ACompleteCharacterizationofUnitaryQuantumSpace

BillFefferman (QuICS,UniversityofMaryland/NIST)

JointwithCedricLin(QuICS)

Basedonarxiv:1604.01384

1.Basics

Quantumspacecomplexity• Mainresult:Givetwoproblems“characterize”unitaryquantumspacecomplexity

• Roughly:Whatproblemscanwesolvebyquantumcomputationwithaboundednumberofqubits?• Forallspaceboundslog(n)≤k(n)≤poly(n) wefindaBQSPACE[k(n)]-completeproblem• Ourreductionswilluseclassical poly(n) timeandO(k(n))-space• Whatisclassical k(n)-space/memory?

• Inputisonitsown“read-only”tape,doesn’tusespace• EachbitoftheoutputcanbecomputedinO(k(n))-space

• k(n)-PreciseSuccinctHamiltonian andk(n)-Well-ConditionedMatrixInversion• BQSPACE[k(n)]istheclassofpromiseproblemsL=(Lyes,Lno)solvablewithaboundederrorquantumalgorithmactingonO(k(n)) qubits:• Existsuniformlygeneratedfamilyofquantumcircuits{Qx}xϵ{0,1}* eachactingonO(k(|x|)) qubits:• “Ifanswerisyes,thecircuitQx acceptswithhighprobability”

• “Ifanswerisno,thecircuitQx acceptswithlowprobability”

• *Uniformlygeneratedmeanspoly-time,O(k)-space

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

Known(andunknown)inspacecomplexity

• Any k≥log(n),NSPACE[k(n)]⊆DSPACE[k(n)2][Savitch ‘70]• Viaalgorithmfordirectedgraphconnectivity,withn verticesinlog2(n) space• (Obvious)Corollary1:NPSPACE=PSPACE• (Obvious)Corollary2:NL=NSPACE[log(n)]⊆DSPACE[log2(n)]

• UndirectedGraphConnectivitywithn verticesiscompleteforDSPACE[log(n)]=L [Reingold ’05]• BQSPACE[k(n)]⊆DSPACE[k(n)2][Watrous’99]

• Inparticular,BQPSPACE=PSPACE• Well-conditionedMatrixinversioninnon-unitaryquantumspacelog(n) [Ta-Shma’14]buildingon[HHL’08]

• Whatisthepowerofintermediatemeasurementsinquantumlogspace?• Notethat“deferring”measurementsinthestandardsenseisnotspaceefficient

• i.e.,aquantumlogspace algorithmcouldmakeasmanyaspoly(n) measurements• Deferringthemeasurementsrequirespoly(n)ancilla qubits

|ψ⟩

QuantumMerlin-Arthur• Problemswhosesolutionscanbeverifiedquantumly givena

quantumstateaswitness• (t(n),k(n))-boundedQMAm(a,b) istheclassofpromiseproblemsL=(Lyes,Lno)sothat:

• WhereVrunsinquantumtimet(n),andquantumspacek(n)• Andthewitness,|Ψ>isanm qubitstate

• QMA=(poly,poly)-boundedQMApoly(2/3,1/3)=(poly,poly)-bounded⋃c>0QMApoly(c,c-1/poly)• preciseQMA=(poly,poly)-bounded⋃c>0QMApoly(c,c-1/exp)• k-LocalHamiltonianproblemisQMA-complete (whenk≥2)[Kitaev ’02]

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

• Promise,for(a,b)sothatb-a≥1/poly(n),either:• ∃|ψ⟩𝑠𝑜𝑡ℎ𝑎𝑡⟨𝜓|H|ψ⟩ ≤ a• ∀|ψ⟩𝑠𝑜𝑡ℎ𝑎𝑡⟨𝜓|H|ψ⟩ ≥ b 5

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

x 2 L

no

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

2.Characterization1:k(n)-PreciseSuccinctHamiltonianproblem

Definitionsandproofoverview• Definition:Succinctencoding

• LetAbea2k(n) x2k(n) matrix.• WesayaclassicalTuringMachineMisasuccinctencodingformatrixAif:

• Oninputi∈{0,1}k(n),M outputsnon-zeroelementsini-th rowofA• Inpoly(n) timeandk(n) space

• k(n)-PreciseSuccinctHamiltonianProblem• Input:Sizen succinctencodingof2k(n) x2k(n) PSDmatrixAsothat:

• |H|=maxs,t(A(s,t))isconstant• Promisedminimumeigenvalueiseithergreaterthanborlessthana,whereb-a>2-O(k(n))

• Whichisthecase?• ProofSketchofBQSPACE[k(n)]-completeness(detailsinnextslides)

• Upperbound:k(n)-P.SHamiltonianProblem∈BQSPACE[k(n)]1. k(n)-P.SHamiltonianProblem∈ (poly,k(n))-bounded QMA(c,c-2-k(n))

• preciseQMA withk(n)-spaceboundedverifier2. (poly,k(n))-bounded QMA(c,c-2-k(n))⊆BQSPACE[k(n)]• Lowerbound:BQSPACE[k(n)]-hardness

