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ACompleteCharacterization ofUnitaryQuantumSpace BillFefferman (QuICS,Universityof Maryland/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)solvablewithaboundederror quantumalgorithmactingonO(k(n)) qubits: • Existsuniformlygeneratedfamilyofquantumcircuits{Qx}xϵ{0,1}* eachactingonO(k(|x|)) qubits: • “Ifanswerisyes,thecircuitQx acceptswithhighprobability” x 2 Lyes ) h0k |Q†x |1ih1|out Qx |0k i 2/3 • “Ifanswerisno,thecircuitQx acceptswithlowprobability” x 2 Lno ) h0k |Q†x |1ih1|out Qx |0k i 1/3 • *Uniformlygeneratedmeanspoly-time,O(k)-space 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 QuantumMerlin-Arthur • Problemswhosesolutionscanbeverifiedquantumly givena quantumstateaswitness • (t(n),k(n))-boundedQMAm(a,b) istheclassofpromiseproblems L=(Lyes,Lno)sothat: x 2 Lyes ) 9| i Pr[V (x, | i) = 1] a |ψ⟩ x 2 Lno ) 8| i Pr[V (x, | i) = 1] b • 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 2.Characterization1:k(n)-PreciseSuccinct Hamiltonianproblem 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)-spaceboundedverifier 2. (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))? | i • AskMerlintosendeigenstatewithminimumeigenvalue | i • Arthurrunsthe“poorman’sphaseestimation”circuitone-iAt and |0i | i • • • • H H e-iAt 1+e 2 i t |0i + 1 e 2 i t |1i | i 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 to withinprecisionε intimeandspacethatscaleswithlog(1/ε) • We’llneedtoimplementuptoprecisionε=2-k(n) • Thiscircuitusespoly(n) timeandO(k(n)) space Upperbound(2/4):QMA amplification • Wehaveshownthatk(n)-PreciseSuccinctHamiltonianisink(n)-space-boundedpreciseQMA • Nextstep:applyspace-efficient“in-place”QMA amplificationtoourpreciseQMA protocol • HowdoweerroramplifyQMA? 1. “Repetition”[Kitaev ’99] 2. “In-place”[MarriottandWatrous ‘04] • AskMerlintosendmanycopiesoftheoriginalwitnessandrunprotocoloneachone,takemajorityvote • Problemwiththis:numberofproofqubitsgrowswithimprovingerrorbounds • Needsr/(c-s)2 repetitionstoobtainerror2-rbyChernoff bound • Definetwoprojectors:⇧0 = |0ih0|anc and ⇧1 = V † |1ih1|out V • Noticethatthemax.acceptanceprobabilityoftheverifierismaximaleigenvalueof⇧0 ⇧1 ⇧0 • Procedure • • InitializeastateconsistingofMerlin’switnessandblankancilla Alternatinglymeasure {⇧1 , 1 ⇧1 } {⇧0 , 1 ⇧0 } andmanytimes • Usepostprocessingtoanalyzeresultsofmeasurements(rejectingiftwoconsecutivemeasurementoutcomesdiffertoomany times) • Analysisrelieson“Jordan’slemma” • • • Giventwoprojectors,there’sanorthogonaldecompositionoftheHilbertspaceinto1and2-dimensionalsubspacesinvariant underprojectors Basicallyallowsverifiertorepeateachmeasurementwithout“losing”Merlin’switness Becauseapplicationoftheseprojectors“stays”inside2Dsubspaces • Asaresult,wecanattainthesametypeoferrorreductionasinrepetition,withoutneedingadditionalwitnessqubits Formanyotherspace-efficientQMAamplificationtechniques,see[F.,Kobayashi,Lin,Morimae, NishimuraarXiv:1604.08192,ICALP’16] Upperbound(3/4):Space-efficientIn-placeamplification • We’renothappywithMarriott-Watrous amplification!! • M-W: k bounded QMAm (c, s) ✓ (k + r s)2 (c ) bounded QMAm (1 2 