Quantum Adversary Lower Bound for Element Distinctness with Small Range

27 Pages·2014·0.32 MB·English

by  Ansis Rosmanis

CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2014,Article04,pages1–27
http://cjtcs.cs.uchicago.edu/
Quantum Adversary Lower Bound for
Element Distinctness with Small Range
Ansis Rosmanis∗
Received March5,2014;Revised May12,2014;Published July9,2014
Abstract: The ELEMENT DISTINCTNESS problem is to decide whether each character
ofaninputstringisunique. Thequantumquerycomplexityof ELEMENT DISTINCTNESS
is known to be Θ(N2/3); the polynomial method gives a tight lower bound for any input
alphabet,whileatightadversaryconstructionwasonlyknownforalphabetsofsizeΩ(N2).
WeconstructatightΩ(N2/3)adversarylowerboundforELEMENT DISTINCTNESSwith
minimalnon-trivialalphabetsize,whichequalsthelengthoftheinput. Thisresultmayhelp
toimprovelowerboundsforotherrelatedqueryproblems.
1 Introduction and motivation
Background. Inquantumcomputation,oneofthemainquestionsthatweareinterestedinis: What
isthequantumcircuitcomplexityofagivencomputationalproblem? Thisquestionishardtoanswer,
andsoweconsideranalternativequestion: Whatisthequantumquerycomplexityoftheproblem? For
many problems, it is seemingly easier to (upper and lower) bound the number of times an algorithm
requirestoaccesstheinputratherthantoboundthenumberofelementaryquantumoperationsrequired
by the algorithm. Nonetheless, the study of the quantum query complexity can give us great insights
forthequantumcircuitcomplexity. Forexample,aquery-efficientalgorithmfor SIMON’S PROBLEM
[26]helpedShortodevelopatime-efficientalgorithmforfactoring[25]. Ontheotherhand, Ω˜(N1/5)
and Ω(N1/2) lower bounds on the (bounded error) quantum query complexity of the SET EQUALITY
[21]andtheINDEX ERASURE[6]problems,respectively,ruledoutcertainapproachesforconstructing
time-efficientquantumalgorithmsforthe GRAPH ISOMORPHISMproblem.
∗SupportedbyMikeandOpheliaLazaridisFellowship,DavidR.CheritonGraduateScholarship,andtheUSARO.
Keywordsandphrases: quantumquerycomplexity,adversarybound,elementdistinctness
©2014AnsisRosmanis
cbLicensedunderaCreativeCommonsAttributionLicense(CC-BY) DOI:10.4086/cjtcs.2014.004
ANSISROSMANIS
Currently, two main techniques for proving lower bounds on quantum query complexity are the
polynomialmethoddevelopedbyBeals,Buhrman,Cleve,Mosca,anddeWolf[7],andtheadversary
method originally developed by Ambainis [2] in what later became known as the positive adversary
method. The adversary method was later strengthened by Høyer, Lee, and Špalek [16] by allowing
negativeweightsintheadversarymatrix. Inrecentresults[22,20],Lee,Mittal,Reichardt,Špalek,and
Szegedyshowedthat,unlikethepolynomialmethod[3],thegeneral(i.e.,strengthened)adversarymethod
cangivetightlowerboundsforallproblems. Thisisastrongincentiveforthestudyoftheadversary
method.
ElementDistinctnessandCollision. Eventhoughweknowthattightadversary(lower)boundsexist
for all query problems, for multiple problems we still do not know how to even construct adversary
bounds that would match lower bounds obtained by other methods. For about a decade, ELEMENT
DISTINCTNESS and COLLISION wereprimeexamplesofsuchproblems. Givenaninputstringz∈ΣN,
the ELEMENT DISTINCTNESS problem is to decide whether each character of z is unique, and the
COLLISIONproblemisitsspecialcasegivenapromisethateachcharacterofziseitheruniqueorappears
inzexactlytwice. Asonecanthinkofzasafunctionthatmaps{1,2,...,N}toΣ,thealphabetΣisoften
alsocalledtherange.
The quantum query complexity of these two problems is known. Brassard, Høyer, and Tapp first
gaveanO(N1/3)quantumqueryalgorithmforCOLLISION[13]. AaronsonandShithengaveamatching
Ω(N1/3)lowerboundforCOLLISIONviathepolynomialmethod,requiringthat|Σ|≥3N/2[1]. Dueto
aparticularreductionfromCOLLISIONtoELEMENT DISTINCTNESS,theirlowerboundalsoimpliedan
Ω(N2/3)lowerboundforELEMENT DISTINCTNESS,requiringthat|Σ|∈Ω(N2). Subsequently,Kutin
(for COLLISION) and Ambainis (for both) removed these requirements on the alphabet size [19, 4].
