ACompleteCharacterizationofUnitaryQuantumSpaceBillFefferman (QuICS,UniversityofMaryland)

JointwithCedricLin(QuICS)

BasedonarXiv:1604.01384

QIP2017,Seattle,Washington

• Ourresults:Givetwonaturalproblemscharacterizethepowerofquantumcomputationwithany boundonthenumberofqubits1. PreciseSuccinctHamiltonianproblem2. Well-conditionedMatrixInversionproblem

• Thesecharacterizationshavemanyapplications• QMA proofsystemsandHamiltoniancomplexity• ThepowerofpreparingPEPS statesvsgroundstatesofLocalHamiltonians• ClassicalLogspace complexity

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Ourmotivation:Howpowerfularequantumcomputerswithasmallnumberofqubits?

Quantumspacecomplexity• BQSPACE[k(n)]istheclassofpromiseproblemsL=(Lyes,Lno)thatcanbedecidedbyaboundederrorquantumalgorithmactingonk(n) qubits.• i.e.,Existsuniformlygeneratedfamilyofquantumcircuits{Qx}xϵ{0,1}* eachactingonO(k(|x|)) qubits:• “Ifanswerisyes,thecircuitQx acceptswithhighprobability”

• “Ifanswerisno,thecircuitQx acceptswithlowprobability”

• OurresultsshowtwonaturalcompleteproblemsforBQSPACE[k(n)]• Foranyk(n) sothatlog(n)≤k(n)≤poly(n)• Ourreductionsuseclassicalk(n)spaceandpoly(n)time

• Subtlety:Thisis“unitaryquantumspace”• Nointermediatemeasurements• Notknownif“deferring”intermediatemeasurementscanbedonespaceefficiently

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x 2 L

yes

) h0k|Q†x

|1ih1|out

Q

x

|0ki � 2/3

x 2 L

no

) h0k|Q†x

|1ih1|out

Q

x

|0ki 1/3

|ψ⟩

QuantumMerlin-Arthur• Problemswhosesolutionscanbeverifiedquantumly givenaquantumstateaswitness• QMA(c,s)istheclassofpromiseproblemsL=(Lyes,Lno)sothat:

• QMA=QMA(2/3,1/3)= ⋃c>0QMA(c,c-1/poly)• k-LocalHamiltonianproblemisQMA-complete (whenk≥2)[Kitaev ’00]

• Input:𝐻 = ∑ 𝐻&'&() ,eachterm𝐻& isk-local

• Promiseeither:• Minimumeigenvalue𝜆min(H)>bor𝜆min(H)<a• Whereb-a≥1/poly(n)

• Whichisthecase?• GeneralizationsofQMA:

1. PreciseQMA=⋃c>0QMA(c,c-1/exp)2. k-boundedQMAm(c,s)

• Arthur’sverificationcircuitactsonk qubits• Merlinsendsan m qubitwitness

x 2 Lyes ) 9| i Pr[V (x, | i) = 1] � c

x 2 L

no

) 8| i Pr[V (x, | i) = 1] s

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Characterization1:PreciseSuccinctHamiltonianproblem

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ThePreciseSuccinctHamiltonianProblem• Definition:“SuccinctEncoding”

• WesayaclassicalTuringmachineMisaSuccinctEncodingfor2k(n) x2k(n) matrixAif:• Oninput i∈{0,1}k(n),M outputsnon-zeroelementsini-th rowofA• Usingatmostpoly(n) timeandk(n) space

• k(n)-PreciseSuccinctHamiltonian problem• Input:Sizen SuccinctEncodingof2k(n) x2k(n) HermitianPSDmatrixA• Promisedeither:

• Minimumeigenvalue𝜆min(A)>bor𝜆min(A)<a• Whereb-a>2-O(k(n))