r ,2 r ) ,2 r ) • Thespacegrowsbecauseweneedtokeeptrackofeachmeasurementoutcome • Forourapplicationwereallywanttobeabletospace-efficientlyamplifyprotocolwith inverseexponentiallysmall(ink)gap • Recall:ourparameters:log(n)≤k≤poly(n),c-s=1/2k andr=k • ThenusingMWthespacecomplexityinamplifiedprotocolisfarlargerthank • Weareabletoimprovethis! k bounded QMAm (c, s) ✓ (k + log r c s I, R1 = 2⇧1 I ) bounded QMAm (1 2 r • NowthesamesettingofparameterspreservesO(k) spacecomplexity! • Proofidea: • Definereflections R0 • UsingJordan’slemma: = 2⇧0 • 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(4/4):(poly,k(n))-bounded QMAk(n)(c,c-2k(n))⊆BQSPACE[k(n)] ✓ tr • Recall: (t, k)-bounded QMAm (c, s) ✓ (O c s • Applyingthisamplificationresult: ◆ ✓ , O k + log ✓ r c s ◆◆ )-bounded QMAm (1 2 r, 2 r) • (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:Considerthesameverificationprocedurethatusesrandomlychosenbasisstate forawitness • Butnowwecanuseouramplificationresultagain(withm=0)! • RHS ⊆BQSPACE[O(k)] Lowerbound:k(n)-PreciseSuccinct Hamiltonianis BQSPACE[k(n)]-hard • Aneasycorollaryofour“space-efficient”amplificationtogetherwithKitaev’s clock construction • LetL=(Lyes,Lno)beanyprobleminBQSPACE[k(n)] • BydefinitionLisdecidedbyuniformfamilyofboundederrorquantumcircuitsusingk(n) space • wlog circuitisofsizeatmost2k(n) • Space-efficientlyamplifythiscircuit(withoutchangingthesizeorspacetoomuch) • Kitaev showshowtotakethiscircuitandproduceaHamiltonianwiththepropertythat: • Inthe“yescase”,theHamiltonian’sminimumeigenvalueislessthansomequantityinvolvingthe completeness andthecircuitsize • Inthe“nocase”,theHamiltonian’sminimumeigenvalueisatleastsomequantityinvolvingthe soundness andthecircuitsize • Byamplifyingthecompletenessandsoundnessofthecircuitwecanensurethatthe promisegapoftheHamiltonianisatleast2-k • EasytoshowthatthisHamiltonianissuccinctlyencoded • FollowsfromsparsityofKitaev’s constructionanduniformityofcircuit Application1:preciseQMA=PSPACE • Question:HowdoesthepowerofQMAscalewiththecompletenesssoundnessgap? • 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)-PreciseSuccinct HamiltonianProblemis 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:checkingifalocalHamiltonianhaszerogroundstateenergyis PSPACE-complete Application2:PreparingPEPSvs LocalHamiltonian • Twoboxes: • 𝓞 PEPS:TakesasinputclassicaldescriptionofPEPSandoutputsthestate • 𝓞 LocalHamiltonian:TakesasinputclassicaldescriptionofLocalHamiltonianandoutputsthe groundstate • Weshowasettinginwhich𝓞 LocalHamiltonian ismorepowerfulthan𝓞 PEPS • BQP𝓞PEPS=PostBQP=PP[Schuch et.al.’07] • Canextendprooftoshowthisisalsotruewithunboundederror • i.e.,PQP𝓞 =PP PEPS • HowpowerfulisPQP𝓞LocalHamiltonian? • PSPACE=PreciseQMA⊆PQP𝓞LocalHamiltonian • So𝓞LocalHamiltonian ismorepowerfulunlessPP=PSPACE 3.Characterization2:k(n)-WellConditioned MatrixInversion OurresultsonMatrixInversion • Classically,weknowthatn xn MatrixInversionisinlog2(n) space,butdon’t believeitcanbesolvedinclassical 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-conditionedMatrixInversioniscompletefor BQSPACE[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!