgroup,symmetricgroup,metabeliangroup sincludingmetacyclicgroups suchasthedihedralandafnegroups. 1.4HiddenSubgroupProblem InthispaperwefocusontheHiddenSubgroupProblem.Therea retwoversionsof theproblem. HiddenSubgroupProblem :Let G beanitegroupand H asubgroup.Givena black-boxfunction f : G S whichisconstanton(left)-cosets gH of H andtakes differentvaluesfordifferentcosets,determineasetofge neratorsfor H Analgorithmisaproceduretosolveacomputationalproblem .Aclassicalalgorithm isonewhichperformsanitenumberofstepsandoutputsthea nswertotheproblem. Sincethebehaviorofaquantumcomputerisnotfullydetermi nedbyitsinput,the 14

PAGE 15

outcomeofaquantumalgorithmisnecessarilyuncertain,bu twemaycalculatethe probabilitythatitproducesthecorrectanswertotheprobl em.Wesaythataquantum algorithmsolvestheproblemifforeveryinputitreturnsth ecorrectanswertothe problemafteranitenumberofstepswithprobability p forsome p > 1 2 Eventhoughan algorithmperformsanitenumberofstepsitmayrunforalon gtimedependingonthe inputs. Itisbelievedthatagoodwaytoestimatetherunningtimeofa nyalgorithmtosolve theHiddenSubgroupProblemsistocountthenumberofquerie smadetotheblackbox function.Thecomplexityoftheproblemsisestimatedby j G j Wesaythatanalgorithm foraclassofproblemsis efcient ifthereissomepolynomial p ( x ) 2 R [ x ] suchthatthe numberofqueriestotheblackboxfunctionrequiredbytheal gorithmisalwaysatmost p (log j G j ). AnobviousalgorithmtosolvetheHiddenSubgroupProblemwo uldbetoevaluate f ( g ) forall g 2 G andtonoticethat H = f g 2 G : f ( g )= f (1) g .Thisrequires j G j queriestothefunction f andsothisalgorithmisnotefcient.Moreefcientquantum algorithmshavebeenobtainedinvolvingQuantumFourierSa mpling. QuantumFouriersamplingisamethodwhichcanbeimplemente dbyaquantum computertoproducewithasinglequerytotheblackboxfunct ionanirreducible representation of G whichhas core G ( H ) initskernelwhere core G ( H ) isthelargest subgroupof H normalin G If affordsthecharacter thentheprobabilityof QuantumFouriersamplingyielding is P H ( )= d j H j j G j h ( ) H ,1 H i where h ( ) H ,1 H i = 1 j H j P h 2 H ( h ). QuantumFouriersamplingcanbeimplementedonaquantumcom puterasfollows. Thesetupisalmostthesameasintheprevioussection.Wehav e S time withbasis B time indexedbytheelements g i 1 i n ofthegroup G FourierTransform Q G sends S time to S space whichisanothercomplexvectorspacewithbasis B space Letusdene acomplexvectorspace S M withorthonormalbasis fj g 1 s 1 i j g 1 s 2 i ,..., j g n s n ig where 15

PAGE 16

g i 2 G and s i 2f 0 g S S Wealsodeneaunitarymap Q f : S time S M givenby Q f ( j g i )= j g f ( g ) i ThemethodofFourierSamplingisnowdetailedbelow: Step 1: Firstprepareastatein S M asbelow. 1 p j G j Xg 2 G j g ,0 i Step 2: Compute Q f denedon S time andgetthestate 1 p j G j Xg 2 G j g f ( g ) i whichisin S M Step 3: Measurethesecondregister.Ifthemeasuredvalueis f m wegetthestate j cH i = 1 p j H j X h 2 H j ch f m i2 S M Nowforeach f m 2 S S time isisomorphictothesubspaceof S M withbasis fj g 1 f m i j g 2 f m i ,..., j g n f m ig Hence j cH i = 1 p j H j P h 2 H j ch f m i canbeidentiedwiththestate j cH i = 1 p j H j P h 2 H j ch i Step 4: QFTisperformedoverthestate j cH i whichyields 1 p j G jj H j X i j p d X h 2 H ij ( ch ) j i j i Step 5: Measure Theprobabilitytoobserveanyparticular underQuantumFouriersamplingis P H ( )= X i j d j G jj H j j X h 2 H ij ( ch ) j 2 Foran n n matrix M welet k M k denotethematrixnormgivenby k M k 2 = trace ( M M )= X i j j M ij j 2 where,foranymatrix M M denotesthetransposeofitscomplexconjugate.Then P H ( )= d j G jj H j k X h 2 H ( ch ) k 2 16

PAGE 17

Now k X h 2 H ( ch ) k 2 = trace (( X h 2 H ( ch )) X h 2 H ( ch ))= trace (( X h 2 H ( h )) ( c ) ( c ) X h 2 H ( h ))= k X h 2 H ( h ) k 2 since ( c ) isunitary.Hence P H ( )= d j H j j G j k 1 j H j X h 2 H ( h ) k 2 Now H isarepresentationof H notnecessarilyirreducible.Itcan,however,be decomposedintoirreduciblerepresentationsover H Withproperchoiceofbasiswecan make ( h ) blockdiagonalwitheachblockcorrespondingtoanirreduci blerepresentation t If m = thenumberofconjugacyclassesof H then 1 t m Henceeachdiagonal entryof P h 2 H ( h ) is P h 2 H t ( h ) forsome 1 t m By(( 22 ), Corollary 3 Chapter 2 ) X h 2 H t ( h )= 8>><>>: j H j for t =1 H 0 otherwise Henceweseethat k 1 j H j P h 2 H ( h ) k 2 = h ( ) H ,1 H i where istheirreduciblecharacter affordedby Thus P H ( )= d j H j j G j h ( ) H ,1 H i AclassofquantumalgorithmstosolvetheHiddenSubgroupPr oblemusing quantumFourierSamplinghavebeendeveloped.Theyarecall edtheWeakStandard Method. WeakStandardMethod :GiventheinputoftheHiddenSubgroupProblem,select somenumber n = Q (log j G j ). ApplyquantumFouriersampling n timesandobtain representations 1 2 ... n Outputthesubgroup T ni =1 ker ( i ). Ithasbeenshownin( 12 )that n =4log j G j isenoughtoretrieve core G ( H ) with probabilitybiggerthan 1 2 Ifthehiddensubgroup H G then core G ( H )= H andtheWeak StandardMethodoutputsthehiddensubgroup H withhighprobability. 17

PAGE 18

Thereisanothermethodwhichisbelievedtobemorepowerful thanWeakStandard Method.Itisknownasthestrongstandardmethod.Inthe StrongStandardMethod both anditsentries i j aresampled.Itisbelievedthatincertaincasessuchsampli nggives moreinformationabout H thantheWeakStandardMethod. 1.5Distinguishability ThereisadecisionversionoftheHiddenSubgroupproblem.I tisstatedasfollows. DecisionVersionoftheHiddenSubgroupProblem :Let G beanitegroupand H asubgroup.Givenablack-boxfunction f : G S whichisconstanton(left)-cosets gH of H andtakesdifferentvaluesfordifferentcosets,determine whether H = f e g or not. Certainsubgroups H of G mayproduceprobabilitiesofrepresentationsunder QuantumFourierSamplingthatareveryclosetothosethatar isefromthetrivial subgroup.Inthiscasewewouldsaythat H isindistinguishablefromthetrivialsubgroup. In2005KempeandShalev( 18 ),proposedthefollowingdenitionofindistinguishable subgroup. Denition 2 Let G beanitegroup.Let Irr( G ) bethesetofirreducible(complex) representationsof G .Wedenoteby thecharacterassociatedtoa 2 Irr( G ) andlet d beit'sdegree.For H < G wedene D H = 1 j G j X 2 Irr( G ) d j X h 6 = e h 2 H ( h ) j Wesaythatasubgroup H isKempe-Shalevdistinguishableif D H log ( j G j ) c forsome c > 0, c beingindependentof G Usingthisdenition,KempeandShalevclassiedthedistin guishablesubgroupsof S n Inchapter 3 weprovethatallthenon-trivialsubgroupsofanabeliangro upare Kempe-Shalevdistinguishable.WealsoclassifytheKempeShalevdistinguishable subgroupsofsomeFrobeniusgroups. 18

PAGE 19

WhiletheKempeShalevdenitioncapturesthedifcultyofd istinguishingthe hiddensubgroupfromthetrivialgroupinsomecases,itisno talwaysobvioushow totranslateitsanswerstopracticalalgorithms.Inchapte r 4 wedene”Algorithmic distinguishability”whichdenesdistinguishabilityofa subgroup H ofanynitegroup G fromthetrivialoneonthebasisofwhenanaturalalgorithmt hatusestheweak standardmethodsucceedsinpolynomialtime.Weprovethatt henon-trivialsubgroups ofanabeliangrouparealgorithmicdistinguishable.Weals oshowthattheKernelsof theFrobeniusgroups,whichareKempe-Shalevdistinguisha ble,arealsoalgorithmic distinguishableandthecomplements,whicharenotKempe-S halevdistinguishable,are alsonotalgorithmicdistinguishable. EventhoughincaseofFrobeniusgroupsthetwoconceptsseem tocoincide,we showinchapter 5 thattheymightnotbethesame.Thoughwedon'thaveageneral proof,computationsshowthatthetwoconceptsmaybediffer entfor G = S 3 S 3 ... S 3 ( ncopies ). 19

PAGE 20

CHAPTER2 HIDDENSUBGROUPSFORALMOSTABELIANGROUPS In( 11 )thenotionofanalmostabeliangroupwasintroduced. Denition 3 (AlmostAbelianGroup):Let G beanitegroup.Let N G ( H ) denotethe normalizerofasubgroup H in G Considerthenormalsubgroup K ( G )= T H N ( H ) of G Wecall G almostabelianif [ G : K ( G )] 2 exp ( r (lg 1 2 ( n ))) where n =lg( j G j ) In( 11 )theauthorsgiveanalgorithmtondthehiddensubgroupsof analmost abeliangroup.Inthischapterwegiveaverysmallvariation oftheiralgorithm.We provethatthisalgorithmsucceedswithasmallnumberofite rationswithprobability greaterthan 1 = 2 oneveryalmostabeliangroup.Weanalyseindetailtheoutco mes ofrunningthisalgorithminthecaseofthegroup G = Z 3 oZ m where m =2 n and anynon-trivialhiddensubgroup.Wendtheprobabilitytha ttheprocesswillyieldany particularsubgroupafter i iterations. 2.1Algorithmtondthehiddensubgroups Thealgorithmtodeterminethehiddensubgroupsforthealmo stabeliangroupsis asfollows: RepeatWeakStandardMethod n = s (log j G j ) times,where s ( x ) 2 R [ x ] is anon-zeropolynomial,forallthesubgroupsof G containing K ( G ). Considerthe intersectionsofthekernelsoftherepresentationsobserv edineachcaseafter n repetitions.Thealgorithmreturnsthelargestsuchinters ection. Thealgorithmcanbeformallywrittenasfollows:StepI:RepeatweakStandardMethod n = s (log j G j ) times,where s ( x ) 2 R [ x ] isa nonzeropolynomialforallsubgroupsof G containing K ( G ). StepII:Taketheintersectionsofthekernelsofthereprese ntationsobservedin eachcaseafter n repeats. StepIII:Returnthelargestsuchintersection. 20

