ACompleteCharacterization
ofUnitaryQuantumSpace
BillFefferman (QuICS,UniversityofMaryland)
JointwithCedricLin(QuICS)
BasedonarXiv:1604.01384
TheorySeminar,UTAustin
Ourmotivation:Howpowerfularequantum
computerswithasmallnumberofqubits?
• Ourresults:Givetwonaturalproblemscharacterizethepowerof
quantumcomputationwithagivenboundonthenumberofqubits
1. PreciseSuccinctHamiltonianproblem
2. Well-conditionedMatrixInversionproblem
• Thesecharacterizationshavemanyapplications
• QMA proofsystemsandHamiltoniancomplexity
• ClassicalLogspace complexity
• Evenconnectionstophysics(e.g.,thepowerofpreparingPEPS states)
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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
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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
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|ψ⟩
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: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)]
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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))?
• Recallalso:Quantumalgorithmfor“phaseestimationproblem”[Kitaev ’95]
• Eigenvaluesofunitarymatricesarerootsofunity,e2𝜋iθ for0≤θ<1
• “Phaseestimationproblem”:GivenunitaryUandeigenstate|𝜓⟩outputanapproximationtothephaseθ
• PreciseQMA protocol:Merlinsendseigenstate|𝜓⟩ withminimumeigenvalue
• Arthurrunsphaseestimationwithoneancilla qubitone-iA and|𝜓⟩
H
H
|0i
e-iA
| i
1+e
2
i
|0i +
1
e
2
i
|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]
•
GivenSuccinctEncodingofA,implemente-iA towithinprecisionε inspacethatscaleswithlog(1/ε)andtimepolylog(1/ε)
• Usingtheseresults,canimplementArthur’scircuitusingO(k(n))
spaceandpoly(n) time
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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))
• We’llactuallydevelopamplificationtechniquethatdoesthis…
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
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QMA amplification
• OurproofneededaparticularlystrongQMA amplificationprocedure
• OnethatpreservesbothMerlin’switnesslengthandArthur’sverificationspace
• Prioramplificationmethods
1.
“Repetition”[Kitaev ’99]
2.
“In-place”Amplification[MarriottandWatrous ‘04]
• Definetwoprojectors:⇧0 = |0ih0|anc and ⇧1 = Vx† |1ih1|out Vx
• AskMerlintosendmanycopiesoftheoriginalwitnessandrunprotocoloneachone,takemajorityvote
• Problemwiththis:numberofwitnessqubitsgrowswithimprovingerrorbounds
• Needsr/(c-s)2 repetitionstoobtainerror2-rbyChernoff bound
• Noticethatthemax.acceptanceprobabilityoftheverifierismaximaleigenvalueof ⇧0 ⇧1 ⇧0
• Procedure
• InitializeastateconsistingofMerlin’switnessandallzeroancilla qubits
• Alternatinglymeasure
{⇧0 , 1 ⇧0 }andmanytimes
{⇧1 , 1 ⇧1 }
• Usepostprocessingtoanalyzeresultsofmeasurements
• Analysisrelieson“Jordan’slemma”
• Giventwoprojectors,there’sanorthogonaldecompositionoftheHilbertspaceinto1and2-dimensional
subspacesinvariantunderprojectors
• Basicallyallowsverifiertorepeateachmeasurementwithout“losing”Merlin’switness
• Becauseapplicationoftheseprojectors“staysinside”2Dsubspaces
• Asaresult,wecanattainthesametypeoferrorreductionasinrepetition,withoutneedingadditional
witnessqubits
Space-efficientIn-placeamplification
ForotherresultsimprovingMarriott-Watrous invariousdirectionsseee.g.,[Nagaj et.al.’09 &F.,
Kobayashi,Lin,Morimae,Nishimura,ICALP’16]
• We’renothappywithMarriott-Watrous amplification!!
k
bounded QMAm (c, s) ✓ (k +
r
)
2
bounded QMAm (1
2
r
,2
r
)
bounded QMAm (1
2
r
,2
r
)
(c s)
• Thespacegrowsbecauseweneedtokeeptrackofeachmeasurementoutcome
• Wewanttobeabletospace-efficientlyamplifyprotocolwithinverseexponentially
smallgap(i.e.,c-s=1/2k)
• Weareabletoimprovethis!
k
bounded QMAm (c, s) ✓ (k + log
r
c s
• NowthesamesettingofparameterspreservesO(k) spacecomplexity!
