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ACompleteCharacterization ofUnitaryQuantumSpace BillFefferman (QuICS,UniversityofMaryland) JointwithCedricLin(QuICS) BasedonarXiv:1604.01384 QIP2017,Seattle,Washington Ourmotivation:Howpowerfularequantum computerswithasmallnumberofqubits? • Ourresults:Givetwonaturalproblemscharacterizethepowerof quantumcomputationwithany boundonthenumberofqubits 1. PreciseSuccinctHamiltonianproblem 2. Well-conditionedMatrixInversionproblem • Thesecharacterizationshavemanyapplications • QMA proofsystemsandHamiltoniancomplexity • ThepowerofpreparingPEPS statesvsgroundstatesofLocalHamiltonians • ClassicalLogspace complexity QIP2017,Seattle,Washington Quantumspacecomplexity • BQSPACE[k(n)]istheclassofpromiseproblemsL=(Lyes,Lno)thatcanbe decidedbyaboundederrorquantumalgorithmactingonk(n) qubits. • i.e.,Existsuniformlygeneratedfamilyofquantumcircuits{Qx}xϵ{0,1}* eachactingon O(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 • OurresultsshowtwonaturalcompleteproblemsforBQSPACE[k(n)] • Foranyk(n) sothatlog(n)≤k(n)≤poly(n) • Ourreductionsuseclassicalk(n)spaceandpoly(n)time • Subtlety:Thisis“unitaryquantumspace” • Nointermediatemeasurements • Notknownif“deferring”intermediatemeasurementscanbedonespaceefficiently QIP2017,Seattle,Washington QuantumMerlin-Arthur • Problemswhosesolutionscanbeverifiedquantumly givenaquantumstate aswitness • QMA(c,s)istheclassofpromiseproblemsL=(Lyes,Lno)sothat: x 2 Lyes ) 9| i Pr[V (x, | i) = 1] c x 2 Lno ) 8| i Pr[V (x, | i) = 1] s • 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. 2. PreciseQMA=⋃c>0QMA(c,c-1/exp) k-boundedQMAm(c,s) • Arthur’sverificationcircuitactsonk qubits • Merlinsendsan m qubitwitness QIP2017,Seattle,Washington |ψ⟩ Characterization1: PreciseSuccinctHamiltonianproblem QIP2017,Seattle,Washington 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 | i | i • Arthurrunsphaseestimationwithoneancilla qubitone-iA and |0i | i H H e-iAt 1+e 2 i t |0i + 1 e 2 i t |1i | i • 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 QIP2017,Seattle,Washington 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-constructionto buildsparseHamiltonianfromtheamplifiedcircuit QIP2017,Seattle,Washington Application: PreciseQMA=PSPACE • Question:HowdoesthepowerofQMAscalewiththecompletenesssoundnessgap? • 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:checkingifalocalHamiltonianhaszerogroundstateenergyisPSPACEcomplete QIP2017,Seattle,Washington Whereisthispowercomingfrom? • CouldQMA=PreciseQMA=PSPACE? • Unlikelysince QMA=PreciseQMA ⇒ PSPACE=PP • UsingQMA ⊆PP • HowpowerfulisPreciseMA,theclassicalanalogueofPreciseQMA? • Crudeupperbound: PreciseMA⊆NPPP ⊆PSPACE • Andbelievedtobestrictlylesspowerful,unlessthe“CountingHiearchy” collapses • SothepowerofPreciseQMA seemstocomefromboththequantum witnessandthesmallgap,together! QIP2017,Seattle,Washington Understanding“Precise”complexityclasses • Wecananswerquestionsinthe“precise”regimethatwehaveno ideahowtoanswerinthe“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 QIP2017,Seattle,Washington Characterization2: Well-ConditionedMatrixInversion QIP2017,Seattle,Washington TheClassicalComplexityofMatrixInversion • TheMatrixInversionproblem • Input:nonsingularn xn matrixAwithintegerentries,promisedeither: an,0 an,1… A-1 = ?... ? … • Whichisthecase? A= a0,0 a0,1… … • A-1[0,0]>2/3or • A-1[0,0]<1/3 ?... ? • ThisproblemcanbesolvedinclassicalO(log2(n)) space[Csanky’76] • NotbelievedtobesolvableclassicallyinO(log(n)) space • Ifitis,thenL=NL (Logspace equivalentofP=NP) QIP2017,Seattle,Washington Canwedobetterquantumly? • “Well-ConditionedMatrixInversion”can besolvedinnon-unitary BQSPACE[log(n)]![Ta-Shma’12]buildingon[HHL’08] • i.e.,sameproblemwithpoly(n)upperboundontheconditionnumber,κ,so thatκ-1I≺A≺I • Appears toattainquadraticspeedupinspaceusageoverclassicalalgorithms • Begsthequestion:howimportantisthis“well-conditioned” restriction? • Canwealsosolvethegeneral MatrixInversionprobleminquantumspace O(log(n))? QIP2017,Seattle,Washington OurresultsonMatrixInversion • Well-conditionedMatrixInversioniscompleteforunitary BQSPACE[log(n)]! 1. WegiveanewquantumalgorithmforWell-conditionedMatrixInversion avoidingintermediatemeasurements • Combinestechniquesfrom[HHL’08]withamplitudeamplification 2. WealsoproveBQSPACE[log(n)]hardness– suggestingthat“well-conditioned” constraintisnecessary forquantumLogspace algorithms QIP2017,Seattle,Washington Cangeneralizefromlog(n)tok(n)qubits… • Result3:k(n)-Well-conditionedMatrixInversion iscompletefor BQSPACE[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,byvaryingthedimensionandtheboundonthe conditionnumber,canuseMatrixInversionproblem tocharacterize thepowerofquantumcomputationwithsimultaneouslybounded timeand space! QIP2017,Seattle,Washington Openquestions • CanweuseourPreciseQMA=PSPACE characterizationtogivea PSPACE upperboundforothercomplexityclasses? • Forexample,QMA(2)? • HowpowerfulisPreciseQIP? • Naturalcompleteproblemsfornon-unitaryquantumspace? QIP2017,Seattle,Washington Thanks! QIP2017,Seattle,Washington