PAGE 21

NoticethatthisalgorithmextendstheWeakStandardMethod tondanyhidden subgroupandnotjustthenormalones.Wenowestimatethenum berofiterations neededsothattheprobabilityofretrievingthehiddensubg roupismorethan 1 = 2. Lemma2.1. Let G beanitegroupoforder a and H thehiddensubgroup.Supposefor eachsubgroup M of G theprobabilityofgetting core M ( H ) be 1 2 e lg( j K ( G ) j ) k Thenthe productoftheseprobabilitiesisgreaterthan 1 = 2 if k < lg j K ( G ) j ln(4)+(lg( a )) 2 Proof. (1 2 e lg( j K ( G ) j ) k ) 2 lg 2 a > 1 2 lg 2 a 2 e lg j K ( G ) j k > 1 e lg 2 a 2 e lg j K ( G ) j k > 1 = 2 implies e (lg( a )) 2 e lg( j K ( G ) j ) k < 1 = 4. Hence ln(4) < lg( j K ( G ) j ) k (lg( a )) 2 Hence k < lg j K ( G ) j ln(4)+(lg( a )) 2 Lemma2.2. Suppose G isanitegroupand H isthehiddensubgroup.Thenwecan retrieve core G ( H ) after m =2 l lg( j G j ) stepswithprobability 1 e lg ( j G j ) 4 l ( l 1) 2 Proof. TheproofisthesameastheproofofTheorem5in( ? )with k =2 l lg( j G j ) and =( l 1)lg( j G j ). Followingtheproofweconcludethattheprobabilityofobta ining core G ( H ) is 1 e lg ( j G j ) 4 l ( l 1) 2 Theorem2.3. Suppose G isanitegroupand a =[ G : K ( G )]. IfwerepeatQuantum FourierSampling m =2 l lg( j G j ) timeswhere l =(8 d ln(4)+lg 2 ( a ) lg( j K ( G ) j ) e +8) forallsubgroups of G containing K ( G ) wecanretrieve H inthe”almostabelian”algorithmwithprobability greaterthan 1 = 2. Proof. Werstnotethat 4 l ( l 1) 2 = 32 d ln(4)+lg 2 ( a ) lg( j K ( G ) j ) e +32 (8 d ln(4)+lg 2 ( a ) lg( j K ( G ) j ) e +7) 2 < 32 d ln(4)+lg 2 ( a ) lg( j K ( G ) j ) e +32 49( d ln(4)+lg 2 ( a ) lg( j K ( G ) j ) e +1) 2 < 32 49( d ln(4)+lg 2 ( a ) lg( j K ( G ) j ) e ) < 32 49 lg j K ( G ) j ln(4)+(lg( a )) 2 Ifthealgorithmisrunfor m stepsforallsubgroupsof G containing K ( G ) thenthe probabilityofretrieving H G foreachofthesesubgroupswillbe 1 e lg ( j G j ) 4 l ( l 1) 2 which,bythe abovecomputation,isbiggerthan 1 e lg ( j G j ) l 0 where l 0 = 32 49 lg j K ( G ) j ln(4)+(lg( a )) 2 Sincethenumber 21

PAGE 22

ofsubgroupsofagroupoforder a cannotexceed 2 lg 2 ( a ) ,byLemma4.1itisclearthatwe canretrieve H attheendofthe”almostabelian”algorithmwithprobabilit ybiggerthan 1 = 2. 2.2AnAlmostAbelianGroupofOrder 3 2 n Wenowanalyzethequantumalgorithmforndingthehiddensu bgroupsofthe almostabeliangroup G = Z 3 oZ m where m =2 n .Thesegroupsarementionedas examplesofalmostabeliangroupsin( 11 ).Foreachnon-trivialsubgroup,wecalculate theprobabilitiesthatthealgorithmyieldsanyparticular subgroupafter i steps.These probabilitiesaregiveninAppendix A Thesubgroupsofthegroup G areasfollows.Wedenoteby Z 0 theuniquenormal subgroupofthesylow2-subgroupsofindex2.Wedenoteby T thesylow3-subgroupof G by SY i where i =1,2,3 thethreesylow2-subgroups.Apartfromthesethesubgroups of TZ 0 and Z 0 arealsosubgroupsof G If N ( H ) denotethenormalizerofasubgroup H of G wehavethat T H N ( H )= Z 0 G has m linearcharactersand m = 2 non-linearcharactersofdegree2inducedfrom thecharactersof TZ 0 where T isthesylow3-subgroupof G Inthissectionweexplicitly calculate,givenanyhiddensubgroup,theprobabilitiesof gettingthesubgroupafter i stepsastheintersectionofthepossiblekernelsoftheirre duciblerepresentations measuredcorrespondingtoallthesubgroupsof G containing Z 0 Theorem2.4. Let G = Z 3 oZ m beanalmostabeliangroup.Let H beanon-trivial subgroupof G Runthealmostabelianalgorithm i times.Thentheprobabilityofthe processyieldinganyparticularsubgroupof G isgivenbytheTablesinAppendix A ( A i where i =1,...15 ). Proof. 1)When H = SY 1 and G = TZ 0 Theprobabilityofgettinganirreducible representation isgivenby d j G j P h 2 H T G ( h ). Hencetheprobabilityisequalto 1 3 m = 2 m = 2=1 = 3 when Z 0 isinthekernelof otherwisetheprobabilityisequalto 22

PAGE 23

0.Thepossiblekernelsare TZ 0 and Z 0 Soaftertherstiterationtheprobabilitiesof gettingkernelssuchthat j ker ( ) j =3 a 2 N are p ( a N )= 8>>>><>>>>: 1 = 3 N = n 1, a =1 2 = 3 N = n 1, a =0 0 otherwise After i iterationstheprobabilitiesaregivenby p ( a N )= 8>>>><>>>>: (1 = 3) i N = n 1, a =1 1 (1 = 3) i N = n 1, a =0 0 otherwise When G = Z 0 Theprobabilityofgettinganon-trivialirreduciblerepre sentationis0. Thetrivialrepresentationismeasuredwithprobability1. Theonlypossiblekernelis Z 0 After i iterationstheprobabilityofgetting Z 0 is1. When G = SY 1 Theprobabilityofgettinganynon-trivialirreduciblerep resentation is0.Theprobabilityofmeasuringthetrivialrepresentati onis1.Thepossiblekernelis SY 1 After i iterationstheprobabilityofgetting SY 1 is1. When G = SY 2 Theprobabilityofmeasuringanirreduciblerepresentatio n is givenby1/2.Soafter i stepstheprobabilitiesofgettingtheintersectionofthek ernelsto beoforder 2 N aregivenby p ( N )= 8>>>><>>>>: (1 = 2) i N = n 1 (1 = 2) i N = n 1 0 otherwise When G = SY 3 Theprobabilitiesthatwegetarethesameasabove. 23

PAGE 24

When G = G If islinearthen d j G j P h 2 H ( h )=1 = 3 if =1 G and0otherwise. Suppose isnon-linear.Then = G where 2 Irr ( TZ 0 ). Now d j G j X h 2 H ( h )=2 j H j j G j (( ) H ,1 H )=2 = 3(( ) H ,1 H )=2 = 3(( G ) H ,1 H ) =2 = 3(( H T TZ 0 ) H ,1 H )=2 = 3( H T TZ 0 ,1 Z 0 ) =2 = 3( Z 0 ,1 Z 0 )=0 if Z 0 6 =1 Z 0 and2/3if Z 0 =1 Z 0 Thepossiblekernelsare G and Z 0 Aftertherstiterationtheprobabilitiesofgetting j ker ( ) j =3 a 2 N regivenby p ( a N )= 8>>>><>>>>: (1 = 3) N = n a =1 (2 = 3) N = n 1, a =0 0 otherwise After i iterationstheprobabilitieswillbe p ( a N )= 8>>>><>>>>: (1 = 3) i N = n a =1 1 (1 = 3) i N = n 1, a =0 0 otherwise H = TZ 0 1)When G = Z 0 Thentheprobabilityofmeasuringthetrivialrepresentati onis1. Thepossiblekernelis Z 0 .Theprobabilitythattheintersectionofthekernelswillb e Z 0 after i iterationsis1. 2)When G = TZ 0 Theprobabilityofmeasuringthetrivialrepresentationis 1.The onlypossiblekernelis TZ 0 .Theprobabilitythatafter i iterationstheintersectionofthe kernelsis TZ 0 is1. 24

PAGE 25

3)When G = SY 1 Theprobabilityofmeasuringanon-trivialrepresentation is givenby 1 j SY 1 j X h 2 Z 0 ( h ). Sotheprobabilityis 1 = 2 if Z 0 2 ker ( ) and0otherwise.Sothepossiblekernelsare Z 0 and SY 1 .Soaftertherstiterationtheprobabilitiesthatthe j ker ( ) j =2 N are p ( a N )= 8>>>><>>>>: (1 = 2) N = n (1 = 2) N = n 1 0 otherwise After i iterationstheprobabilitiesare p ( a N )= 8>>>><>>>>: (1 = 2) i N = n (1 = 2) i (2 i 1) N = n 1 0 otherwise 4)When G = G Thentheprobabilityofmeasuringanirreduciblerepresent ation is 1/2.Henceifthekernelsaregivenby j ker ( ) j =3.2 N thentheprobabilitiesofgetting theintersectiontobeoneofthemafter i iterationsare p ( a N )= 8>>>><>>>>: (1 = 2) i N = n (1 = 2) i (2 i 1) N = n 1 0 otherwise When H = Z 0 1) G = TZ 0 Thentheprobabilityofmeasuringanirreduciblerepresent ation of TZ 0 is1/3when Z 0 2 ker ( ) and0otherwise.Thepossiblekernelsare TZ 0 and Z 0 25

PAGE 26

After i iterationstheprobabilitiesofgetting j ker ( ) j =3 a 2 N are p ( a N )= 8>>>><>>>>: (1 = 3) i N = n 1, a =1 1 (1 = 3) i N = n 1, a =0 0 otherwise 2) G = Z 0 Theprobabilityofmeasuringthetrivialrepresentationis 1and0 otherwise.Soafter i iterationstheprobabilitiesthat j ker ( ) j =2 N are p ( a N )= 8><>: 1 N = n 1 0 otherwise 3) G = SY 1 Theprobabilityofmeasuringanirreduciblerepresentatio n of SY 1 is 1/2if Z 0 isinkernelof and0otherwise.Thepossiblekernelsof are SY 1 and Z 0 theprobabilitiesafter i iterationsofgetting j ker ( ) j =2 N are p ( a N )= 8>>>><>>>>: (1 = 2) i N = n (1 = 2) i (2 i 1) N = n 1 0 otherwise If G = g 1 SY 1 ( g 1 ) 1 theprobabilitiesaregoingtobethesame. 4) G = G Theprobabilityofmeasuringanirreduciblerepresentatio n is1/6if islinearandcontains Z 0 initskerneland2/3ifitisnon-linearandcontains Z 0 inits kerneland0otherwise.Thepossiblekernelsare Z 0 TZ 0 and G Sotheprobabilities that j ker ( ) j =3 a 2 N are p ( a N )= 8>>>>>>><>>>>>>>: (1 = 6) N = n a =1 (1 = 6) N = n 1, a =1 2 = 3 N = n 1, a =0 0 otherwise 26

PAGE 27

Soafter i iterationstheprobabilitiesare p ( a N )= 8>>>>>>><>>>>>>>: (1 = 6) i N = n a =1 (1 = 6) i (2 i 1) N = n 1, a =1 (1 1 3 i ) N = n 1, a =0 0 otherwise When H < TZ 0 and j H j =3.2 N 1)When G = TZ 0 Theprobabilityofmeasuringanirreduciblerepresentatio n is 1 j TZ 0 j X h 2 H ( h )=2 k +1 = 2 n =2 k +1 n if H 2 ker ( ) andequalto0otherwise.Sotheprobabilitiesthat j ker ( ) j =3.2 N are p ( N )= 8>>>><>>>>: 0 N < k 2 k +1 n N = n 1 1 2 N k +1 k N < n 1 After i iterationstheprobabilitythattheintersectionoftheker nelsis H isgivenby 1 [ p ( N 6 = k )] i =1 (1 = 2) i Theprobabilitythattheintersectionis TZ 0 isgivenby [ p ( N = n 1)] i Theprobability thattheintersectionisasubgroupof TZ 0 oforder 3.2 m where k < m < n 1 isgivenby p ( N m ) p ( N > m )=[ 2 2 m k +1 ] i [ 1 2 m k +1 ] i =[ 1 2 m k +1 ] i (2 i 1) Hencewehaveafter i iterations p ( N )= 8>>>>>>><>>>>>>>: 0 N < k (2 k +1 n ) i N = n 1 ( 1 2 m k +1 ) i (2 i 1) k < m < n 1 (1 [1 = 2] i ) N = k 27