• Proofidea:
• Definereflections R0 = 2⇧0 I, R1 = 2⇧1 I
)
• UsingJordan’slemma:
• Within2Dsubspaces,theproductR0R1 isarotationbyananglerelatedtoacceptanceprobabilityof
verifierVx
• UsephaseestimationonR0R1withMerlin’sstate|𝜓⟩andancillas setto0
• Keypoint:Phaseestimationtoprecisionj uses O(log(1/j)) ancilla qubits
• Acceptifthephaseislargerthanfixedthreshold,rejectotherwise
Lowerbound:k(n)-PreciseSuccinct
Hamiltonianis BQSPACE[k(n)]-hard
• Followsfromspace-efficientQMA amplificationandKitaev’s “clock-construction”
• AnylanguageinBQSPACE[k(n)]canbedecidedbyuniformfamilyofquantum
circuits{Qx}xϵ{0,1}* ofsizeatmost2k(|x|)
• Byouruniformitycondition
• Kitaev showshowtotakethiscircuitandbuildaHamiltonian𝐻 = ∑'
&() 𝐻& with
thepropertythat:
• Inthe“yescase”,theHamiltonian’sminimumeigenvalueislessthansomequantitya
involvingthecompleteness andthecircuitsize
• Inthe“nocase”,theHamiltonian’sminimumeigenvalueisatleastsomequantityb involving
thesoundness andthecircuitsize
• Byamplifyingthecompletenessandsoundnessofthecircuitwecanensurethat
thepromisegapoftheHamiltonian,b-a,isatleast2-k
• EasytoshowthatthisHamiltonianissuccinctlyencoded
• FollowsfromsparsityofKitaev’s constructionanduniformityofcircuit
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Application: PreciseQMA=PSPACE
• Question:HowdoesthepowerofQMAscalewiththecompleteness-soundness
gap?
• Recall: PreciseQMA=Uc>0QMA(c,c-2-poly(n))
• Bothupperandlowerboundsfollowfromourcompletenessresult,together
withBQPSPACE=PSPACE[Watrous’03]
• Upperbound(PreciseQMA⊆PSPACE):
• Showedpoly(n)-P.SHam. ⊆ BQSPACE[poly(n)]=PSPACE
• Lowerbound(PSPACE⊆PreciseQMA):
1.
2.
Showedpoly(n)-P.S.Ham. ishardforBQSPACE[poly(n)]=PSPACE
Butalsoit’sinPreciseQMA by“poorman’sphaseestimation”
• Corollary:“precisek-LocalHamiltonianproblem”isPSPACE-complete
• Extension:“PerfectCompletenesscase”: QMA(1,1-2-poly(n))=PSPACE
• Corollary:checkingifalocalHamiltonianhaszerogroundstateenergyisPSPACEcomplete
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Whereisthispowercomingfrom?
• CouldQMA=PreciseQMA=PSPACE?
• Unlikelysince QMA=PreciseQMA ⇒ PSPACE=PP
• UsingQMA ⊆PP
• HowpowerfulisPreciseMA,theclassicalanalogueofPreciseQMA?
• Crudeupperbound: PreciseMA⊆NPPP ⊆PSPACE
• Andbelievedtobestrictlylesspowerful,unlessthe“CountingHierarchy”
collapses
• SothepowerofPreciseQMA seemstocomefromboththequantum
witnessandthesmallgap,together!
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Understanding“Precise”complexityclasses
• Wecananswerquestionsinthe“precise”regimethatwehaveno
ideahowtoanswerinthe“bounded-error”regime
• Example1:HowpowerfulisQMA(2)?
• PreciseQMA=PSPACE(ourresult)
• PreciseQMA(2)=NEXP [Blier &Tapp‘07]
• 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:
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[Csansky’76]
• NotbelievedtobesolvableclassicallyinO(log(n)) space
• Ifitis,thenL=NL (Logspace equivalentofP=NP)
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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))?
• OrcouldtheWell-ConditionedcasebeinclassicalLogspace?
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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
• SothisisanotherreasontobelieveMatrixInversioncan’tbesolvedin
classicalLogspace (becauseotherwiseL=BQL)
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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!
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Openquestions
• CanweuseourPreciseQMA=PSPACE characterizationtogivea
PSPACE upperboundforothercomplexityclasses?
• Forexample,QMA(2)?
• HowpowerfulisPreciseQIP?
• Naturalcompleteproblemsfornon-unitaryquantumspace?
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Thanks!
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