• Whichisthecase?• ComparedtotheLocalHamiltonianproblem…

• InputisSuccinctlyEncodedinsteadofLocal• Precisionneededtodeterminethepromiseis1/2kinsteadof1/poly(n)

• OurResult:k(n)-P.SHamiltonian problemiscomplete forBQSPACE[k(n)]QIP2017,Seattle,Washington

Upperbound(1/2):k(n)-P.SHam.∈k(n)-bounded QMAk(n)(c,c-2-k(n))• Recall:k(n)-PreciseSuccinctHamiltonian problem

• GivenSuccinctEncodingof2k(n) x2k(n) HermitianPSDmatrixA,isλmin(A)≤aor λmin(A)≥b whereb-a≥2-O(k(n))?• Merlinsendeigenstatewithminimumeigenvalue

• Arthurrunsphaseestimationwithoneancilla qubitone-iA and

• Measureancilla andacceptiff “0”• Easytoseethatweget“0”outcomewithprobabilitythat’sslightly(2-O(k))higherifλmin(A)<a thanifλmin(A)>b• Butthisisexactlywhat’sneededtoestablishtheclaimedbound!

• Remainingquestion:howdoweimplemente-iA ?• Weneedtoimplementthisoperatorwithprecision2-k,sinceotherwisetheerrorinsimulationoverwhelmsthegap!• Luckily,wecaninvokerecent“preciseHamiltoniansimulation”resultsof[Childset.al’14]

• Implemente-iA towithinprecisionε inspacethatscaleswithlog(1/ε)andtimepolylog(1/ε)• SeealsoGuang Hao Low’stalkonThursday!

• Usingtheseresults,canimplementArthur’scircuitinpoly(n) timeandO(k(n)) space

H H|0i

| i e-iAt | i

1 + e�i�t

2|0i+ 1� e�i�t

2|1i

| i| i

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Upperbound(2/2):k(n)-bounded QMAk(n)(c,c-2-k(n))⊆BQSPACE[k(n)]1. ErroramplifythePreciseQMA protocol

• Goal:Obtainaprotocolwitherrorinverseexponentialinthewitnesslength,k(n)• WewanttodothiswhilesimultaneouslypreservingverifierspaceO(k(n))• Wedevelopnew“space-preserving”QMA amplificationprocedures

• Bycombiningideasfrom“in-place”amplification[Marriott&Watrous ‘04]withphaseestimation

2. “Guessthewitness”!• Considerthisamplifiedverificationprotocolrunonamaximallymixedstateonk(n)qubits• Nothardtoseethatthisnew“nowitness”protocolhasa“precise”gapofO(2-k(n))!

3. Amplifyagain!• Useour“space-efficient”QMA erroramplificationtechniqueagain!• Obtainboundederror,atacostofexponentialtime• ButthespaceremainsO(k(n)),establishingtheBQSPACE[k(n)]upperbound

• Space-efficientamplificationalsousedtoprovehardness!• k(n)-P.SHamiltonianisBQSPACE[k(n)]-hard• Followsfromfirstusingourspace-boundedamplification,andthenKitaev’s clock-constructiontobuildsparseHamiltonianfromtheamplifiedcircuit

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Application: PreciseQMA=PSPACE• Question:HowdoesthepowerofQMAscalewiththecompleteness-soundnessgap?• Recall: PreciseQMA=Uc>0QMA(c,c-2-poly(n))• Bothupperandlowerboundsfollowfromourcompletenessresult,togetherwithBQPSPACE=PSPACE[Watrous’03]• Corollary:“precisek-LocalHamiltonianproblem”isPSPACE-complete• Extension:“PerfectCompletenesscase”: QMA(1,1-2-poly(n))=PSPACE• Corollary:checkingifalocalHamiltonianhaszerogroundstateenergyisPSPACE-complete

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Whereisthispowercomingfrom?

