QMA(2)workshop— Tutorial1

BillFefferman(QuICS)

Agenda

I. BasicsII. KnownresultsIII. Openquestions/Nexttutorialoverview

I.Basics

I.1ClassicalComplexityTheory

• P• Classofproblemsefficientlysolvedonclassicalcomputer

• NP• Classofproblemswithefficiently

verifiablesolutions• Characterizedby3SAT

• Input:Ψ:{0,1}n→{0,1}• n-variable3-CNFformula

• E.g.,(x1∨x2∨x3)∧(x1∨−x2∨x6)∧...• Problem:∃x1,x2,...,xn sothatΨ(x)=1?

• Coulduseaboxsolving3SAT tosolveanyprobleminNP

4

NP

P

I.2Merlin-Arthur

• “Randomizedgeneralization”ofNP• Canthinkofagamebetweenall-knowingbutpotentiallydishonestMerlintryingtoprovestatementtoefficientrandomizedclassicalcomputer (Arthur)

• Ifstatementistrue,thereexistsapolynomiallengthclassicalbitstring or“witness”toconvinceArthurtoacceptwithhighprobability(Completeness)

• Ifstatementisfalse,thenevery“witness”isrejectedbyArthurwithhighprobability(Soundness)

• UndercommonlybelievedderandomizationhypothesisMA=NP

𝜋∈{0,1}p(n)

5

I.3 QuantumMerlin-Arthur

• QMA:Samesetup,nowArthurisBQPmachine,witnessispolynomialqubitquantumstate

• Formally:QMAm istheclassofpromiseproblemsL=(Lyes,Lno)sothat:• Thereexistsauniformverifier{𝑉#}#∈{&,(}* ofpolynomialsizethatactsonO(m(|x|)+k(|x|))qubits(fork∈poly(n)):

• “Quantumanalogue”ofNP• k-LocalHamiltonianproblemisQMA-complete (whenk≥2)[Kitaev ’02]

• Input:𝐻 = ∑ 𝐻./.0( ,eachterm𝐻. isk-local• Promise,for(a,b)sothatb-a≥1/poly(n),either:

• ∃|ψ⟩𝑠𝑜𝑡ℎ𝑎𝑡⟨𝜓|H|ψ⟩ ≤ aOR• ∀|ψ⟩𝑤𝑒ℎ𝑎𝑣𝑒⟨𝜓|H|ψ⟩ ≥ b

|ψ⟩

6

x 2 Lyes

) 9| i�h |⌦ h0k|

�V

†x

|1ih1|out

V

x

�| i ⌦ |0ki

�� 2/3

x 2 Lno

) 8| i�h |⌦ h0k|

�V

†x

|1ih1|out

V

x

�| i ⌦ |0ki

� 1/3

I.4Entangledquantumstates

• LetAandBbetwofinitedimensionalcomplexvectorspaces• Abipartitedensitymatrix,orstate,isapositivesemidefinitematrixρAB onA⊗Bthathasunittrace• ρAB iscalledseparableifitcanbewrittenas

• Forlocalstates{ρA,k}and{ρB,k} andprobabilitiespk• Statesthatarenotseparable areentangled

⇢AB =X

k

pk⇢A,k ⌦ ⇢B,k

I.5 QMA(2):Thepowerofseparablewitness• Ourquestion:IsthereanadvantagetoMerlinsendingunentangledstates?• QMA(2):

• Completeness:ThereexiststatethatconvincesArthurtoacceptwithhighprobability

• Soundness:AllstatesarerejectedbyArthurwithhighprobability• QMA(k):Sameclasswithk witnesses

• Trivialbounds:QMA⊆QMA(2)⊆NEXP• Whyisn’tQMA(2)obviously containedinQMA?

• Merlincancheatbyentangling,andcheckingseparability ishard• E.g.,“Weak-membership(ε)”isNP-hard[e.g.,Gharibian’09]

• GivenρAB isitseparableor|ρAB-Sep|>ε ?• Whereε=1/poly(|A|,|B|) relativetothetracenorm

• Erroramplificationisnon-trivial• Repetitiondoesn’twork(Measurementsononesetofcopiescancreateentanglement

betweenwitnesses)

| 1i ⌦ | 2i

| 1i ⌦ | 2i

I.6Whyshouldyou careaboutQMA(2)?

• Therearemanymulti-proverquantumcomplexityclasses,whyshouldwecareaboutthisone?