PAGE 28

2)When G = Z 0 Theprobabilityofmeasuringanirreduciblerepresentatio n is 1 j Z 0 j X h 2 H T Z 0 ( h )=2 k +1 = 2 n =1 = 2 n k 1 if H T Z 0 2 ker ( ) and0otherwise.Sotheprobabilitiesthat j ker ( ) j =2 N are p ( N )= 8>>>><>>>>: 0 N < k 2 k +1 n N = n 1 1 2 N k +1 k N < n 1 After i iterationstheprobabilitythattheintersectionoftheker nelsis Z 0 is [1 = 2 n k 1 ] i Theprobabilitythattheintersectionisasubgroupof Z 0 oforder 2 m where k < m < n 1 is [1 = 2 m k +1 ] i (2 i 1) .Henceafter i iterationswehave p ( N )= 8>>>>>>><>>>>>>>: 0 N < k (2 k +1 n ) i N = n 1 ( 1 2 m k +1 ) i (2 i 1) k < m < n 1 (1 [1 = 2] i ) N = k 3)When G = SY 1 Theprobabilityofmeasuringanirreduciblerepresentatio n is givenby 1 j SY 1 j X h 2 H T SY 1 ( h )=2 k = 2 n =1 = 2 n k if H T SY 1 2 ker ( ) and0otherwise.Sotheprobabilitiesthat j ker ( ) j =2 N are p ( N )= 8>>>><>>>>: 0 N < k 1 = 2 n k N = n 1 2 N k +1 k N < n 28

PAGE 29

Theprobabilitiesafter i iterationsare p ( N )= 8>>>>>>><>>>>>>>: 0 N < k (1 = 2 n k ) i N = n ( 1 2 m k +1 ) i (2 i 1) k < m < n (1 [1 = 2] i ) N = k 4)When G = G Thentheprobabilityofmeasuringanirreduciblerepresent ation of G is 2 k = 2 n =1 = 2 n k if H 2 ker ( ) and0otherwise.Theprobabilitiesthat j ker ( ) j =3.2 N aregivenby p ( N )= 8>>>><>>>>: 0 N < k 1 = 2 n k N = n 1 2 N k +1 k N < n After i stepstheprobabilitiesare p ( N )= 8>>>>>>><>>>>>>>: 0 N < k (1 = 2 n k ) i N = n ( 1 2 m k +1 ) i (2 i 1) k < m < n (1 [1 = 2] i ) N = k When H < Z 0 and j H j =2 k 1)When G = TZ 0 Theprobabilityofmeasuringanirreduciblerepresentatio n is givenby d j TZ 0 j X h 2 H ( h )= 1 3 2 k n +1 if H 2 ker ( ) and0otherwise.Theprobabilitiesthat j ker ( ) j =3 a 2 N aregivenby 29

PAGE 30

p ( a N )= 8>>>>>>>>>><>>>>>>>>>>: 0 N < k 1 3.2 n k 1 N = n 1, a =1 2 k N 1 3 k N < n 1, a =1 2 k N 3 k N < n 1, a =0 1 3.2 n k 2 N = n 1, a =0 Thisprobabilitydistributionisaproductoftwoindepende ntprobabilitydistributions p 0,3 ( i ) and p 0,2 ( i ) where p 0,3 ( i )= 8><>: 2 = 3 i =0 1 = 3 i =1 and p 0,2 ( i )= 8>>>><>>>>: 0 i < k 1 = 2 n k 1 i = n 1 2 k i 1 i < n 1 After i iterationstheprobabilitythattheintersectionoftheker nelsis TZ 0 is 1 3 i 1 (2 n k 1 ) i Theprobabilitythattheintersectionis Z 0 is (1 1 3 i )( 1 (2 n k 1 ) i Theprobability thattheintersectionis H is (1 1 3 i )(1 1 2 i ). Theprobabilitythattheintersectionisa subgroupof TZ 0 oforder 3.2 m isgivenby 1 3 i (2 i 1) 1 (2 m k +1 ) i andtheprobabilitythatthe intersectionofkernelsisasubgroupof Z 0 oforder 2 m isgivenby (1 1 3 i )(2 i 1) 1 (2 m k +1 ) i Hencewehave p ( a N )= 8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>: 0 N < k 1 3 i 1 (2 n k 1 ) i N = n 1, a =1 (1 1 3 i ) 1 (2 n k 1 ) i N = n 1, a =0 (1 1 3 i )(1 1 2 i ) N = k a =0 1 3 i (1 1 2 i ) N = k a =1 1 3 i (2 i 1)( 1 2 m k +1 ) i N = m a =1, k < m < n 1 (1 1 3 i )(2 i 1)( 1 2 m k +1 ) i N = m a =0 30

PAGE 31

2)When G = SY 1 Theprobabilityofmeasuringanyirreduciblerepresentati on is 1 = 2 n k if H isinthekernelof and0otherwise.Sotheprobabilitiesthat j ker ( ) j =2 N aregivenby p ( N )= 8>>>><>>>>: 0 N < k (1 = 2 n k ) N = n ( 1 2 N k +1 ) k N < n After i iterationstheprobabilitiesthattheintersectionofthek ernelshaveorder 2 N are givenby p ( N )= 8>>>>>>><>>>>>>>: 0 N < k (1 = 2 n k ) i N = n (1 1 2 i ) N = k 1 (2 m k +1 ) i (2 i 1) k < m < n 3)When G = Z 0 Theprobabilityofmeasuringanyirreduciblerepresentati on of Z 0 isgivenby 1 = 2 n k 1 if H iscontainedinthekernelof and0otherwise.The probabilitiesthat j ker ( ) j =2 N aregivenby p ( N )= 8>>>><>>>>: 0 N < k (1 = 2 n k 1 ) N = n 1 1 2 N k +1 k N < n 1 After i stepstheprobabilitiesoftheintersectionofthekernelsa re p ( N )= 8>>>>>>><>>>>>>>: 0 N < k (1 = 2 n k 1 ) i N = n 1 (1 1 2 i ) N = k 1 (2 m k ) i (1 1 2 i ) k < m < n 1 4)When G = G Theprobabilityofmeasuringanirreduciblerepresentatio n is givenby d j G j X h 2 H ( h ). 31

PAGE 32

When islinearthentheprobabilityis 2 k = 3.2 n = 1 3 2 k n if ker ( ) contains H andit is 1 32 n k 2 if isnonlinearanditskernelcontains H Sowehavethefollowingforthe probabilitiesofgetting j ker( ) j =3 a 2 N p ( a N )= 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>: 0 N < k 1 3 2 k N 1 k N n 1, a =1 1 3 2 k n N = n a =1 0 N = n a =0 1 3 2 k n +2 N = n 1, a =0 1 3 2 k N k N < n 1, a =0 After i iterationstheprobabilitythatintersectionofkernelsha veorder 3 a 2 N is ( p ( a N )) i ( p ( a +1, N )) i ( p ( a N +1)) i +( p ( a +1, N +1)) i Hencewehave p ( a N )= 8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>: 0 N < k ( 2 k n +1 3 ) i ( 2 k n 3 ) i N = n 1, a =1 (2 k m ) i ( 2 k m 3 ) i (2 k m 1 ) i +( 2 k m 1 3 ) i k m < n 1, a =0 ( 2 k m 3 ) i ( 2 k m 1 3 ) i k m < n 1, a =1 (2 k n +1 ) i ( 2 k n +1 3 ) i N = n 1, a =0 0 N = n a =0 1 3 i ( 1 2 n k ) i N = n a =1 32

PAGE 33

CHAPTER3 KEMPE-SHALEVDISTINGUISHABLITY Fortheremainderofthiswork,weconsiderthedecisionvers ionoftheHidden Subgroupproblem.Aswesaw,itisstatedasfollows. DecisionVersionoftheHiddenSubgroupProblem :Let G beanitegroupand H asubgroup.Givenablack-boxfunction f : G S whichisconstanton(left)-cosets gH of H andtakesdifferentvaluesfordifferentcosets,determine whether H = f e g or not. Therearetwowaystoformalizetheexistenceofsolutionsto thisproblem.Inthis chapter,wediscussthedenitiongivenbyKempeandShalev( 18 ).Adifferentway todenethiswillbediscussedinthenextchapter.Kempeand Shalevproposethatif thesubgroup H of G producesprobabilitiesofrepresentationsunderQuantumF ourier Samplingthatareveryclosetothosethatarisefromthetriv ialsubgroup,thenthe subgroupshouldbecalledindistinguishablefromthetrivi alsubgroup.Moreprecisely, thedenitionproposedbyKempeandShalev( 18 )in2005isasfollows. Denition 4 Let G beanitegroup.Let Irr( G ) bethesetofirreducible(complex) representationsof G .Wedenoteby thecharacterassociatedtoa 2 Irr( G ) andlet d beit'sdegree.For H < G wedene D H = 1 j G j X 2 Irr( G ) d j X h 6 = e h 2 H ( h ) j Wesaythatasubgroup H isdistinguishableif D H log ( j G j ) c forsome c > 0, c being independentof G Webeginthissectionbyprovingthatallsubgroupsofanabel iangroupare Kempe-Shalevdistinguishable. Theorem3.1. Subgroupsofanabeliangroup G areKempe-Shalevdistinguishable. 33

PAGE 34

Proof. Let H beahiddensubgroupoftheabeliangroup G Then D H = 1 j G j X d j X h 6 = e ( h ) j = 1 j G j X j X h 6 = e ( h ) j = 1 j G j [ X ker ( ) H j X h 6 = e ( h ) j + X ker ( ) + H j X h 6 = e ( h ) j ] = 1 j G j [ X ker ( ) H jj H j < ( ) H ,1 H > (1) j + X ker ( ) + H jj H j < ( ) H ,1 H > (1) j ] = 1 j G j [ X ker ( ) H jj H j (1) (1) j + X ker ( ) + H j (1) j ] = 1 j G j [ X ker ( ) H (1)[ j H j 1]+ X ker ( ) + H (1)] = 1 j G j [ X ker ( ) H [ j H j 1]+ X ker ( ) + H 1] = 1 j G j [ j G j = j H j [ j H j 1]+( j G jj G j = j H j )]=2 1 j G j ( j G jj G j = j H j )=2(1 1 j H j ) > 1 > (log( j G j ) c foreach c > 0 as j G j!1 Inthischapterwestudythedistinguishabilityofsubgroup sofFrobeniusgroups usingtheolddenitionofdistinguishability.Herewecons iderFrobeniusgroupswith abeliankernel.Wegivenecessaryandsufcientconditions forthedistinguishabilityof thesubgroupsoftheFrobeniusgroupsundercertainconditi ons. Denition 5 Let G beanitegroup.Let H G and H isanontrivialpropersubgroupof G .Assumethat H \ H g =1 whenever g 2 G H .Then H isaFrobeniuscomplementin G .AgroupwhichcontainsaFrobeniuscomplementiscalledaFr obeniusgroup. 34