Finally,AmbainisgaveanO(N2/3)quantumqueryalgorithmforELEMENT DISTINCTNESSbasedona
quantumwalk[5],thusimprovingthebestpreviouslyknownO(N3/4)upperbound[14].
Hence,theproofoftheΩ(N2/3)lowerboundforELEMENT DISTINCTNESSwithminimalnon-trivial
alphabet size N (and, thus, any alphabet size) consists of three steps: an Ω(N1/3) lower bound for
COLLISION,areductionfromanΩ(N1/3)lowerboundforCOLLISIONtoanΩ(N2/3)lowerboundfor
ELEMENT DISTINCTNESS withthealphabetsizeΩ(N2),andareductionofthealphabetsize. Inthis
paperweprovethesameresultdirectlybyprovidinganΩ(N2/3)generaladversaryboundfor ELEMENT
DISTINCTNESSwiththealphabetsizeN.
Theproblemsof SET EQUALITY,k-DISTINCTNESS,andk-SUM arecloselyrelatedto COLLISION
andELEMENT DISTINCTNESS. SET EQUALITYisaspecialcaseofCOLLISIONgivenanextrapromise
that each character of the first half (and, thus, the second half) of the input string is unique. Given a
constantk,thek-DISTINCTNESSproblemistodecidewhethertheinputstringcontainssomecharacterat
leastk times. Fork-SUM, weassumethatΣisanadditivegroupandtheproblemistodecideifthere
existknumbersamongN thatsumuptoaprescribednumber.
Recentadversarybounds. Duetothecertificatecomplexitybarrier[30,28],thepositiveweightadver-
sarymethodfailstogiveabetterlowerboundforELEMENTDISTINCTNESSthanΩ(N1/2). Andsimilarly,
due to the property testing barrier [16], it fails to give a better lower bound for COLLISION than the
trivialΩ(1). Recently,BelovsgaveanΩ(N2/3)generaladversaryboundforELEMENT DISTINCTNESS
CHICAGOJOURNALOFTHEORETICALCOMPUTERSCIENCE2014,Article04,pages1–27 2
QUANTUMADVERSARYLOWERBOUNDFORELEMENTDISTINCTNESSWITHSMALLRANGE
withalargeΩ(N2)alphabetsize[8]. Inaseriesofworksthatfollowed,tightgeneraladversarybounds
weregivenforthek-SUM[12], CERTIFICATE-SUM[10],andCOLLISIONand SET EQUALITYproblems
[11], all of them requiring that the alphabet size is large. Ω(Nk/(k+1)) and Ω(N1/3) lower bounds for
k-SUM and SET EQUALITY, respectively, were improvements over the best previously known lower
bounds. (TheΩ(N1/3)lowerboundfor SET EQUALITYwasalsoindependentlyprovenbyZhandry[29];
heusedacompletelydifferentmethod,whichdidnotrequireanyassumptionsonthealphabetsize.)
The adversary lower bound for a problem is given via the adversary matrix (Section 2.2). The
constructionoftheadversarymatrixinalltheserecent(general)adversaryboundsmentionedhasone
ideaincommon: theadversarymatrixisextractedfromalargermatrixthathasbeenconstructedusing,
essentially,theHammingassociationscheme[15]. Thefactthatweinitiallyembedtheadversarymatrix
inthislargermatrixisthereasonbehindtherequirementofthelargealphabetsize. Moreprecisely,dueto
thebirthdayparadox,theseadversaryboundsrequirethealphabetΣtobelargeenoughsothatarandomly
chosenstringinΣN withconstantprobabilityisanegativeinputoftheproblem.
Also, for these problems, all the negative inputs are essentially equally hard. However, for k-
DISTINCTNESS,forexample,thehardestnegativeinputsseemtobetheonesinwhicheachcharacter
appears k−1 times, and a randomly chosen negative input for k-DISTINCTNESS is such only with a
minusculeprobability. ThismightbeareasonwhyanΩ(N2/3)adversaryboundfork-DISTINCTNESS
[27] based on the idea of the embedding does not narrow the gap to the best known upper bound,
O(N1−2k−2/(2k−1))[9]. (TheΩ(N2/3)lowerboundwasalreadyknownpreviouslyviathereductionfrom
ELEMENT DISTINCTNESSattributedtoAaronsonin[5].)
Motivation for our work. In this paper we construct an explicit adversary matrix for ELEMENT
DISTINCTNESSwiththealphabetsize|Σ|=N (and,thus,anyalphabetsize)yieldingthetightΩ(N2/3)
lowerbound. Wealsoprovidecertain“tight”conditionsthateveryoptimaladversarymatrixforELEMENT
DISTINCTNESS mustsatisfy,1 thereforesuggestingthateveryoptimaladversarymatrixfor ELEMENT
DISTINCTNESSmighthavetobe,insomesense,closetotheadversarymatrixthatwehaveconstructed.
