Download A Complete Characterization of Unitary Quantum

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!