PAGE 35

Theorem3.2 (Frobenius) Let G beaFrobeniusgroupwithcomplement H .Thenthere exists K G with HK = G and H \ K =1 Proof. See( 16 ). Thissubgroup K iscalledtheFrobeniuskernelof G .ItisclearthataFrobenius groupwithkernel K andcomplement H canbewrittenas K o H .Italsofollowsthatany non-trivialelementof H xesnoelementof K i.e H actssemiregularlyon K Theorem3.3. Let G = Z ( ) g ( ) oZ beaFrobeniusgroupwhere g ( ) isafunctionof and ( ) iscoprimeto ,andforsome c > 0 andforallsufcientlylarge g ( ) > 1 c .Let H beasubgroupoforder ( ) f ( ) or ( ) f ( ) where f ( ) isafunctionof such that 0 f ( ) g ( ) then H isKempe-Shalevdistinguishableifandonlyif f ( ) > 0 for all >> 0. Proof. Suppose f ( ) > 0 for >> 0, andtake sufcientlylarge.Then f ( ) > 0 and H hasorderdivisibleby whichiscoprimeto Furthermoreforeach 2 Irr ( G ) j X h 6 = e h 2 H ( h ) j = jj H j [( ) H ,1 H ] (1) j 1, since alpha divides j H j andiscoprimeto (1). Hence X = non-linear j X h 6 = e h 2 H ( h ) j ( ) g ( ) 1 Hence D H ( j H j 1)+ ( g ( ) 1 ) ( ) g ( ) > ( ( ) g ( ) 1)+1 ( ) g ( ) = 1 > 1 ( log j G j ) c forsome c > 0. Hence H isdistinguishable.If f ( ) =0then H hasordereither f ( ) =1 or ( ) f ( ) = .Inthiscase D H 2( 1) ( ) g ( ) 35

PAGE 36

andforeach c > 0 andsufcientlylarge ,wehave 2( 1) ( ) g ( ) < ( log ( j G j )) c Therefore,if f ( )=0 innitelyoften, H isindistinguishable.Hencetheproof. Corollary3.4. TheFrobeniuscomplementof G isindistinguishable. Proof. TheFrobeniuscomplementof G hasorder .Hence f ( )=0 .Fromtheprevious theoremthecomplementisindistinguishable. Corollary3.5. TheFrobeniuskernelof G isKempe-Shalevdistinguishable. Proof. TheorderoftheFrobeniuskernelis ( ) g ( p ) .Hencethecorollaryfollowsfrom theprevioustheorem. Proposition3.6. Forthefrobeniusgroup Z 2 p 1 oZ p thesmallest D S for S anon-trivial subgroupisobtainedwhenthesubgroupSistheSylow p –subgroupofthegroup. Proof. Fromthecharactertableitisevidentthatwhen S istheSylow p –subgroupthen P d j P h 6 = e h 2 S ( ) j =2 ( p 1) thecontributionfromnon-linearcharactersbeingzero. So D S = 2( p 1) 2 p 1 p .NowifHisasubgroupwith j H j =2 k then D H = p (2 k 1)+.... 2 p 1 p where p (2 k 1) isthecontributionfromlinearcharacters.This D H isobviouslybiggerthan D S when k > 1.If k =1then j H j =2and D H = ( p 1)+ p x 2 p 1 p where x = P = non-linear j P h 6 = e h 2 H ( ) j > 0. Thissumisequalto P = non-linear jj H j [( ) H ,1 H ] (1) j whichisbiggerthan1 because j H j [( ) H ,1 H ] (1) isneverequaltozeroandbiggerthan1.Hencethis isalsobiggerthan D S .Anyothersubgroupof G willbeoftheform H w Z 2 k oZ p This subgroup H has 2 k 1 elementsoforder2andso D H = p (2 k 1)+.... 2 p 1 p .Since k > 1thisis greaterthan D S Lemma3.7. Let > 1 beaconstant,andlet m n g benon-negativerealvalued functionsof N .Assumethat m ( i ), n ( i ) !1 as i !1 .Alsoassumethatthereexists 36

PAGE 37

some c 0 > 0 suchthat g ( i ) c 0 > m ( i ) n ( i ) forallsufcientlylarge i .Thenforeach c > 0 g ( i ) m ( i ) n ( i ) > g ( i ) c forallsufcientlylarge i Proof. Let c > 0 .Forallsufcientlylarge i ,wehave g ( i ) > g ( i ) c + c 0 .Henceforall sufcientlylarge i wehave g ( i ) > g ( i ) c + c 0 = g ( i ) c g ( i ) c 0 > g ( i ) c m ( i ) n ( i ) Hence g ( i ) m ( i ) n ( i ) > g ( i ) c Theorem3.8. Let G = Z g ( i ) oZ m ( i ) oZ n ( i ) beaFrobeniusgroupwhere ( m ( i ), n ( i ))=1 and iscoprimeto m ( i ) and n ( i ). Assumethat m ( i ), n ( i ) !1 as i !1 Alsoassume thatforsome c > 0 g ( i ) c m ( i ) n ( i ) andforsufcientlylarge i .Let H beasubgroup oforder f ( i ) or f ( i ) m ( i ) or f ( i ) n ( i ) or f ( i ) m ( i ) n ( i ) .Then H isdistinguishableiff f ( i ) > 0 forallsufcientlylarge i > 0 Proof. Let f ( i ) > 0 andtake i sufcientlylarge.Then H hasorderdivisibleby whichis coprimeto m ( i ) and n ( i ) .Thereforeforeach 2 Irr ( G ) j X h 6 = e h 2 H ( h ) j = jj H j [( ) H ,1 H ] (1) j 1. Hence X = non-linearofdegreem(i)n(i) j X h 6 = e h 2 H ( h ) j g ( n ) 1 mn Hence D H = P = linear j P h 6 = e h 2 H ( h ) j + P = non-linear d j P h 6 = e h 2 H ( h ) j g ( i ) m ( i ) n ( i ) Hence D H 1+ P = non-linear d j P h 6 = e h 2 H ( h ) j g ( i ) m ( i ) n ( i ) 1+ m ( i ) n ( i ) P = non-linearofdegreem(i)n(i) 1 g ( i ) m ( i ) n ( i ) 37

PAGE 38

1+ m ( i ) n ( i ). g ( i ) 1 m ( i ) n ( i ) g ( i ) m ( i ) n ( i ) = 1 m ( i ) n ( i ) whichisevidentlybiggerthan log ( j G j ) c fromhypotheses.If f ( i )=0 then j H j = m ( i ) or n ( i ) or m ( i ) n ( i ). If j H j = m ( i ) then D H n ( i )( m ( i ) 1)+ n ( i )( m ( i ) 1) n ( i )( m ( i ) 1) g ( i ) m ( i ) n ( i ) = n ( i )( m ( i ) 1) g ( i ) m ( i ) n ( i ) + n ( i ) 2 ( m ( i ) 1) 2 g ( i ) m ( i ) n ( i ) 1 g ( i ) + m ( i ) n ( i ) g ( i ) ( log j G j ) c foreach c > 0 andsufcientlylarge i fromLemma 3.7 .If j H j = n ( i ) then D H 2( n ( i ) 1)+ n ( i )( m ( i ) 1) n ( i )( n ( i ) 1) g ( i ) m ( i ) n ( i ) = 2 g ( i ) m ( i ) 2 g ( i ) m ( i ) n ( i ) + n ( i ) 2 g ( i ) ( log j G j ) c foreach c > 0 andsufcientlylarge n ( i ). If j H j = m ( i ) n ( i ) then D H ( m ( i ) n ( i ) 1)+( n ( i ) 1) m ( i ) n ( i )+ n ( i ) m ( i ) 1 n ( i ) n ( i ) g ( i ) m ( i ) n ( i ) n ( i ) g ( i ) + 1 g ( i ) 1 g ( i ) m ( i ) ( log j G j ) c foreach c > 0 andsufcientlylarge i fromLemma 3.7 .Hencetheproof. Theorem3.9. Let G = Z g ( i ) o ( SL (2,5) ( Z m ( i ) oZ n ( i ) )) beaFrobeniusgroupwhere i 2 N ( m ( i ), n ( i ))=1 and > 1 isaconstantandcoprimeto m ( i ) and n ( i ). Assume that m ( i ), n ( i ) !1 as i !1 Alsoassumethatforsome c > 0 g ( i ) c m ( i ) n ( i ) forallsufcientlylarge i .Let H beasubgroupoforder f ( i ) or f ( i ) m ( i ) or f ( i ) n ( i ) or f ( i ) m ( i ) n ( i ) or f ( i ) 120 m ( i ) or f ( i ) 120 n ( i ) .Then H isKempe-Shalevdistinguishableiff f ( i ) > 0 forallsufcientlylarge i 38

PAGE 39

Proof. Let f ( i ) > 0 andtake i sufcientlylarge.Then H hasorderdivisibleby whichis coprimeto m ( i ) and n ( i ). Thereforeforeach 2 Irr ( G ) j X h 6 = e h 2 H ( h ) j = jj H j [( ) H ,1 H ] (1) j 1. Hence X = non-linearofdegree120m(i)n(i) j X h 6 = e h 2 H ( h ) j g ( i ) 1 120 m ( i ) n ( i ) Hence D H = P = linear j P h 6 = e h 2 H ( h ) j + P = non-linear d j P h 6 = e h 2 H ( h ) j g ( i ) .120 m ( i ) n ( i ) Hence D H 1+ P = non-linear d j P h 6 = e h 2 H ( h ) j g ( i ) .120 m ( i ) n ( i ) 1+ m ( i ) n ( i ) P = non-linearofdegree120m(i)n(i) 1 g ( i ) .120 m ( i ) n ( i ) 1+ m ( i ) n ( i ). g ( i ) 1 120 m ( i ) n ( i ) g ( i ) .120 m ( i ) n ( i ) = 1 120 m ( i ) n ( i ) whichisevidentlybiggerthan log ( j G j ) c forsufcientlylarge i .Nowlet f ( i ) > 0 .If theorderof H is 120 m ( i ) thensincethereare n ( i ) linearcharacters P = linear P h 6 = e h 2 H j ( h ) j n ( i )(120 m ( i ) 1) .Also jj H j [( ) H ,1 H ] (1) j (1)(120 m ( i ) 1) ( D H ) g ( i ) 120 m ( i ) n ( i ) n ( i )(120 m ( i ) 1)+2 n ( i ) 2( m ( i ) 1) n ( i ) 2 n ( i )(120 m ( i ) 1) +3 n ( i )2( m ( i ) 1)3 n ( i )(120 m ( i ) 1) +4 n ( i )2( m ( i ) 1)4 n ( i )(120 m ( i ) 1) +5 n ( i )2( m ( i ) 1)5 n ( i )(120 m ( i ) 1) 39

PAGE 40

+6 n ( i )2( m ( i ) 1)6 n ( i )(120 m ( i ) 1). Hence D H < 1 g ( n ) + (8+18+16+25+36) m ( i ) n ( i ) g ( i ) < ( log j G j ) c foreach c > 0 andsufcientlylarge i .Iforderof H is 120 n ( i ) then jj H j [( ) H ,1 H ] (1) j (1)(120 n ( i ) 1). Sincethereare n ( i ) linearcharactersthecontributionfromthemislessthanor equalto n ( i )(120 n ( i ) 1) .Then ( D H )120 g ( i ) m ( i ) n ( i ) n ( i )(120 n ( i ) 1) +2 n ( i )2( m ( i ) 1)2 n ( i )(120 n ( i ) 1) +3 n ( i )2( m ( i ) 1)3 n ( i )(120 n ( i ) 1) +4 n ( i )2( m ( i ) 1)4 n ( i )(120 n ( i ) 1) +5 n ( i )2( m ( i ) 1)5 n ( i )(120 n ( i ) 1) +6 n ( i )2( m ( i ) 1)6 n ( i )(120 n ( i ) 1). Hence ( D H )120 g ( i ) m ( i ) n ( i ) < 120( n ( i ) 3 +8 n ( i ) 3 m ( i )+18 n ( i ) 3 m ( i ) +32 n ( i ) 3 m ( i )+25 n ( i ) 3 m ( i )+36 n ( i ) 3 m ( i )) < 120 2 n ( i ) 3 m ( i ). Hence D H < 120 n ( i ) 3 m ( i ) g ( i ) m ( i ) n ( i ) < ( log j G j ) c foreach c > 0 40