The tight Ω(Nk/(k+1)) adversary bound for k-SUM by Belovs and Špalek [12] is an extension of
Belovs’Ω(N2/3)adversaryboundfor ELEMENT DISTINCTNESS [8],anditrequires|Σ|∈Ω(Nk). We
constructtheadversarymatrixforELEMENT DISTINCTNESSdirectly,withouttheembedding,therefore
wedonotrequirethecondition|Σ|∈Ω(N2)asinBelovs’adversarybound. Wehopethatthismighthelp
toreducetherequiredalphabetsizeintheΩ(Nk/(k+1))lowerboundfork-SUM.
Aswementionedbefore,anadversarymatrixfork-DISTINCTNESSbasedontheideaoftheembed-
dingmightnotbeabletogivetightlowerbounds. Ontheotherhand,inourconstructionweonlyassume
thattheadversarymatrixisinvariantunderallindexandallalphabetpermutations,andthatissomething
we can always do without loss of generality due to the automorphism principle [16]—for ELEMENT
DISTINCTNESS,k-DISTINCTNESS,andmanyotherproblems. Hence,duetotheoptimalityofthegeneral
adversarymethod,weknowthatonecanconstructatightadversaryboundfork-DISTINCTNESS that
satisfiesthesesymmetries,andwehopethatourconstructionforELEMENT DISTINCTNESSmightgive
insightsinhowtodothat.
1Assuming,withoutlossofgenerality,thattheadversarymatrixhasthesymmetrygivenbytheautomorphismprinciple.
CHICAGOJOURNALOFTHEORETICALCOMPUTERSCIENCE2014,Article04,pages1–27 3
ANSISROSMANIS
Structureofthepaper. Thispaperisstructuredasfollows. InSection2wepresentthepreliminariesof
ourwork,includingtheadversarymethod,theautomorphismprinciple,andthebasicsoftherepresentation
theoryofthesymmetricgroup. InSection3weshowthattheadversarymatrixΓcanbeexpressedas
alinearcombinationofspecificmatrices. InthissectionwealsopresentClaim3.2,whichstateswhat
conditions every optimal adversary matrix for ELEMENT DISTINCTNESS must satisfy; we prove this
claimintheappendix. InSection4weshowhowtospecifytheadversarymatrixΓviaitsubmatrixΓ ,
1,2
whichwillmaketheanalysisoftheadversarymatrixsimpler. InSection5wepresenttoolsforestimating
the spectral norm of the matrix entrywise product of Γ and the difference matrix ∆, a quantity that is
i
essentialtotheadversarymethod. InSection6weusetheconditionsgivenbyClaim3.2toconstructan
adversarymatrixforELEMENT DISTINCTNESSwiththealphabetsizeN,andweshowthatthismatrix
indeedyieldsthedesiredΩ(N2/3)lowerbound. WeconcludeinSection7withopenproblems.
2 Preliminaries
2.1 Elementdistinctnessproblem
Let N be the length of the input and let Σ be the input alphabet. Let [i,N] = {i,i+1,...,N} and
[N]=[1,N]forshort. Givenaninputstringz∈ΣN,theELEMENT DISTINCTNESSproblemistodecide
whetherzcontainsacollisionornot,namely,weatherthereexisti,j∈[N]suchthati(cid:54)= jandz =z . We
i j
onlyconsideraspecialcaseoftheproblemwherewearegivenapromisethattheinputcontainsatmost
onecollision. Thispromisedoesnotchangethecomplexityoftheproblem[5].
LetD andD bethesetsofpositiveandnegativeinputs,respectively,thatis,inputswithaunique
1 0
collision and inputs without a collision. If |Σ|<N, then D =0/, and the problem becomes trivial,
0
thereforeweconsiderthecasewhen|Σ|=N. Wehave
(cid:18) (cid:19) (cid:18) (cid:19)
N |Σ|! N |Σ|!
|D |= = N! and |D |= =N!.
1 0
2 (|Σ|−N+1)! 2 (|Σ|−N)!
2.2 Adversarymethod
The general adversary method gives optimal bounds for any quantum query problem. Here we only
considertheELEMENTDISTINCTNESSproblem,soitsufficestodefinetheadversarymethodfordecision
problems. Letusthinkofadecisionproblem pasaBoolean-valuedfunction p:D→{0,1}withdomain
D⊆ΣN,andletD = p−1(1)andD = p−1(0).