• ApplicationofKitaev’s clock-construction

Upperbound(1/4):k(n)-P.SHam.∈(poly,k(n))-bounded QMAk(n)(c,c-2-k(n))• Recall:k(n)-PreciseSuccinctHamiltonianproblem

• Givensuccinctencodingof2k(n) x2k(n) matrixA,isλmin ≥b or≤a whereb-a≥2-O(k(n))?• AskMerlintosendeigenstatewithminimumeigenvalue

• Arthurrunsthe“poorman’sphaseestimation”circuitone-iAt and

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

• Canusehigh-precisionsparseHamiltoniansimulationof[Childset.al.’14]toimplemente-iAt towithinprecisionε intimeandspacethatscaleswithlog(1/ε)• We’llneedtoimplementuptoprecisionε=2-k(n)

• Thiscircuitusespoly(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

Upperbound(2/4):QMA amplification• Wehaveshownthatk(n)-PreciseSuccinctHamiltonianisink(n)-space-boundedpreciseQMA• Nextstep:applyspace-efficient“in-place”QMA amplificationtoourpreciseQMA protocol• HowdoweerroramplifyQMA?

1. “Repetition”[Kitaev ’99]• AskMerlintosendmanycopiesoftheoriginalwitnessandrunprotocoloneachone,takemajorityvote• Problemwiththis:numberofproofqubitsgrowswithimprovingerrorbounds• Needsr/(c-s)2 repetitionstoobtainerror2-rbyChernoff bound

2. “In-place”[MarriottandWatrous ‘04]• Definetwoprojectors: and• Noticethatthemax.acceptanceprobabilityoftheverifierismaximaleigenvalueof• Procedure

• InitializeastateconsistingofMerlin’switnessandblankancilla• Alternatinglymeasure andmanytimes

• Usepostprocessingtoanalyzeresultsofmeasurements(rejectingiftwoconsecutivemeasurementoutcomesdiffertoomanytimes)

• Analysisrelieson“Jordan’slemma”• Giventwoprojectors,there’sanorthogonaldecompositionoftheHilbertspaceinto1and2-dimensionalsubspacesinvariant

underprojectors• Basicallyallowsverifiertorepeateachmeasurementwithout“losing”Merlin’switness• Becauseapplicationoftheseprojectors“stays”inside2Dsubspaces

• Asaresult,wecanattainthesametypeoferrorreductionasinrepetition,withoutneedingadditionalwitnessqubits

⇧0 = |0ih0|anc

{⇧0, 1�⇧0} {⇧1, 1�⇧1}

⇧1 = V †|1ih1|out

V⇧0⇧1⇧0

• We’renothappywithMarriott-Watrous amplification!!• M-W:

• Thespacegrowsbecauseweneedtokeeptrackofeachmeasurementoutcome• Forourapplicationwereallywanttobeabletospace-efficientlyamplifyprotocolwithinverseexponentiallysmall(ink)gap• Recall:ourparameters:log(n)≤k≤poly(n),c-s=1/2k andr=k• ThenusingMWthespacecomplexityinamplifiedprotocolisfarlargerthank

• Weareabletoimprovethis!

• NowthesamesettingofparameterspreservesO(k) spacecomplexity!• Proofidea:

• Definereflections• UsingJordan’slemma:

• Within2Dsubspaces,theproductR0R1 isarotationbyananglerelatedtoacceptanceprobabilityofverifierVx

• UsephaseestimationonR0R1withMerlin’sstateandancillias setto0• Keypoint:Phaseestimationtoprecisionj withfailureprobabilityα usesO(log(1/jα)) ancilla qubits

• “Succeed”ifthephaseislargerthanfixedthreshold,rejectotherwise• Repeatthismanytimesanduseclassicalpost-processingontheoutcomestodetermineacceptance

• Relatedtoolderresultof[NWZ’11]butimprovesonspacecomplexity

Upperbound(3/4):Space-efficientIn-placeamplification

R0 = 2⇧0 � I, R1 = 2⇧1 � I

Formanyotherspace-efficientQMAamplificationtechniques,see[F.,Kobayashi,Lin,Morimae,NishimuraarXiv:1604.08192,ICALP’16]

k� bounded QMAm(c, s) ✓ (k+ log

r

c� s)� bounded QMAm(1� 2

�r, 2�r)

k � bounded QMAm(c, s) ✓ (k +

r

(c� s)2)� bounded QMAm(1� 2

�r, 2�r)

Upperbound(4/4):(poly,k(n))-boundedQMAk(n)(c,c-2k(n))⊆BQSPACE[k(n)]• Recall:• Applyingthisamplificationresult:

• (poly,k(n))-bounded QMAk(c,c-2-k(n))⊆(2O(k),k(n))-bounded QMAk(1-2-O(k),2-O(k))

• Removingthewitness![MarriottandWatrous ‘04]• Thm.RHS⊆QSPACE[O(k)](3/4(2-O(k)),1/4(2-O(k)))