PAGE 41

andsufcientlylarge i fromLemma 3.7 .Iforderof H is m ( i ) then jj H j [( ) H ,1 H ] (1) j (1)( m ( i ) 1). Thecontributionfromlinearchararctersislessthanorequ alto n ( i )( m ( i ) 1) .Thus ( D H )120 g ( i ) m ( i ) n ( i ) n ( i )( m ( i ) 1) +2 n ( i )2( m ( i ) 1)2 n ( i )( m ( i ) 1) +3 n ( i )2( m ( i ) 1)3 n ( i )( m ( i ) 1) +4 n ( i )2( m ( i ) 1)4 n ( i )( m ( i ) 1) +5 n ( i )( m ( i ) 1)5 n ( i )( m ( i ) 1) +6 n ( i )( m ( i ) 1)6 n ( i )( m ( i ) 1) ( D H )120 g ( i ) m ( i ) n ( i ) < n ( i ) m ( i )+8 m ( i ) 2 n ( i ) 2 +18 m ( i ) 2 n ( i ) 2 +32 m ( i ) 2 n ( i ) 2 +25 m ( i ) 2 n ( i ) 2 +36 m ( i ) 2 n ( i ) 2 Hence D H < 1 120 g ( i ) + 119 m ( i ) n ( i ) 120 g ( i ) < ( log j G j ) c foreach c > 0 andsufcientlylarge i fromLemma 3.7 .Iforderof H is n ( i ) thenthecontributionfromlinearcharactersislessthanor equalto 2( n ( i ) 1) ( D H )120 g ( i ) m ( i ) n ( i ) 2( n ( i ) 1) +2 n ( i )2( m ( i ) 1)2 n ( i )( n ( i ) 1) +3 n ( i )2( m ( i ) 1)3 n ( i )( n ( i ) 1) +4 n ( i )2( m ( i ) 1)4 n ( i )( n ( i ) 1) +5 n ( i )2( m ( i ) 1)5 n ( i )( n ( i ) 1) +6 n ( i )2( m ( i ) 1)6 n ( i )( n ( i ) 1). 41

PAGE 42

Hence ( D H )120 g ( i ) m ( i ) n ( i ) < ( n ( i ) 3 +8 n ( i ) 3 m ( i )+18 n ( i ) 3 m ( i ) +32 n ( i ) 3 m ( i )+25 n ( i ) 3 m ( i )+36 n ( i ) 3 m ( i )) < 120 n ( i ) 3 m ( i ). Hence D H < n ( i ) 3 m ( i ) g ( i ) m ( i ) n ( i ) < ( log j G j ) c foreach c > 0 andsufcientlylarge i fromLemma 3.7 Iftheorderof H is m ( i ) n ( i ) then jj H j [( ) H ,1 H ] (1) j (1)( m ( i ) n ( i ) 1). Sothecontributionfromlinearcharactersislessthanoreq ualto n ( i )( m ( i ) n ( i ) 1) ( D H )120 g ( i ) m ( i ) n ( i ) n ( i )( m ( i ) n ( i ) 1) +2 n ( i )2( m ( i ) n ( i ) 1)2 n ( i ) +3 n ( i )2( m ( i ) n ( i ) 1)3 n ( i ) +4 n ( i )2( m ( i ) n ( i ) 1)4 n ( i ) +5 n ( i )( m ( i ) n ( i ) 1)5 n ( i )+6 n ( i )( m ( i ) n ( i ) 1)6 n ( i ) Hence D H < n ( i ) 2 g ( i ) < ( log j G j ) c foreach c > 0 and sufcientlylarge i fromLemma 3.7 .Henceinallthesecases H isindistinguishable. Hencetheproof. 42

PAGE 43

CHAPTER4 ALGORITHMICDISTINGUISHABILITY InChapter 3 wehavediscussedindetailKempe-Shalevdistinguishabili tyas denedbyKempeandShalevin( 18 ).WhiletheKempeShalevdenitioncapturesthe difcultyofdistinguishingthehiddensubgroupfromthetr ivialgroupinsomecases, itisnotalwaysobvioushowtotranslateitsanswerstopract icalalgorithms.Inthis chapterwedene”Algorithmicdistinguishability”.Algor ithmicdistinguishabilitydenes distinguishabilityofasubgroup H ofanynitegroup G fromthetrivialoneonthe basisofwhenanaturalalgorithmthatusestheweakstandard methodsucceedsin polynomialtime.Westudythisnewdenitionasappliedtoan umberofexamples.We alsoshowthatthenewconceptcoincidesfortheFrobeniusgr oupswiththeKempe Shalevdenitiondiscussedinthepreviouschapter. 4.1Analgorithmfordistinguishability Anaturalalgorithmtotellwhetherornotthehiddensubgrou pistrivialisas follows.ApplyQuantumFourierSampling m times.Considertheresultingsequenceof representations 1 ,.., m Askwhetheritismorelikelytoobtainthisparticularseque nce ifthehiddensubgroupis H orifitistrivial.Return 1 ifitis H andreturn 0 ifitistrivial. Thisalgorithmisgivenbyafunction 4 : Irr ( G ) m !f 0,1 g whichcanbeobtainedasfollows: Let G beanitegroup,andlet H beanon-trivialsubgroupof G .Wedenote P H ,1 :Irr( G ) [0,1] thefunctiondenedby P H ,1 ( )= (1) j H j j G j j H ,1 h j 43

PAGE 44

forall 2 Irr( G ) .Foreachpositiveinteger m ,wedenoteby P H m :Irr( G ) m [0,1] thefunctiondenedby P H m ( 1 ,..., m )= m Y i =1 P H ,1 ( i ). If m isunderstoodfromthecontext,wedenote P H m simply P H .Let 4 :Irr( G ) m !f 0,1 g beafunctiondenedby 4 ( 1 ,.., m )= 8><>: 0 P 1, m ( 1 ,.., m ) P H m ( 1 ,.., m ) 1 P 1, m ( 1 ,.., m ) < P H m ( 1 ,.., m ) Thustheformalalgorithmisasfollows:Algorithm : StepI:RepeatQuantumFourierSampling m times. StepII:Getasequenceof i where 1 i m StepIII:Applythefunction 4 tothetuple ( 1 ,.., m ). StepIV:Ifthefunctionreturns 1 then H isnon-trivialandifitreturns 0 then otherwise. Denition 6 Let G beanitegroup,let H beanon-trivialsubgroupof G ,andlet A and b beconstants.Wesaythat H is distinguishablewithconstants A and b > 0 ifthereexists some m suchthatthefollowinghold. 1. m A log 2 ( j G j ) b 2. X ( 1 ,..., m ) 24 1 (0) P 1 ( 1 ,..., m ) > 1 = 2, 44

PAGE 45

3. Wealsohavethefollowing: X ( 1 ,..., m ) 24 1 (1) P H ( 1 ,..., m ) > 1 = 2. Hence,wearesayingthatthereisanalgorithm f whichgivenasequenceofresults returnsitsguessofeither trivial or nontrivial ,andthisfunctioniscorrectwithmorethan probability 1 = 2 bothwhenthehiddensubgroupis H andwhenthehiddensubgroupis 1 .Thisdenitionseemstocapturetheessenceofwhatitmeans forasubgrouptobe distinguishable .MostauthorsonQuantumComputingthinkintermsofsequenc esof groups,andforthisreasontheconstants A and b neednotbeexplicitlymentioned. Denition 7 Let ( G i ) 1i =1 beasequenceofnitegroupsandlet ( H i ) 1i =1 beasequenceof non-trivialsubgroupswhere H i G i Wesaythat ( H i ) 1i =1 isdistinguishableifthereexists someconstants A and b ,suchthat H i isdistinguishablewithconstants A and b inthe previoussenseforall i 4.2AbelianGroups Usingthisdenitionwewillrstshowthatthenon-trivials ubgroupsofanabelian group G areallalgorithmicallydistinguishable. Theorem4.1. Allnon-trivialsubgroups H ofanabeliangroup G arealgorithmically distinguishable. Proof. Let G beanabeliangroupand H anynon-trivialsubgroupof G Since G is abelianallirreduciblecharactersof G arelinear. H isnormalin G andiscontainedinthe kernelsof j G = H j linearcharacters.Now P H ( )= (1) j H j < H ,1 H > j G j = 8><>: 0 H ker ( ) 1 [ G : H ] otherwise So P H m ( 1 ,.., m )= 8><>: 0 H ker ( i ) forsomei ( 1 [ G : H ] ) m otherwise 45

PAGE 46

Also P 1, m ( 1 ,.., m )= m Y i =1 1 j G j =( 1 j G j ) m Weset 4 ( 1 ,.., m )= 8><>: 0 P 1, m ( 1 ,.., m ) > P H m ( 1 ,.., m ) 1 P 1, m ( 1 ,.., m ) < P H m ( 1 ,.., m ) Weobservethat X ( 1 ,.., m ) 24 1 (1) P 1, m ( 1 ,.., m )=( 1 j G j ) m [( j G = H j ) m ]=( 1 j H j ) m < 1 = 2 forall m > 0. Hence X ( 1 ,.., m ) 24 1 (0) P 1, m ( 1 ,.., m ) > 1 = 2 forall m > 0. Also X ( 1 ,.., m ) 24 1 (1) P H m ( 1 ,.., m )=1 forall m > 0. So H isdistinguishable. 4.3FrobeniusGroups Inthissectionwestudythealgorithmicditinguishability ofthekernelsand complementsoftheFrobeniusgroupsdiscussedinChapter3. Wehavethefollowing lemma. Lemma4.2. If 0 < x < 1 and k > 1 then (1 x ) k > 1 kx Proof. Let f ( x )=(1 x ) k (1 kx ). Differentiatingbothsideswithrespectto x weget g ( x )= k (1 x ) k 1 + k = k [1 (1 x ) k 1 ] > 0. 46

PAGE 47

Hence f ( x ) isincreasingandalso f (0)=0. Hence (1 x ) k > 1 kx Theorem4.3. Let G = Z g ( ) ( ) o Z beaFrobeniusgroupwhere ( ), g ( ) arefunctions of and ( ) iscoprimeto and ( ), g ( ) !1 as !1 ThentheKernelof G is algorithmicallydistinguishableandthecomplementisnot Proof. Let H bethekernelof G Then H hasorder ( ) g ( ) Then P H ( )= (1) j H j j G j = (1) ( ) g ( ) ( ) g ( ) = (1) if H ker ( ) andisequalto 0 otherwise.Since H isinthekernelofonlylinear characterswehave P H ( )= 1 when H ker ( ) andisequalto 0 otherwise.So P H m ( 1 ,.., m )= 8><>: ( 1 ) m i (1)=1 andH ker ( i ) 0 otherwise Also P 1, m ( 1 ,.., m )= m Y i =1 i (1) 2 ( ) g ( ) Soweseethat P 1, m ( 1 ,.., m ) > P H m ( 1 ,..., m ) if P H m ( 1 ,.., m )=0. and P 1, m ( 1 ,.., m ) < P H m ( 1 ,..., m ) otherwise.Weset 4 ( 1 ,.., m )= 8><>: 1 P H m ( 1 ,.., m ) > P 1, m ( 1 ,.., m ) 0 P H m ( 1 ,.., m ) < P 1, m ( 1 ,.., m ) Now X ( 1 ,.., m ) 24 1 (1) P 1, m ( 1 ,.., m )=( 1 ( ) g ( ) ) m m < 1 = 2. 47