1. Connectionstoseparability testing(i.e.,givenaquantumstateisitseparableorfarfromseparable?)

2. Connectionstoentanglementmeasuresand“quantumdeFinettitheorems”

3. Closeconnectionstohardnessofapproximationandclassicalcomplexitytheory:“UniqueGamesConjecture”andthe“ExponentialTimeHypothesis”

I.7Classesofbipartitemeasurementoperatorse.g.,HM’12• There’saninterestinglineofworkattemptingtounderstandQMA(2)withrestrictedverificationprotocols• WesayaPOVM (M,I-M)isin:• BELL :“systemsaremeasuredlocallywithnoconditioning”

• Where and• Sissetofpairsofoutcomes(indices)

• i.e.,systemsaremeasuredlocallygetoutcome(i,j)andacceptiff (i,j)∈S• 1LOCC:“choosemeasurementonsystemBconditionedonoutcomeofmeasurementonsystemA”

• Where andforeachMi• CanbegeneralizedtoLOCC byallowingforfinitenumberofroundsofalternatingmeasurementsonthetwosubsystems

• SEP istheclassofmeasurementsMsothat• Forpositivesemidefinitematrices{αi}and{βi}

• NoticethatBELL⊆LOCC1⊆LOCC⊆SEP⊆ALL

M =X

(i,j)2S

↵i ⌦ �jXi

↵i = IX

i

�i = I

M =X

i

↵i ⌦MiX

i

↵i = I 0 Mi I

M =X

i

↵i ⌦ �i

II.ResultsonQMA(2)

II.1.SAT protocol:Aaronson,Beigi,Drucker,F.,Shor‘09• Conjecture1:3SATn cannotbesolvedinclassical poly(n) time

• EquivalenttoNP⊄P• Conjecture2:3SATn cannotbesolvedinclassical 2o(n) time

• “Exponential-timeHypothesis”[Impagliazzo &Paturi ‘99]• Seemsreasonableevenquantumly – “Quantum ETH”

• Ourresult:• i.e.,sqrt(n) witnesses,eachonlog(n) qubits(*herenisnumberofclauses)• Noticetotalnumberofwitnessqubitsiso(n)• SameresultclassicallywouldshowExponential-timeHypothesistobefalse

• Proofidea:• Supposex1,x2,…,xn∈{0,1}n isMerlin’sclaimedsatisfyingassignment• AskallMerlins tosendthesamestate:• NeedmanyMerlins tocheckthathesentthisstate!

1pn

nX

i=1

(�1)xi |ii

3SATn 2 QMAlogn(Õ(pn))

II.1(partii).RelatedQMA(2)protocols

• Relatedprotocols:• [Blier &Tapp ’09]

• Viaprotocolfor3Coloring• IfsoundnesswasconstantthenNEXP⊆QMA(2)

• [Chen&Drucker’10]• Verifieruseslocalmeasurements• Matchesparametersof[ABDFS’09]

• Perfectcompletenessandconstantsoundness

NP ✓ QMAlogn(2, 1, 1�

1

poly(n))

3SATn 2 QMABELLlogn (Õ(

pn))

II.2.“Producttest”:Harrow &Montanaro ’12

• Forallk≤poly(n) QMA(k)=QMA(2)• Usesthe“producttest”• AskbothMerlins tosend• Pr ½:Arthur“swaptests”oneachofthekpairsofcorrespondingsubsystemsandacceptsiff theyallaccept• Swaptestonstatesρ andσ acceptswithprobability1/2+1/2Tr[ρσ]

• Pr ½:Arthurrunsverificationprotocolononeofthestates• Mainresult:

• Supposewearegiventwocopiesofk-partitestate• Let• ThenProducttestacceptswithprobability

• Infact,QMA(k)=QMASep(2)• Becausethe“accept”measurementofproducttestisseparableoperator

| i

1�⇥(✏)1� ✏ = max

|�i2Sep(k){|h |�i|2}

| i = | 1i ⌦ | 2i ⌦ | 3i ⌦ ...⌦ | ki

II.2(partii).Moreconsequencesof[HM’12]1. ImprovestheSAT protocolfrombefore[ABDFS’09]

1. Resultasstated:2. Resulttogetherwith[HM’12]:3. Don’tknowhowtoextendthistoChen&Druckerresult

2. Hardnessconsequencesfor“ε-BestSeparableState”problem• Input:HermitianmatrixMonA⊗B• Output:EstimateofhSep(M)=maxσ∈SepTr[Mσ]towithinadditiveerrorε• “Equivalent”inhardnesstoWeakMembershipproblem

• SothisproblemisNP-hardforε=1/poly(d)• NoticethatthisproblemisatleastashardasdecidingalanguageinQMA(2)

• Therefore,SATn canbecastasaBSS problemwith|A|=|B|≈2O(sqrt(n))• Givessubexponential boundsonthecomplexityofε-BestSeparableStateforconstantε• Supposethere’sanalgorithmrunsintimeexp(O(log1-ɣ|A|log1-ν|B|)thenETHisfalse!