1 0
Anadversarymatrixforadecisionproblem pisareal|D |×|D |matrixΓwhoserowsarelabeled
1 0
by the positive inputs D and columns by the negative inputs D . Let Γ[[x,y]] denote the entry of Γ
1 0
correspondingtothepairofinputs(x,y)∈D ×D . Fori∈[N],thedifferencematrices∆ and∆ arethe
1 0 i i
matricesofthesamedimensionsandthesamerowandcolumnlabelingasΓthataredefinedby
(cid:40) (cid:40)
0, ifx =y, 1, ifx =y,
i i i i
∆[[x,y]]= and ∆[[x,y]]=
i i
1, ifx (cid:54)=y, 0, ifx (cid:54)=y.
i i i i
CHICAGOJOURNALOFTHEORETICALCOMPUTERSCIENCE2014,Article04,pages1–27 4
QUANTUMADVERSARYLOWERBOUNDFORELEMENTDISTINCTNESSWITHSMALLRANGE
Theorem2.1(Adversarybound[16,20]). Thequantumquerycomplexityofthedecisionproblem pis
Θ(Adv(p)),whereAdv(p)istheoptimalvalueofthesemi-definiteprogram
maximize (cid:107)Γ(cid:107)
(2.1)
subjectto (cid:107)∆ ◦Γ(cid:107)≤1foralli∈[N],
i
wherethemaximizationisoveralladversarymatricesΓfor p,(cid:107)·(cid:107)isthespectralnorm(i.e.,thelargest
singularvalue),and◦istheentrywisematrixproduct.
Everyfeasiblesolutiontothesemi-definiteprogram(2.1)yieldsalowerboundonthequantumquery
complexity of p. Note that we can choose any adversary matrix Γ and scale it so that the condition
(cid:107)∆ ◦Γ(cid:107)≤1holds. Inpractice,weusethecondition(cid:107)∆ ◦Γ(cid:107)∈O(1)insteadof(cid:107)∆ ◦Γ(cid:107)≤1. Alsonote
i i i
that∆ ◦Γ=Γ−∆ ◦Γ.
i i
2.3 Symmetriesoftheadversarymatrix
It is known that we can restrict the maximization in Theorem 2.1 to adversary matrices Γ satisfying
certainsymmetries. LetS bethesymmetricgroupofafinitesetA,thatis,thegroupwhoseelementsare
A
allthepermutationsofelementsofAandwhosegroupoperationisthecompositionofpermutations. The
automorphismprinciple[16]impliesthat,withoutlossofgenerality,wecanassumethatΓfor ELEMENT
DISTINCTNESS is fixed under all index and all alphabet permutations. Namely, index permutations
π ∈S andalphabetpermutationsτ ∈S actoninputstringsz∈ΣN inthenaturalway:
[N] Σ
π ∈S : z=(z ,...,z ) (cid:55)→ z =(cid:0)z ,...,z (cid:1),
[N] 1 N π π−1(1) π−1(N)
τ ∈S : z=(z ,...,z ) (cid:55)→ zτ =(cid:0)τ(z ),...,τ(z )(cid:1).
Σ 1 N 1 N
The actions of π and τ commute: we have (z )τ = (zτ) , which we denote by zτ for short. The
π π π
automorphismprincipleimpliesthatwecanassume
Γ[[x,y]]=Γ[[xτ,yτ]] (2.2)
π π
forallx∈D ,y∈D ,π ∈S ,andτ ∈S .
1 0 [N] Σ
Let X∼=R|D1| and Y∼=R|D0| be the vector spaces corresponding to the positive and the negative
inputs, respectively. (We can view Γ as a linear operator that maps Y to X.) Let Uτ and Vτ be the
π π
permutationmatricesthatrespectivelyactonthespacesXandYandthatmapeveryx∈D toxτ and
1 π
everyy∈D toyτ. Then(2.2)isequivalentto
0 π
UτΓ=ΓVτ (2.3)
π π
forallπ ∈S ,andτ ∈S . BothU andV arerepresentationsofS ×S .
[N] Σ [N] Σ
2.4 Representationtheoryofthesymmetricgroup
Letuspresentthebasicsoftherepresentationtheoryofthesymmetricgroup. (Foradetailedstudyofthe
representationtheoryofthesymmetricgroup,referto[17,23];forthefundamentalsoftherepresentation
theoryoffinitegroups,referto[24].)
CHICAGOJOURNALOFTHEORETICALCOMPUTERSCIENCE2014,Article04,pages1–27 5
ANSISROSMANIS
Uptoisomorphism,thereisone-to-onecorrespondencebetweentheirreps(i.e.,irreduciblerepresen-
tations)ofS and|A|-boxYoungdiagrams,andweoftenusethesetwotermsinterchangeably. Weuseζ,
A
η,andθ todenoteYoungdiagramshavingo(N)boxes,λ,µ,andν todenoteYoungdiagramshavingN,
N−1,andN−2boxes,respectively,andρ andσ forgeneralstatementsandotherpurposes.