PAGE 48

Hence X ( 1 ,.., m ) 24 1 (0) P 1, m ( 1 ,.., m ) > 1 = 2 forall m > 0 and !1 Also X ( 1 ,.., m ) 24 1 (1) P H m ( 1 ,.., m )=1. Hence H isalgorithmicallydistinguishable. Let H nowbetheFrobeniuscomplement.Then j H j = and P H ( )= (1) j H j j G j < H ,1 H > = 8>>>><>>>>: j H j j G j =1 G (1) 2 j G j (1)= 0 otherwise = 8>>>><>>>>: 1 g ( ) =1 G g ( ) (1)= 0 otherwise = 8><>: (1) g ( ) =1 G (1)= 0 otherwise So P H k ( 1 ,.., k )= 8><>: Q ki =1 i (1) g ( ) i =1 G i (1)= 0 otherwise Also P 1, k ( 1 ,.., k )= k Y i =1 i (1) 2 g ( ) Set 4 ( 1 ,.., k )= 8>>>><>>>>: 0 P 1, k ( 1 ,.., k )= P H k ( 1 ,.., k ) 0 P 1, k ( 1 ,.., k ) > P H k ( 1 ,.., k ) 1 P 1, k ( 1 ,.., k ) < P H k ( 1 ,.., k ) 48

PAGE 49

Wenoticethat X ( 1 ,.., k ) 24 1 (1) P H k ( 1 ,.., k )= X ( 1 ,.., k ) 24 1 (1) k Y i =1 i (1) g ( ) = 1 ( g ( ) ) k [(1+( g ( ) 1)) k ( g ( ) 1) k ]=[1 (1 1 g ( ) ) k ]. Nowsince k < c lg( j G j ) b where c > 0 and b > 0, forlarge wehave k ( ) g ( ) < 1 = 2. BythepreviousLemmaweget (1 1 g ( ) ) k > 1 k 1 1 ( ) g ( ) > 1 = 2. Thisinturnimpliesthat X ( 1 ,.., k ) 24 1 (1) P H k ( 1 ,.., k ) < 1 = 2. Hence H isnotalgorithmicallydistinguishable. Corollary4.4. Let G = Z p o Z 2 beaFrobeniusgroup.Thekernelof G isalgorithmically distinguishableandthecomplementisnot. Corollary4.5. Let G = Z p 1 2 o Z p beaFrobeniusgroup.Thekernelof G isalgorithmicallydistinguishableandthecomplementisnot. Theorem4.6. Let G = Z g ( i ) o Z m ( i ) o Z n ( i ) beaFrobeniusgroupwhere ( m ( i ), n ( i ))=1 and iscoprimeto m ( i ) and n ( i ). Wealsoassume m ( i ), n ( i ), g ( i ) !1 as i !1 Thenthekernelof G isalgorithmicallydistinguishablebutthecomplementisn ot. Proof. If G = Z g ( i ) o Z m ( i ) o Z n ( i ) and j H j = g ( i ) Now P H ( )= (1) j H j j G j < H ,1 H > 49

PAGE 50

Also P 1 ( )= (1) 2 j G j So P H ( )= 8><>: (1) 2 m ( i ) n ( i ) H Ker ( ) 0 otherwise Hence P H m ( 1 ,.., k )= 8><>: Q ki =1 i (1) 2 m ( i ) n ( i ) H Ker ( i ) 0 otherwise Also P 1, k ( 1 ,.., k )= k Y i =1 i (1) 2 g ( i ) m ( i ) n ( i ) Set 4 ( 1 ,.., k )= 8><>: 0 P 1, k ( 1 ,.., k ) > P H m ( 1 ,.., k ) 1 P 1, k ( 1 ,.., k ) < P H m ( 1 ,.., k ) So X ( 1 ,.., k ) 24 1 (0) P 1, k ( 1 ,.., k )= 1 ( g ( i ) m ( i ) n ( i )) k X ( 1 ,.., k ) 24 1 (0) k Y i =1 i (1) 2 1 ( g ( i ) m ( i ) n ( i )) k X ( 1 ,.., k ) 24 1 (0) m ( i ) 2 k n ( i ) 2 k = m ( i ) k n ( i ) k g ( i ) k ( g ( i ) 1) k m ( i ) k n ( i ) k =(1 1 g ( i ) ) k > 1 = 2 as i !1 Also X ( 1 ,.., k ) 24 1 (1) P H k ( 1 ,.., k ) > 1 = 2. Hence H isdistinguishable. If j H j = m ( i ) n ( i ) then P H ( )= (1) j H j j G j < H ,1 H > = 8>>>><>>>>: j H j j G j =1 G (1) 2 j G j (1)= m ( i ) n ( i ) 0 otherwise 50

PAGE 51

= 8>>>><>>>>: 1 g ( i ) =1 G m ( i ) n ( i ) g ( i ) (1)= m ( i ) n ( i ) 0 otherwise = 8><>: (1) g ( i ) =1 G (1)= m ( i ) n ( i ) 0 otherwise So P H k ( 1 ,.., k )= 8><>: Q ki =1 (1) g ( i ) =1 G (1)= m ( i ) n ( i ) 0 otherwise Also P 1, k ( 1 ,.., k )= k Y i =1 (1) 2 g ( i ) m ( i ) n ( i ) Set 4 ( 1 ,.., k )= 8>>>><>>>>: 0 P 1, k ( 1 ,.., k )= P H k ( 1 ,.., k ) 0 P 1, k ( 1 ,.., k ) > P H k ( 1 ,.., k ) 1 P 1, k ( 1 ,.., k ) < P H k ( 1 ,.., k ) Wenoticethat X ( 1 ,.., k ) 24 1 (1) P H k ( 1 ,.., k )= X ( 1 ,.., k ) 24 1 (1) k Y i =1 i (1) g ( i ) = 1 ( g ( i ) ) k [(1+( g ( i ) 1)) k ( g ( i ) 1) k ]=[1 (1 1 g ( i ) ) k ]. Theaboveexpressionislessthan 1 = 2 for i !1 andsmall k So H isnotalgorithmically distinguishable. Theorem4.7. Let G = Z g ( i ) o ( SL (2,5) Z m ( i ) o Z n ( i ) ) beaFrobeniusgroupwhere i 2 N ( m ( i ), n ( i ))=1 and > 1 isaconstantandcoprimeto m ( i ), n ( i ). Wealsoassumethat m ( i ), n ( i ), g ( i ) !1 as i !1 ThenthekernelofthisFrobeniusgroupisalgorithmically distinguishablebutthecomplementisnot. 51

PAGE 52

Proof. Let H betheFrobeniuskernel.So j H j = g ( i ) Now P H ( )= (1) j H j j G j < H ,1 H > Also P 1 ( )= (1) 2 j G j So P H ( )= 8><>: (1) 2 120 m ( i ) n ( i ) H Ker ( ) 0 otherwise So P H m ( 1 ,.., k )= 8><>: Q ki =1 i (1) 2 120 m ( i ) n ( i ) H Ker ( i ) 0 otherwise Also P 1, k ( 1 ,.., k )= k Y i =1 i (1) 2 120 g ( i ) m ( i ) n ( i ) Set 4 ( 1 ,.., k )= 8><>: 0 P 1, k ( 1 ,.., k ) > P H m ( 1 ,.., k ) 1 P 1, k ( 1 ,.., k ) < P H m ( 1 ,.., k ) So X ( 1 ,.., k ) 24 1 (0) P 1, k ( 1 ,.., k )= 1 (120 g ( i ) m ( i ) n ( i )) k X ( 1 ,.., k ) 24 1 (0) k Y i =1 i (1) 2 1 (120 g ( i ) m ( i ) n ( i )) k X ( 1 ,.., k ) 24 1 (0) m ( i ) 2 k n ( i ) 2 k = m ( i ) k n ( i ) k 120 k g ( i ) k ( g ( i ) 1) k 120 k m ( i ) k n ( i ) k =( 1 120 1 120 g ( i ) ) k > 1 = 2 as i !1 Also X ( 1 ,.., k ) 24 1 (1) P H k ( 1 ,.., k ) > 1 = 2. Hence H isalgorithmicallydistinguishable. 52

PAGE 53

If j H j =120 m ( i ) n ( i ) then P H ( )= (1) j H j j G j < H ,1 H > = 8>>>><>>>>: j H j j G j =1 G (1) 2 j G j (1)=120 m ( i ) n ( i ) 0 otherwise = 8>>>><>>>>: 1 g ( i ) =1 G 120 m ( i ) n ( i ) g ( i ) (1)=120 m ( i ) n ( i ) 0 otherwise = 8><>: (1) g ( i ) =1 G (1)=120 m ( i ) n ( i ) 0 otherwise So P H k ( 1 ,.., k )= 8><>: Q ki =1 (1) g ( i ) =1 G (1)=120 m ( i ) n ( i ) 0 otherwise Also P 1, k ( 1 ,.., k )= k Y i =1 (1) 2 120 g ( i ) m ( i ) n ( i ) Set 4 ( 1 ,.., k )= 8>>>><>>>>: 0 P 1, k ( 1 ,.., k )= P H k ( 1 ,.., k ) 0 P 1, k ( 1 ,.., k ) > P H k ( 1 ,.., k ) 1 P 1, k ( 1 ,.., k ) < P H k ( 1 ,.., k ) Wenoticethat X ( 1 ,.., k ) 24 1 (1) P H k ( 1 ,.., k )= X ( 1 ,.., k ) 24 1 (1) k Y i =1 i (1) g ( i ) = 1 ( g ( i ) ) k [(1+( g ( i ) 1)) k ( g ( i ) 1) k ]=[1 (1 1 g ( i ) ) k ]. Thisexpressionislessthan 1 = 2 for i !1 andsmall k Hence H isnotalgorithmically distinguishable. 53

PAGE 54

Wenotethatinallthecaseswehavelookedatsofar,Kempe-Sh alevdistinguishability coincideswithAlgorithmicdistinguishability. 54

PAGE 55

CHAPTER5 SOMECALCULATIONS EventhoughKempe-ShalevdistinguishabilityandAlgorith micdistinguishability coincideinthecasesdescribedinChapter 3 andChapter 4 thesetwoconceptsappear tobedifferent.Thefollowingisanexamplewheretheyseemn ottocoincide. Herewepicksomepositiveinteger n andset G = S 3 ... S 3 where S 3 isthe symmetricgrouponthreelettersandthereareexactly n copiesofitintheproduct.We x t tobeanelementoforder 3 in S 3 andwelet H bethesubgroupof G generatedby ( t t ,..., t ). Then H hasorder 3. Now Irr ( S 3 )= f 1 S 3 sgn g where 1 S 3 istheprincipalcharacter, sgn isthesign character,and istheuniquenon-linearcharacter.Both 1 S 3 and sgn arelinear,and (1)=2. Furthermore,wenoticethatthevaluesofthesecharacterso n t areasfollows. Bothlinearcharactershavevalue 1 on t and ( t )= 1. Now Irr ( G ) canbethoughtof asthecartesianproductof n copiesof Irr ( S 3 ). Hence j Irr ( G ) j =3 n andthedegreesof theseirreduciblecharactersareoftheform 2 i forsome i with 0 i n Anycharacterof degree 2 i canbeobtaineduniquelybychoosing i locationsoutof n ( wherethecharacter willbetakentobe ) andthenchoosingoneorotherofthelinearcharactersforal lthe otherlocations.Hence,thereareexactly n i 2 n i charactersin Irr ( G ) ofdegree 2 i 5.1Kempe-Shalevdistinguishability UsingthedenitionofKempe-Shalevdistinguishabilitywe seethat D H = 1 6 n X d j X h 6 = e ( h ) j = 1 6 n [ X d = linear j X h 6 = e ( h ) j + X = non linear d j X h 6 = e ( h ) j ]= 1 6 n [2 n 2+2( j 1 1 j n 1 2 n 1 )+4( j 1+1 j n 2 2 n 2 )+..+2 n nn ]= 1 6 n [2 n +1 +2 n +1 [ n1 +...+ nn ]]= 1 6 n [2 n +1 (1+(2 n 1))]= 2 n +1 2 n 6 n =( 2 3 ) n 2 < ( n log(6)) c forall c > 0 as n !1 55