• ε-BestSeparableStateturnsouttoalsobepolynomial-timeequivalenttomanyotherproblems• ConnectionstoUniqueGamesconjecturevia“2-to-4normproblem”(see[HM’12]fordetails)

3. IsQMA(2)⊆QMA?• QMAm(1)⊆BQTIME[O(2m)][Marriott&Watrous ‘04]• So,ifQMAm(2)=QMA𝒎2−ν theQuantumETHisfalse

4. QMASep(2)characterizationallowsustoerroramplifyusingrepetition!

SATn 2 QMAlogn(Õ(pn))

SATn 2 QMAÕ(pn)(2)

II.3.QMA(2)with1LOCC measurements[BCY’11]• “QuantumdeFinetti”Theorem

• Definition:WesayabipartitestateρAB isk-extendible if:• Thereexistsa(k+1)-partite𝜌CDEDF…DG sothat

• Separable statesarek-extendibleforallk>0[e.g.,DPS’08]• [Christandl et.al‘07]showsthatk-extendiblestatesareclosetoseparableinawell-definedsense:

• [BCY’11]showsmuchtighterrelationfor1LOCC norm:

• Asaconsequence,QMAm1LOCC(2)=QMAm2(1)• Proofidea:InQMA(1)protocol,ArthurasksMerlintosendk-extensionofhisbipartitewitness• UsedeFinetti theoremfor1LOCC toboundsoundnessprobability(i.e.,theadvantageMerlingetsfrom

entanglinghisstatesincasetheansweris‘No’)

• There’saninterestinglineofworktryingtoimprovethisresultinvariouswayse.g.,[Brandao &Harrow ‘11],[Lancien &Winter’16]

||⇢AB � Sep||1 4|B|2

k

||⇢AB � Sep||1LOCC r

log |A|k

⇢AB = ⇢AB1 = ⇢AB2 = ... = ⇢ABk

II.4.CompleteproblemforQMA(2)[Chailloux &Sattath ’12]• Recall:“k-localHamiltonianproblem”isQMA-complete• “SeparablesparseHamiltonianproblem”

• Definition:Anoperatorovern qubitsisrow-sparseif:• EachrowinAhasatmostpoly(n)non-zeroentries• There’sclassicalalgorithmthattakesarowindexandoutputsthenon-zeroentriesthisrow

• Input:Row-sparseHamiltonian,H,onnqubits• Promise:for(a,b)sothatb-a≥1/poly(n),either:

• ∃|ψ⟩ = |ψ⟩H ⊗ |ψ⟩J𝑠𝑜𝑡ℎ𝑎𝑡⟨𝜓|H|ψ⟩ ≤ aOR• ∀|ψ⟩ = |ψ⟩H ⊗ |ψ⟩J𝑤𝑒ℎ𝑎𝑣𝑒⟨𝜓|H|ψ⟩ ≥ b

• Proofuses “clock”constructionofKitaev and“Producttest”ofHarrow-Montanaro• Infact,samepapershowsthatSeparablelocal HamiltonianisQMA-complete• StartingpointforrecentattemptatprovingQMA(2)upperbound[Schwarz’15]

III.Openquestions/Previewofthingstocome

III.OpenQuestions

• CanweputanontrivialupperboundonQMA(2)?• CanChen&Drucker’s3SATprotocolwithBELLmeasurementsbeimprovedtouseonly2witnesses?• Canthe1LOCC deFinetti theorembeextendedtoSEP?measurements?ThiswouldimplyQMA(2)⊆QMA(1)• QMA(1)=QMA1LOCC(2)vsQMASEP(2)=QMA(k)• OtherQMA(2)-completeproblems?

III.Nexttime!

• Classicalcomplexityoftheε-BestSeparableStateproblem1. SDPhierarchiesanditsrelationtoBSS

• Givealgorithmsfor(specialcases)ofBSS• “Sum-of-Squares”

2. ε-nets