Let ρ (cid:96) M denote that ρ is an M-box Young diagram. For a Young diagram ρ, let ρ(i) and
ρ(cid:62)(j) denote the number of boxes in the i-th row and j-th column of ρ, respectively. We write
ρ = (ρ(1),ρ(2),...,ρ(r)), where r = ρ(cid:62)(1) is the number of rows in ρ, and, given M ≥ ρ(1), let
(M,ρ)beshortfor(M,ρ(1),ρ(2),...,ρ(r)).
Wesaythatabox(i,j)ispresentinρ andwrite(i,j)∈ρ ifρ(i)≥ j(equivalently,ρ(cid:62)(j)≥i). The
hook-lengthh (b)ofaboxbisthesumofthenumberofboxesontherightfrombinthesamerow(i.e.,
ρ
ρ(i)− j)andthenumberofboxesbelowbinthesamecolumn(i.e.,ρ(cid:62)(j)−i)plusone(i.e.,theboxb
itself). Thedimensionoftheirrepcorrespondingtoρ isgivenbythehook-lengthformula:
(cid:14)
dimρ =|ρ|! h(ρ), where h(ρ)=∏ h (i,j) (2.4)
(i,j)∈ρ ρ
and|ρ|isthenumberofboxesinρ.
Letσ <ρ andσ (cid:28)ρ denotethataYoungdiagramσ isobtainedfromρ byremovingexactlyone
boxandexactlytwoboxes,respectively. Givenσ (cid:28)ρ,letuswriteσ (cid:28) ρ orσ (cid:28) ρ ifthetwoboxes
r c
removedfromρ toobtainσ are, respectively, indifferentrowsordifferentcolumns. Letσ (cid:28) ρ be
rc
shortfor(σ (cid:28) ρ)&(σ (cid:28) ρ). Thedistancebetweentwoboxesb=(i,j)andb(cid:48)=(i(cid:48),j(cid:48))isdefinedas
r c
|i(cid:48)−i|+|j−j(cid:48)|. Givenσ (cid:28) ρ,letd ≥2bethedistancebetweenthetwoboxesthatweremovefrom
rc ρ,σ
ρ toobtainσ.
Thebranchingrulestatesthattherestrictionofanirrepρ ofS toS ,wherea∈A,is
A A\{a}
ResSA ρ ∼=(cid:77) σ.
S
A\{a} σ<ρ
ThemoregeneralLittlewood–Richardsonruleimpliesthattherestrictionofanirrepρ ofS toS ×
A {a,b}
S ,wherea,b∈A,is
A\{a,b}
ResSA ρ ∼=(cid:77) (id×σ)⊕(cid:77) (sgn×σ(cid:48)),
S{a,b}×SA\{a,b} σ(cid:28)cρ σ(cid:48)(cid:28)rρ
whereid=(2)andsgn=(1,1)arethetrivialandthesignrepresentationofS ,respectively. Frobenius
{a,b}
reciprocitythentellsusthatthe“opposite”happenswhenweinduceanirrepofS orS ×S
A\{a} {a,b} A\{a,b}
toS .
A
Given(cid:96)∈{0,1,2,3},asetA=[N]orA=Σ,itssubsetA\{a ,...,a },andρ (cid:96)N−(cid:96),letuswrite
1 (cid:96)
ρ ifwewanttostressthatwethinkofρ asanirrepofS . Weomitthesubscriptif(cid:96)=0
a1...a(cid:96) A\{a1,...,a(cid:96)}
or when {a ,...,a } is clear from the context. To lighten the notations, given k∈o(N)and η (cid:96)k, let
1 (cid:96)
η¯ =(N−(cid:96)−k,η) (cid:96)N−(cid:96);hereweomitthesubscriptifandonlyif(cid:96)=0.
a1...a(cid:96) a1...a(cid:96)
2.5 Transporters
Suppose we are given a group G, and let ξ and ξ be two isomorphic irreps of G acting on spaces
1 2
Z andZ , respectively. Uptoaglobalphase(i.e., ascalarofabsolutevalue1), thereexistsaunique
1 2
CHICAGOJOURNALOFTHEORETICALCOMPUTERSCIENCE2014,Article04,pages1–27 6
QUANTUMADVERSARYLOWERBOUNDFORELEMENTDISTINCTNESSWITHSMALLRANGE
isomorphismT fromξ toξ thatsatisfies(cid:107)T (cid:107)=1. Wecallthisisomorphismatransporterfrom
2←1 1 2 2←1
ξ toξ (or,fromZ toZ ).