• Pf.Idea:Considerthesameverificationprocedurethatusesrandomlychosenbasisstateforawitness

• Butnowwecanuseouramplificationresultagain(withm=0)!• RHS⊆BQSPACE[O(k)]

(t, k)-bounded QMAm(c, s) ✓ (O✓

tr

c� s

◆,O

✓k+ log

✓r

c� s

◆◆)-bounded QMAm(1� 2�r, 2�r

)

• Aneasycorollaryofour“space-efficient”amplificationtogetherwithKitaev’s clockconstruction

• LetL=(Lyes,Lno)beanyprobleminBQSPACE[k(n)]• BydefinitionLisdecidedbyuniformfamilyofboundederrorquantumcircuitsusingk(n)space• wlog circuitisofsizeatmost2k(n)

• Space-efficientlyamplifythiscircuit(withoutchangingthesizeorspacetoomuch)• Kitaev showshowtotakethiscircuitandproduceaHamiltonianwiththepropertythat:

• Inthe“yescase”,theHamiltonian’sminimumeigenvalueislessthansomequantityinvolvingthecompleteness andthecircuitsize

• Inthe“nocase”,theHamiltonian’sminimumeigenvalueisatleastsomequantityinvolvingthesoundness andthecircuitsize

• ByamplifyingthecompletenessandsoundnessofthecircuitwecanensurethatthepromisegapoftheHamiltonianisatleast2-k

• EasytoshowthatthisHamiltonianissuccinctlyencoded• FollowsfromsparsityofKitaev’s constructionanduniformityofcircuit

Lowerbound:k(n)-PreciseSuccinctHamiltonianis BQSPACE[k(n)]-hard

Application1:preciseQMA=PSPACE• Question:HowdoesthepowerofQMAscalewiththecompleteness-soundnessgap?• Recall: preciseQMA=Uc>0QMA(c,c-2-poly(n))• Upperbound: preciseQMA⊆BQPSPACE=PSPACE

• Priorslidesshowedsomethingstronger!

• Lowerbound:PSPACE⊆preciseQMA• Wejustshowedk(n)-Precise SuccinctHamiltonianProblemis BQSPACE[k(n)]-hard• SinceBQPSPACE=PSPACE [Watrous’03]wehavepoly(n)-PreciseSuccinctHamiltonianProblemis PSPACE-hard

Application1:preciseQMA=PSPACE

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

• UsingQMA⊆PP

• WhatistheclassicalanalogueofpreciseQMA?• Certainly NPPP⊆PPPP⊆PSPACE• PPPP=PSPACE⇒CHcollapse!

• Corollary:“precisek-LocalHamiltonianproblem”isPSPACE-complete• Extension:“PerfectCompleteness”: QMA(1,1-2-poly(n))=PSPACE

• Corollary:checkingifalocalHamiltonianhaszerogroundstateenergyisPSPACE-complete

Application2:PreparingPEPSvs LocalHamiltonian

• Twoboxes:• 𝓞 PEPS:TakesasinputclassicaldescriptionofPEPSandoutputsthestate• 𝓞 LocalHamiltonian:TakesasinputclassicaldescriptionofLocalHamiltonianandoutputsthegroundstate

• Weshowasettinginwhich𝓞 LocalHamiltonian ismorepowerfulthan𝓞 PEPS

• BQP𝓞PEPS=PostBQP=PP[Schuch et.al.’07]• Canextendprooftoshowthisisalsotruewithunboundederror• i.e.,PQP𝓞PEPS=PP

• HowpowerfulisPQP𝓞LocalHamiltonian?• PSPACE=PreciseQMA⊆PQP𝓞LocalHamiltonian

• So𝓞LocalHamiltonian ismorepowerfulunlessPP=PSPACE

3.Characterization2:k(n)-WellConditionedMatrixInversion

OurresultsonMatrixInversion• Classically,weknowthatn xn MatrixInversionisinlog2(n) space,butdon’tbelieveitcanbesolvedinclassical log(n)space• k(n)-Well-conditionedMatrixInversion

• Input:Efficientencodingof2k x2k PSDmatrixA,ands,t∈{0,1}k :• Upperboundκ<2O(k(n)) ontheconditionnumbersothatκ-1I≺A≺I• Promisedeither|A-1(s,t)|≥b or≤a wherea,b areconstantsbetween0and1

• Decidewhichisthecase?• Ourresult: k(n)-Well-conditionedMatrixInversioniscompleteforBQSPACE[k(n)]• ImprovesonTa-Shma:

1. Nointermediatemeasurements!2. Wealsohavehardness!

• Complexityimplications:• IfMatrixinversioncanbesolvedinL thenBQL=L (seemsunlikely)• Evidenceofquantumspacehierarchytheorem?

• k(n)-Well-conditionedMatrixinversionseemsstrictlyharderwithlargerk• Seemsclosetoshowingiff(n)=o(g(n))thenBQSPACE[g(n)]⊄BQSPACE[f(n)]

Thanks!