PAGE 56

Hence H isKempe-Shalevindistinguishable. 5.2AlgorithmicDistinguishability Let 2 Irr ( G ) have (1)=2 i Then (( t t ,.., t ))=( 1) i Itfollowsthat P H ,1 ( )= 2 i 3 6 n 1 3 (2 i +2( 1) i ). Furthermore,wehave P 1,1 ( )= (1) 6 n < 1 ,1 1 > = (1) 2 6 n Fixsomepositiveinteger m Theabovecomputationyieldsthat,forevery ( 1 ,.., m ) 2 Irr ( G ) m wehave P H m ( 1 ,.., m )= m Y i =0 2 i 6 n (2 i +2( 1) i ), and P 1, m ( 1 ,.., m )= m Y i =0 2 2 i 6 n for i =1,.., m Itthenfollowsthat P H m ( 1 ,.., m )= P 1, m ( 1 ,.., m ) m Y i =0 (1+ ( 1) i 2 i 1 ). Inviewofthesecalculations,itisnaturaltodeneafuncti on 4 : Irr ( G ) m !f 0,1 g byfor ( 1 ,.., m ) 2 Irr ( G ), weset 4 ( 1 ,.., m )= 8>>>><>>>>: 0 P 1, m ( 1 ,.., m )= P H m ( 1 ,.., m ) 0 P 1, m ( 1 ,.., k ) > P H k ( 1 ,.., m ) 1 P 1, m ( 1 ,.., k ) < P H k ( 1 ,.., m ) Usingthisfunction f theprobabilityofitgivingthecorrectanswerwhenthehidd en subgroupis 1 is X ( 1 ,.., m ) 24 1 (0) P 1 ( 1 ,.., m ) 56

PAGE 57

andtheprobabilityofitgivingthecorrectanswerwhentheh iddensubgroupis H is X ( 1 ,.., m ) 24 1 (1) P H ( 1 ,.., m ). Thesenumberscanbecomputedforsmallvaluesof n and m using GAP Hereisa GAP programthatcalculatestheseprobabilities. C_1:=function(n,m)localprob,tuples,reltuples,pp,a,s;prob:=0;reltuples:=[];tuples:=Tuples([0..n],m);foraintuplesdopp:=Product(a,x->(1+(-1)^x/(2^(x-1))));ifpp<=1thenAppend(reltuples,[a]);fi;od;forainreltuplesdos:=Product(a,x->Binomial(n,x));s:=s*Product(a,x->2^(n-x));s:=s*Product(a,x->2^(2*x)/(6^n));prob:=prob+s;od;returnprob;end;C_2:=function(n,m)localprob,tuples,reltuples,pp,a,s;prob:=0;reltuples:=[];tuples:=Tuples([0..n],m);foraintuplesdopp:=Product(a,x->(1+(-1)^x/(2^(x-1)))); 57

PAGE 58

ifpp>1thenAppend(reltuples,[a]);fi;od;forainreltuplesdos:=Product(a,x->Binomial(n,x));s:=s*Product(a,x->2^(n-x));s:=s*Product(a,x->2^(x)*(2^x+2*(-1)^x)/(6^n));prob:=prob+s;od;returnprob;end; Thecomputationappearstoshowthatthesubgroup H isdistinguishableaccording toourdenitioninthecaseswetried.Forsomecomputations seechartsinappendixB. 58

PAGE 59

APPENDIXA TABLESFORCHAPTER4 HerearethetablesforChapter 4. TableA-1. H = SY 1 TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 1 3 i 0 0 0 0 0 Z 0 11 3 i 1 0 11 2 i 11 2 i 11 3 i SY 1 0 0 1 0 0 0 SY 2 0 0 0 1 2 i 0 0 SY 3 0 0 0 0 1 2 i 0 H t ( k ) 0 0 0 0 0 0 H ( k ) 0 0 0 0 0 0 G 0 0 0 0 0 1 3 i TableA-2. H = SY 1 Intermediate TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 1 1 3 i 1 1 1 1 11 3 i Z 0 0 0 0 0 0 0 SY 1 11 3 i 1 0 11 2 i 11 2 i 11 3 i SY 2 11 3 i 1 0 1 1 2 i 1 1 2 i 1 1 3 i SY 3 11 3 i 1 0 1 1 2 i 1 1 2 i 1 1 3 i H t ( k ) 0 0 0 0 0 0 H ( k ) 0 0 0 0 0 0 G 1 1 1 1 1 1 1 3 i 59

PAGE 60

TableA-3. H = SY 1 Final H = SY 1 TZ 0 1 3 i (1 1 3 i ) Z 0 0 SY 1 (1 1 3 i ) 2 (1 1 2 i + 1 3.2 2 i ) SY 2 1 2 (1 1 3 i ) 2 (1 1 2 i ) 1 2 i + 1 3 (1 1 3 i ) 2 1 2 2 i SY 3 1 2 (1 1 3 i ) 2 (1 1 2 i ) 1 2 i + 1 3 (1 1 3 i ) 2 1 2 2 i H t ( k ) 0 H ( k ) 0 G 1 3 i TableA-4. H = TZ 0 TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 1 0 0 0 0 11 2 i Z 0 0 1 11 2 i 11 2 i 11 2 i 0 SY 1 0 0 1 2 i 0 0 0 SY 2 0 0 0 1 2 i 0 0 SY 3 0 0 0 0 1 2 i 0 H t ( k ) 0 0 0 0 0 0 H ( k ) 0 0 0 0 0 0 G 0 0 0 0 0 1 2 i 60

PAGE 61

TableA-5. H = TZ 0 Intermediate TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 0 1 1 1 1 0 Z 0 0 0 0 0 0 0 SY 1 0 1 1 1 2 i 1 1 2 i 1 1 2 i 0 SY 2 0 1 1 1 2 i 1 1 2 i 1 1 2 i 0 SY 3 0 1 1 1 2 i 1 1 2 i 1 1 2 i 0 H t ( k ) 0 0 0 0 0 0 H ( k ) 0 0 0 0 0 0 G 1 1 1 1 1 1 1 2 i TableA-6. H = TZ 0 Final H = TZ 0 TZ 0 (1 1 2 i ) Z 0 0 SY 1 0 SY 2 0 SY 3 0 H t ( k ) 0 H ( k ) 0 G 1 2 i 61

PAGE 62

TableA-7. H = Z 0 TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 1 3 i 0 0 0 0 1 3 i 1 6 i Z 0 11 3 i 1 1 1 2 i 11 2 i 11 2 i 11 3 i SY 1 0 0 1 2 i 0 0 0 SY 2 0 0 0 1 2 i 0 0 SY 3 0 0 0 0 1 2 i 0 H t ( k ) 0 0 0 0 0 0 H ( k ) 0 0 0 0 0 0 G 0 0 0 0 0 1 6 i TableA-8. H = Z 0 Intermediate TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 1 1 3 i 1 1 1 1 1 1 3 i Z 0 0 0 0 0 0 0 SY 1 11 3 i 1 1 1 2 i 1 1 2 i 1 1 2 i 1 1 3 i SY 2 11 3 i 1 1 1 2 i 1 1 2 i 1 1 2 i 1 1 3 i SY 3 11 3 i 1 1 1 2 i 1 1 2 i 1 1 2 i 1 1 3 i H t ( k ) 0 0 0 0 0 0 H ( k ) 0 0 0 0 0 0 G 1 1 1 1 1 1 1 6 i 62

PAGE 63

TableA-9. FinalProbabilities H = Z 0 TZ 0 1 3 i (2 1 2 i 1 3 i ) Z 0 (1 1 3 i )(1 1 2 i ) 3 (1 1 3 i ) SY 1 (1 1 3 i ) 2 (1 1 2 i ) 2 1 2 i +(1 1 3 i ) 2 (1 1 2 i ) 1 2 2 i + 1 3 (1 1 3 i ) 2 1 2 3 i SY 2 (1 1 3 i ) 2 (1 1 2 i ) 2 1 2 i +(1 1 3 i ) 2 (1 1 2 i ) 1 2 2 i + 1 3 (1 1 3 i ) 2 1 2 3 i SY 3 (1 1 3 i ) 2 (1 1 2 i ) 2 1 2 i +(1 1 3 i ) 2 (1 1 2 i ) 1 2 2 i + 1 3 (1 1 3 i ) 2 1 2 3 i H t ( k ) 0 H ( k ) 0 G 1 6 i 63

PAGE 64

TableA-10. H = H t ( k ) TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 ( 1 2 n k 1 ) i 0 0 0 0 ( 1 2 n k ) i (2 i 1) Z 0 0 ( 1 2 n k 1 ) i ( 1 2 n k ) i (2 i 1) ( 1 2 n k ) i (2 i 1) ( 1 2 n k ) i (2 i 1) 0 SY 1 0 0 ( 1 2 n k ) i 0 0 0 SY 2 0 0 0 ( 1 2 n k ) i 0 0 SY 3 0 0 0 0 ( 1 2 n k ) i 0 H t ( k ) 11 2 i 0 0 0 0 11 2 i H t ( m ) ( 1 2 m k +1 ) i (2 i 1) 0 0 0 0 ( 1 2 m k +1 ) i (2 i 1) H ( m ) 0 ( 1 2 m k +1 ) i (2 i 1) ( 1 2 m k +1 ) i (2 i 1) ( 1 2 m k +1 ) i (2 i 1) ( 1 2 m k +1 ) i (2 i 1) 0 H ( k ) 0 11 2 i 11 2 i 11 2 i 11 2 i 0 G 0 0 0 0 0 ( 1 2 n k ) i 64

PAGE 65

TableA-11. H = H t ( k ) Intermediate TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 1 ( 1 2 n k 1 ) i 1 1 1 1 1 ( 1 2 n k 1 ) i Z 0 1 ( 1 2 n k 2 ) i 1 ( 1 2 n k 1 ) i 1 ( 1 2 n k 1 ) i 1 ( 1 2 n k 1 ) i 1 ( 1 2 n k 1 ) i 1 ( 1 2 n k 2 ) i SY 1 1 ( 1 2 n k 1 ) i 1 1 ( 1 2 n k ) i 1 ( 1 2 n k ) i 1 ( 1 2 n k ) i 1 ( 1 2 n k 1 ) i SY 2 1 ( 1 2 n k 1 ) i 1 1 ( 1 2 n k ) i 1 ( 1 2 n k ) i 1 ( 1 2 n k ) i 1 ( 1 2 n k 1 ) i SY 3 1 ( 1 2 n k 1 ) i 1 1 ( 1 2 n k ) i 1 ( 1 2 n k ) i 1 ( 1 2 n k ) i 1 ( 1 2 n k 1 ) i H t ( k ) 0 11 2 2 i 11 2 2 i 11 2 2 i 11 2 2 i 0 H t ( m ) 1 ( 1 2 m k ) i 1( 1 2 m k +2 ) i 1( 1 2 m k +2 ) i 1( 1 2 m k +2 ) i 1( 1 2 m k +2 ) i 1( 1 2 m k ) i H ( m ) 1( 1 2 m k 1 ) i 1 ( 1 2 m k ) i 1 ( 1 2 m k ) i 1( 1 2 m k ) i 1( 1 2 m k ) i 1( 1 2 m k 1 ) i H ( k ) 0 0 0 0 0 0 G 1 1 1 1 1 1( 1 2 n k ) i 65

PAGE 66

TableA-12. Finalprobabilities H = H t ( k ) Z 0 (1 ( 1 2 n k 2 ) i ) 2 (1 ( 1 2 n k ) i ) 3 (1 ( 1 2 n k 2 ) i ) 2 (1 ( 1 2 n k 1 ) i ) 4 TZ 0 (1 ( 1 2 n k ) i ) (1 ( 1 2 n k 1 ) i ) 2 SY 1 (1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) i (1 ( 1 2 n k ) i ) 2 +(1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) 2 i (1 ( 1 2 n k ) i )+ 1 3 (1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) 3 i SY 2 (1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) i (1 ( 1 2 n k ) i ) 2 +(1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) 2 i (1 ( 1 2 n k ) i )+ 1 3 (1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) 3 i SY 3 (1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) i (1 ( 1 2 n k ) i ) 2 +(1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) 2 i (1 ( 1 2 n k ) i )+ 1 3 (1 ( 1 2 n k 1 ) i ) 2 ( 1 2 n k ) 3 i H t ( k ) (1 1 2 i ) 2 (1 1 2 2 i ) 4 H t ( m ) (1 ( 1 2 m k +1 ) i ) 2 (1 ( 1 2 m + k +2 ) i ) 4 (1 ( 1 2 m k ) i ) 2 (1 ( 1 2 m + k +2 ) i ) 4 H ( k ) 0 H ( m ) (1 ( 1 2 m k 1 ) i ) 2 (1 ( 1 2 m k +1 ) i ) 5 (1 ( 1 2 m k 1 ) i ) 2 (1 ( 1 2 m k ) i ) 4 G ( 1 2 n k ) i 66