1 2 1 2
Inthispaperweonlyconsiderunitaryrepresentationsandrealvectorspaces,thereforeallsingular
valuesofT areequalto1and,fortheglobalphase,wehavetochooseonlybetween±1. Wealways
2←1
choosetheglobalphasessothattheyrespectinversionandcomposition,namely,sothatT T isthe
1←2 2←1
identitymatrixonZ andT T =T ,whereξ isanirrepisomorphictoξ andξ .
1 3←2 2←1 3←1 3 1 2
3 Building blocks of Γ
3.1 DecompositionofU andV intoirreps
Withoutlossofgenerality,letusassumethattheadversarymatrixΓfortheELEMENT DISTINCTNESS
problemsatisfythesymmetry(2.3)givenbytheautomorphismprinciple. BothU andV arerepresenta-
tionsofS ×S and,duetoSchur’slemma,wewanttoseewhatirrepsofS ×S occurinbothU and
[N] Σ [N] Σ
V. ItisalsoconvenienttoconsiderU andV asrepresentationsofjustS orjustS .
[N] Σ
Claim3.1. V decomposesintoirrepsofS ×S asV ∼=(cid:76) λ×λ.
[N] Σ λ(cid:96)N
Proof. AsarepresentationofS andS ,respectively,V isisomorphictotheregularrepresentationof
[N] Σ
S andS . Foreveryy∈D andeveryπ ∈S ,thereisauniqueτ ∈S suchthaty =yτ,andπ andτ
[N] Σ 0 [N] Σ π
belongtoisomorphicconjugacyclasses. Thus,foreveryλ,theisotypicalsubspaceofYcorresponding
to λ (i.e., the subspace corresponding to all irreps isomorphic to λ) is the same for both S and S
[N] Σ
[24,Section2.6]. SinceV isisomorphictotheregularrepresentation,thedimensionofthissubspaceis
(dimλ)2,whichisexactlythedimensionoftheirrepλ×λ ofS ×S .
[N] Σ
NowletusaddressU,whichactsonthespaceXcorrespondingtothepositiveinputsx∈D . Letus
1
decomposeD asadisjointunionof(cid:0)N(cid:1)setsD , where{i,j}⊂[N]andD isthesetofallx∈D
1 2 i,j i,j 1
suchthatx =x . LetusfurtherdecomposeD asadisjointunionof(cid:0)N(cid:1)setsDs,t,where{s,t}⊂Σand
i j i,j 2 i,j
Ds,t isthesetofallx∈D thatdoesnotcontainsandcontainst twiceorviceversa. LetX andXs,t
i,j i,j i,j i,j
bethesubspacesofXthatcorrespondtothesetsD andDs,t,respectively. ThespaceXs,t isinvariant
i,j i,j i,j
undertheactionofthesubgroupSs,t definedas
i,j
Ss,t =(S ×S )×(S ×S ),
i,j {i,j} [N]\{i,j} {s,t} Σ\{s,t}
namely,UτXs,t =Xs,t forall(π,τ)∈Ss,t. ThereforeU restrictedtothesubspaceXs,t isarepresentation
π i,j i,j i,j i,j
ofSs,t,and,similarlytoClaim3.1,itdecomposesintoirrepsas
i,j
(cid:77) (cid:0) (cid:1) (cid:0) (cid:1)
id×ν × (id⊕sgn)×ν . (3.1)
ν(cid:96)N−2
To see how U decomposes into irreps of S ×S , we induce the representation (3.1) from Ss,t to
[N] Σ i,j
S ×S .
[N] Σ
TheLittlewood–RichardsonruleimpliesthatanirrepofS ×S isomorphictoλ×λ canoccurin
[N] Σ
U duetooneofthefollowingscenarios.
CHICAGOJOURNALOFTHEORETICALCOMPUTERSCIENCE2014,Article04,pages1–27 7
ANSISROSMANIS
• Ifν (cid:28) λ andν(cid:28)(cid:54) λ (i.e.,ν isobtainedfromλ byremovingtwoboxesinthesamerow),then
c r
λ ×λ occurs once in the induction of (id×ν)×(id×ν). Let Xλ denote the subspace of X
id,ν
correspondingtothisinstanceofλ×λ.
• Ifν (cid:28) λ,thenλ×λ occursonceintheinductionof(id×ν)×(id×ν)andonceintheinduction
rc
of(id×ν)×(sgn×ν). LetXλ andXλ denotetherespectivesubspacesofXcorresponding
id,ν sgn,ν
totheseinstancesofλ×λ.
