Download A Complete Characterization of Unitary Quantum Space

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
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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: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)]
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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))?
• 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
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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-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
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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 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,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:
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?
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Thanks!
QIP2017,Seattle,Washington