PAGE 67

TableA-13. H = H ( k ) TZ 0 Z 0 SY 1 SY 2 SY 3 G TZ 0 1 3 i ( 1 2 n k 1 ) i 0 0 0 0 ( 2 k n +1 3 ) i ( 2 k n 3 ) i Z 0 (1 1 3 i )( 1 2 n k 1 ) i (1 = 2 n k 1 ) i ( 1 2 n k ) i (2 i 1) ( 1 2 n k ) i (2 i 1) ( 1 2 n k ) i (2 i 1) (2 k n +1 ) i ( 2 k n +1 3 ) i SY 1 0 0 (1 = 2 n k ) i 0 0 0 SY 2 0 0 0 ( 1 2 n k ) i 0 0 SY 3 0 0 0 0 ( 1 2 n k ) i 0 H t ( k ) 1 3 i (1 1 2 i ) 0 0 0 0 1 3 i (1 ( 1 2 ) i H t ( m ) 1 3 i (2 i 1)( 1 2 m k +1 ) i 0 0 0 0 ( 2 k m 3 ) i ( 2 k m 1 3 ) i H ( m ) (1 1 3 i )(2 i 1)( 1 2 m k +1 ) i 1 (2 m k ) i (1 1 2 i ) 1 (2 m k +1 ) i (2 i 1) ( 1 2 m k +1 ) i (2 i 1) ( 1 2 m k +1 ) i (2 i 1) (2 k m 1 ) i (1 1 3 i )(2 i 1) H ( k ) (1 1 3 i )(1 1 2 i ) (1 1 2 i ) (1 1 2 i ) (1 1 2 i ) 11 2 i (1 1 3 i )(1 1 2 i ) G 0 0 0 0 0 1 3 i ( 1 2 n k ) i 67

PAGE 68

TableA-14. H = H ( k ) Intermediate TZ0 Z0 SY1 SY2 SY3 G TZ0 1-1 3 i(1 2 n k 1)i 1 1 1 1 1 1 3 i(1 2 n k 1)i Z0 1 1 3 i(1 2 n k 2)i 1 (1 = 2n k 1)i 1 (1 2 n k 1)i 1 (1 2 n k 1)i 1 (1 2 n k 1)i 1 (2 k n +2 3)i SY1 1 1 3 i(1 2 n k 1)i 1 1 (1 = 2n k)i 1 (1 = 2n k)i 1 (1 = 2n k)i 1 (1 2 n k 2)i 1 3 i SY2 1 1 3 i(1 2 n k 1)i 1 1 (1 = 2n k)i 1 (1 = 2n k)i 1 (1 = 2n k)i 1 (1 2 n k 2)i 1 3 i SY3 1 1 3 i(1 2 n k 1)i 1 1 (1 = 2n k)i 1 (1 = 2n k)i 1 (1 = 2n k)i 1 (1 2 n k 2)i 1 3 i Ht( k ) (1 1 3 i)(1 1 2 2 i) 1 1 2 2 i 1 1 2 2 i 1 1 2 2 i 1 1 2 2 i (1 1 2 2 i)(1 1 3 i) Ht( m ) 1 3 i(1 2 m k)i(1 1 4 i+3 i 4 i)+1 1 (1 2 m k +2)i) 1 (1 2 m k +2)i) 1 (1 2 m k +2)i) 1 (1 2 m k +2)i) 1 3 i(1 2 m k)i(1 1 4 i+3 i 4 i)+1 H ( m ) 1 3 i(1 (1 2 m k 1)i)+(1 1 3 i)(1 (1 2 m k)i) 1 (1 2 m k)i 1 (1 2 m k)i 1 (1 2 m k)i 1 (1 2 m k)i (1 2 m k)i(1+2 i 3 i1 3 i)+1 H ( k ) 0 0 0 0 0 0 G 1 1 1 1 1 1-1 3 i(1 2 n k)i 68

PAGE 69

TableA-15. Final H = H ( k ) Z0 (1 (2 k n +2 3)i+(2k n +1)i (2 k n +1 3)i)2 (1 (2 k n +2 3)i)2(1 (1 2 k n +1)i)4 TZ0 (1 (2 k n 3)i) (1 (2 k n +1 3)i)2 SY1 (1 (2 k n +1 3)i)2(1 2 n k)i(1+(1 2 n k)2 i (1 2 n k)i) SY2 (1 (2 k n +1 3)i)2(1 2 n k)i(1+(1 2 n k)2 i (1 2 n k)i) SY3 (1 (2 k n +1 3)i)2(1 2 n k)i(1+(1 2 n k)2 i (1 2 n k)i) Ht( k ) (1+1 2 i(1 6 i1 2 i1 3 i))2(1 1 2 2 i)4 (1 1 3 i)2(1 1 2 2 i)6 Ht( m ) (1 (1 2 m k +2)i)4((1 (2 k m 1 3)i+(2 k m 2 3)i (2k m 2)i)2 (1 (2 k m 3)i(1 1 4 i+3 i 4 i))2) H ( k ) (1 1 3 i)2(1 1 2 i)6 H ( m ) (1 (1 2 m k 1)i (2 k m +1 3)i+(2 k m 1 3)i)2(1 (1 2 m k +1)i)4 (1 (1 2 m k)i)4(1 (1 2 m k)i(1+2 i 3 i1 3 i))2 G (1 2 n k)i 69

PAGE 70

APPENDIXB TABLEFORCHAPTER5 TableB-1.Tablefor P P 1, m ( 1 ,.., m ) C 1 n m sum 4 3 > 1 = 2 5 4 > 1 = 2 6 5 > 1 = 2 7 6 > 1 = 2 TableB-2.Tablefor P P H m ( 1 ,.., m ) C 2 n m sum 4 3 > 1 = 2 5 4 > 1 = 2 6 5 > 1 = 2 7 6 > 1 = 2 70

PAGE 71

REFERENCES [1] DaveBacon,AndrewM.Childs,andWimvanDam, Fromoptimalmeasurement toefcientquantumalgorithmsforthehiddensubgroupprob lemoversemidirect productgroups ,arXiv:quant-ph/0504083v2.,2005. [2] R.Beals, Quantumcomputationoffouriertransformsoversymmetricg roups Proceedingsofthetwenty-ninthannualACMsymposiumonThe oryofcomputing. 29 (1997),48–53. [3] EthanBernsteinandUmeshVazirani, Quantumcomplexitytheory ,SIAMJournal onComputing 26(5) (1997),1411–1473. [4] ThomasBeth,MarkusPuschel,andMartinRotteler, Fastquantumfouriertransformsforaclassofnon-abeliangroups ,LectureNotesInComputerScience. 1719 (1999),148–159. [5] DongPyoChi,JeongSanKim,andSoojoonLee, Notesonthehiddensubgroup problemonsomesemi-directproductgroups ,arXiv:quant-ph/0604172v1.,2006. [6] DavidDeutsch, Quantumcomputationalnetworks ,ProceedingsoftheRoyal SocietyofLondonA 425 (1989),73–90. [7] DavidDeutschandRichardJozsa, Rapidsolutionsofproblemsbyquantum computation ,ProceedingsoftheRoyalSocietyofLondonA 439 (1992),553–558. [8] M.EttingerandP.Hoyer, Onquantumalgorithmsfornoncommutativehidden subgroups ,Adv.inAppl.Math 25(3) (2000),239–251. [9] RichardP.Feynman, Simulatingphysicswithcomputers ,InternationalJournalof TheoreticalPhysics. 21:6/7 (1982),467–488. [10] K.Friedl,G.Ivanyos,F.Magniez,M.Santha,andP.Sen, Hiddentranslationand orbitcosetinquantumcomputing ,Proceedingsof35thACMSymposiumonTheory ofComputing 35 (2003),1–9. [11] M.Grigni,L.Schulman,M.Vazirani,andU.Vazirani, Quantummechanicalalgorithmsforthenonabelianhiddensubgroupproblem ,Proc.Symp.PureMath. 46 (1987),111–138. [12] S.Hallgren,A.Russell,andA.Ta-Shma, Thehiddensubgroupproblemand quantumcomputationusinggrouprepresentations ,Proceedingsof32thACM SymposiumonTheoryofComputing. 32 (2000),627–635. [13] MikaHirvensalo(ed.), Quantumcomputing ,Springer.,NewYork,2001. [14] PeterHoyer, Efcientquantumtransforms ,quant-ph/9702028.,1997. 71

PAGE 72

[15] YoshifumiInuiandFrancoisLeGall, Efcientquantumalgorithmsforthe hiddensubgroupproblemoveraclassofsemi-directproduct groups arXiv:quant-ph/0412033v3.,2004. [16] I.M.Isaacs, Charactertheoryofnitegroups ,Dover,NewYrok,1994. [17] G.Ivanyos,F.Magniez,andM.Santha, Efcientquantumalgorithmsforsome instancesofthenon-abelianhiddensubgroupproblem ,Proceedingsof13thACM SymposiumonParallelisminAlgorithmsandArchitectures. 13 (2001),263–270. [18] J.KempeandA.Shalev, Thehiddensubgroupproblemandpermutationgroup theory ,Proceedingsofthe16thACM-SIAMSymposiumonDiscreteAlg orithms. 16 (2005),1118–1125. [19] G.Kuperberg, Asubexponential-timealgorithmforthedihedralhiddensu bgroup problem ,SIAMJ.Comput. 35(1) (2005),170–188. [20] ChristopherMoore,DanielRockmore,andAlexanderRussell Genericquantum ffts ,ProceedingsoftheFifteenthAnnualACM-SIAMSymposiumon Discrete Algorithms. 15 (2004),778–787. [21] O.Regev, Asubexponentialtimealgorithmforthedihedralhiddensub group problemwithpolynomialspace. ,http://xxx.lanl.gov/abs/quant-ph/0406151,2004. [22] J.P.Serre, Linearrepresentationsofnitegroups ,Springer-Verlag,NewYork,1977. [23] P.W.Shor, Algorithmsforquantumcomputation:Discretelogandfacto ring Proceedingsofthe35thAnnualSymposiumontheFoundations ofComputer Science. 35 (1994),124–134. [24] DanielR.Simon, Onthepowerofquantumcomputation ,Proceedingsofthe35th AnnualIEEESymposiumontheFoundationsofComputerScienc e 35 (1994), 116–123. [25] ChristofZalka, Onaparticularnon-abelianhiddensubgroupproblem http://qso.lanl.gov/zalka/QC/QC.html,1999. 72

PAGE 73

BIOGRAPHICALSKETCH AnalesDebhaumikwasbornintheyear1972inCalcutta,India .Hegraduated withbachelor'sinmathematicsfromCalcuttaUniversity.H ealsograduatedwitha master'sdegreeinappliedmathematicsfromCalcuttaUnive rsityandamaster'sdegree incomputerapplicationsfromBangaloreUniversity,India .HecametoUnitedStatesin 2003asaPhDstudentinmathematicsDepartmentofUniversit yofFlorida.Hereceived M.SinmathematicsfromUniversityofFloridaintheyear200 5.HegraduatedwithPhD inMay2010.Hisresearchinterestisnitegrouptheory. 73