Note: thesubspacesXλ andXλ areindependentfromthechoiceof{i,j}⊂[N]and{s,t}⊂Σ.
id,ν sgn,ν
3.2 Γasalinearcombinationoftransporters
Let Ξλ and Ξλ denote the transporters from the unique instance of λ ×λ in Y to the subspaces
id,ν sgn,ν
Xλ and Xλ , respectively. We will specify the global phases of these transporters in Section 4.3.
id,ν sgn,ν
WeconsiderΞλ andΞλ asmatricesofdimensions(cid:0)N(cid:1)N!×N!andrank(dimλ)2. Schur’slemma
id,ν sgn,ν 2
impliesthat,dueto(2.3),wecanexpressΓasalinearcombinationofthesetransporters. Namely,
(cid:16) (cid:17)
Γ= ∑ ∑ βλ Ξλ + ∑ βλ Ξλ , (3.2)
id,ν id,ν sgn,ν sgn,ν
λ(cid:96)N ν(cid:28)cλ ν(cid:28)rcλ
wherethecoefficientsβλ andβλ arereal.
id,ν sgn,ν
(cid:72)
(cid:72) ν
(cid:72)(cid:72) (N−2) (N−3,1) (N−4,2) (N−4,1,1)
λ (cid:72)(cid:72)
(N) (cid:88)
0
(N−1,1) (cid:88)(cid:88) (cid:88)
1 0
(N−2,2) (cid:88) (cid:88)(cid:88) (cid:88)
2 1 0
(N−2,1,1) (cid:88)(cid:88) (cid:88)
1 0
(N−3,3) (cid:88) (cid:88)(cid:88)
2 1
(N−3,2,1) (cid:88)(cid:88) (cid:88)(cid:88) (cid:88)(cid:88)
2 1 1
(N−3,1,1,1) (cid:88)(cid:88)
1
(N−4,4) (cid:88)
2
(N−4,3,1) (cid:88)(cid:88) (cid:88)
2 2
(N−4,2,2) (cid:88)
2
(N−4,2,1,1) (cid:88)(cid:88)
2
Table1: AvailableoperatorsfortheconstructionofΓ. Wedistinguishthreecases: bothλ andν arethe
samebelowthefirstrow(label“(cid:88) ”),λ hasoneboxmorebelowthefirstrowthanν (label“(cid:88)(cid:88) ”),λ has
0 1
twoboxesmorebelowthefirstrowthanν (labels“(cid:88) ”and“(cid:88)(cid:88) ”).
2 2
Thus we have reduced the construction of the adversary matrix Γ to choosing the coefficients β
of the transporters in (3.2). To illustrate what are the available transporters, let us consider the last
four(N−2)-boxYoungdiagramsν ofthelexicographicalorder—(N−2),(N−3,1),(N−4,2),and
(N−4,1,1)—andallλ thatareobtainedfromtheseν byaddingtwoboxesindifferentcolumns. Table
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QUANTUMADVERSARYLOWERBOUNDFORELEMENTDISTINCTNESSWITHSMALLRANGE
1 shows pairs of λ and ν for which we have both Ξλ and Ξλ available for the construction of Γ
id,ν sgn,ν
(doublecheckmark“(cid:88)(cid:88)”)orjustΞλ available(singlecheckmark“(cid:88)”).
id,ν
Duetothesymmetry,(cid:107)∆ ◦Γ(cid:107)isthesameforalli∈[N],so,fromnowon,letusonlyconsider∆ ◦Γ.
i 1
Wewanttochoosethecoefficientsβ sothat(cid:107)Γ(cid:107)∈Ω(N2/3)and(cid:107)∆ ◦Γ(cid:107)∈O(1). Theautomorphism
1
principlealsoimplies(see[16])thatwecanassumethattheprincipalleftandrightsingularvectorsofΓ
aretheall-onesvectors,whichcorrespondtoΞ(N) . Wethuschooseβ(N) ∈Θ(N2/3).
id,(N−2) id,(N−2)
Inordertounderstandhowtochoosethecoefficientsβ,inAppendixAweprovethefollowingclaim,
whichrelatesallthecoefficientsoftransportersofTable1andmore.
Claim 3.2. Suppose Γ is given as in (3.2) and β(N) =N2/3. Consider λ (cid:96)N that has O(1) boxes
id,(N−2)
belowthefirstrowandν (cid:28) λ. Inorderfor(cid:107)∆ ◦Γ(cid:107)∈O(1)tohold,weneedtohave
c 1
1. βλ =N2/3+O(1)ifλ andν arethesamebelowthefirstrow,
id,ν
2. βλ ,βλ =cλN1/6+O(1) if λ has one box more below the first row than ν, where cλ is a
id,ν sgn,ν ν ν
constantdependingonlyonthepartofλ andν belowthefirstrow,2
3. βλ ,βλ =O(1)ifλ hastwoboxesmorebelowthefirstrowthanν.
id,ν sgn,ν
Notethatwealwayshavethefreedomofchanging(aconstantnumberof)coefficientsβ uptoan
additivetermofO(1)becauseofthefactthat
(cid:8) (cid:9)
γ (∆ )=max (cid:107)∆ ◦B(cid:107) : (cid:107)B(cid:107)≤1 ≤2 (3.3)
2 1 1
B
(see[16]forthisandotherfactsabouttheγ norm). WewillusethisfactagaininSection6.