• CouldQMA=PreciseQMA=PSPACE?• Unlikelysince QMA=PreciseQMA⇒ PSPACE=PP

• UsingQMA⊆PP

• HowpowerfulisPreciseMA,theclassicalanalogueofPreciseQMA?• Crudeupperbound: PreciseMA⊆NPPP⊆PSPACE• Andbelievedtobestrictlylesspowerful,unlessthe“CountingHiearchy”collapses

• SothepowerofPreciseQMA seemstocomefromboththequantumwitnessandthesmallgap,together!

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Understanding“Precise”complexityclasses

• Wecananswerquestionsinthe“precise”regimethatwehavenoideahowtoanswerinthe“bounded-error”regime• Example1:HowpowerfulisQMA(2)?• PreciseQMA=PSPACE(ourresult)• PreciseQMA(2)=NEXP [Blier &Tapp‘07,Pereszlényi‘12]• So,PreciseQMA(2)≠PreciseQMA,unlessNEXP=PSPACE

• Example2:Howpowerfularequantumvsclassicalwitnesses?• PreciseQCMA⊆NPPP• So,PreciseQMA ≠PreciseQCMA,unlessPSPACE⊆NPPP

• Example3:HowpowerfulisQMA withperfectcompleteness?• PreciseQMA=PreciseQMA1=PSPACE

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Characterization2:Well-ConditionedMatrixInversion

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TheClassicalComplexityofMatrixInversion

• TheMatrixInversionproblem• Input:nonsingularn xn matrixAwithintegerentries,promisedeither:

• A-1[0,0]>2/3or• A-1[0,0]<1/3

• Whichisthecase?

• ThisproblemcanbesolvedinclassicalO(log2(n)) space[Csanky’76]• NotbelievedtobesolvableclassicallyinO(log(n)) space• Ifitis,thenL=NL (Logspace equivalentofP=NP)

a0,0 a0,1…

an,0 an,1…A=

?... ?

?... ?A-1 =… …

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Canwedobetterquantumly?

• “Well-ConditionedMatrixInversion”can besolvedinnon-unitaryBQSPACE[log(n)]![Ta-Shma’12]buildingon[HHL’08]• i.e.,sameproblemwithpoly(n)upperboundontheconditionnumber,κ,sothatκ-1I≺A≺I• Appears toattainquadraticspeedupinspaceusageoverclassicalalgorithms

• Begsthequestion:howimportantisthis“well-conditioned”restriction?• CanwealsosolvethegeneralMatrixInversionprobleminquantumspaceO(log(n))?

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OurresultsonMatrixInversion

• Well-conditionedMatrixInversioniscompleteforunitaryBQSPACE[log(n)]!1. WegiveanewquantumalgorithmforWell-conditionedMatrixInversion

avoidingintermediatemeasurements• Combinestechniquesfrom[HHL’08]withamplitudeamplification

2. WealsoproveBQSPACE[log(n)]hardness– suggestingthat“well-conditioned”constraintisnecessary forquantumLogspace algorithms

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Cangeneralizefromlog(n)tok(n)qubits…

• Result3:k(n)-Well-conditionedMatrixInversion iscompleteforBQSPACE[k(n)]• Input:SuccinctEncodingof2k x2k PSDmatrixA

• Upperboundκ<2O(k(n)) ontheconditionnumbersothatκ-1I≺A≺I• Promisedeither|A-1[0,0]|≥2/3 or≤1/3• Decidewhichisthecase?

• Additionally,byvaryingthedimensionandtheboundontheconditionnumber,canuseMatrixInversionproblem tocharacterizethepowerofquantumcomputationwithsimultaneouslyboundedtimeand space!

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Openquestions

• CanweuseourPreciseQMA=PSPACE characterizationtogiveaPSPACE upperboundforothercomplexityclasses?• Forexample,QMA(2)?

• HowpowerfulisPreciseQIP?• Naturalcompleteproblemsfornon-unitaryquantumspace?

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Thanks!

QIP2017,Seattle,Washington