2
4 Specification of Γ via Γ
1,2
Duetothesymmetry(2.2),itsufficestospecifyasinglerowoftheadversarymatrixΓinordertospecify
thewholematrix. Fortheconvenience,letusinsteadspecifyΓviaspecifyingits(N!×N!)-dimensional
submatrix Γ —for {i,j}⊂[N], we define Γ to be the submatrix of Γ that corresponds to the rows
1,2 i,j
labeled by x∈D , that is, positive inputs x with x =x . We think of Γ both as an N!×N! square
i,j i j i,j
matrix and as a matrix of the same dimensions as Γ that is obtained from Γ by setting to zero all the
(cid:0)(cid:0)N(cid:1)−1(cid:1)N!rowsthatcorrespondtox∈/ D .
2 i,j
4.1 NecessaryandsufficientsymmetriesofΓ
1,2
Forall(π,τ)∈(S ×S )×S , wehaveUτX =X and, therefore,UτΓ =Γ Vτ. Thisis
{1,2} [3,N] Σ π 1,2 1,2 π 1,2 1,2 π
thenecessaryandsufficientsymmetrythatΓ mustsatisfyinorderforΓtobefixedunderallindexand
1,2
alphabetpermutations. SinceU Γ =Γ ,wealsohaveΓ V =Γ . Wehave
(12) 1,2 1,2 1,2 (12) 1,2
(cid:18) (cid:19)
N 1
Γ= ∑ Γ = ∑U Γ V = ∑ U Γ V , (4.1)
i,j π 1,2 π−1 2 N! π 1,2 π−1
{i,j}⊂[N] π∈R π∈S[N]
(cid:113)
2Letλˆ andνˆ bethepartofλandνbelowthefirstrow,respectively.Thencλ= h(λˆ)/h(νˆ)=(cid:112)Ndimν/dimλ+O(1/N).
ν
CHICAGOJOURNALOFTHEORETICALCOMPUTERSCIENCE2014,Article04,pages1–27 9
ANSISROSMANIS
whereR=Rep(S /(S ×S ))isatransversaloftheleftcosetsofS ×S inS .
[N] {1,2} [3,N] {1,2} [3,N] [N]
Let f beabijectionbetweenD andD definedas
0 1,2
f: D →D : (y ,y ,y ,...,y )(cid:55)→(y ,y ,y ,...,y ),
0 1,2 1 2 3 N 1 1 3 N
andletF bethecorrespondingpermutationmatrixmappingYtoX . Letusorderrowsandcolumnsof
1,2
Γ sothattheycorrespondto f(y)andy,respectively,wherewetakey∈D inthesameorderforboth
1,2 0
(seeFigure1). Hence,F becomestheidentitymatrixonY(fromthispointonward,weessentiallythink
ofX andYasthesamespace). LetusdenotethisidentitymatrixbyI.
1,2
(12) π τ
a a b c c d e
b b c b b a d
c c ... e a e b ... c
d e a d a e b
e d d e d c a
aacde
aaced
.
.
.
bbead
ccade
π
Γ1,2
(12) ccead
τ
ddbec
.
.
.
eecba
Figure1: SymmetriesofΓ forN =5andΣ={a,b,c,d,e}. Withrespecttothebijection f,theorder
1,2
ofrowsandcolumnsmatches. ThesolidarrowsshowthatUτ andVτ actsymmetricallyonΓ (herewe
1,2
useτ =(aeb)(cd)∈S ),andsodoU andV forπ ∈S (hereweuseπ =(354)). However,asshown
Σ π π [3,N]
bythedash-dottedarrows,U actsastheidentityontherows,whileV transposesthecolumns.
(12) (12)
Forall(π,τ)∈S ×S wehave f(yτ)=(f(y))τ and,thus,Vτ =FVτ =UτF =Uτ,wherewe
[3,N] Σ π π π π π π
considertherestrictionofUτ toX . NotethatU =IonX ,whileV (cid:54)=I. Hencenowthetwo
π 1,2 (12) 1,2 (12)
necessaryandsufficientsymmetriesthatΓ mustsatisfyare
1,2
VτΓ =Γ Vτ forall (π,τ)∈S ×S and Γ V =Γ . (4.2)
π 1,2 1,2 π [3,N] Σ 1,2 (12) 1,2
Figure1illustratesthesesymmetries.
CHICAGOJOURNALOFTHEORETICALCOMPUTERSCIENCE2014,Article04,pages1–27 10