Space-efficientErrorReductionforUnitaryQuantum
ComputationsBillFefferman
QuICS,UniversityofMaryland/NISTJointwithHirotada Kobayashi,CedricYen-YuLin,Tomoyuki
Morimae,andHarumichi Nishimura
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Overview
1. BasicDefinitions2. Pastwork:QMA erroramplification3. Ourresults
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1.BasicDefinitions
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Unitaryquantumspacecomplexity• Wesaythatafamilyofquantumcircuits{Qx}x∈{0,1}n actingonk(n) qubitssolves apromiseproblemL=(Lyes,Lno)if:
• BQTIME[t(n)]istheclassofpromiseproblemssolvableinquantumtimet(n):• i.e.,byauniformlygeneratedfamilyofquantumcircuits{Qx},eachcomposedofO(t(n))gates
• BQSPACE[k(n)]istheclassofpromiseproblemssolvableink(n)quantumspace• i.e.,byauniformlygeneratedfamilyofquantumcircuits{Qx}eachactingonO(k(n))qubits
• Subtletiesindefiningquantumspaceboundedcomputation• Powerofintermediatemeasurements?• Ourfocus:unitary case
x 2 L
yes
) h0k|Q†x
|1ih1|out
Q
x
|0ki � 2/3
x 2 L
no
) h0k|Q†x
|1ih1|out
Q
x
|0ki 1/3
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|ψ⟩
QuantumMerlin-Arthur• Problemswhosesolutionscanbeverifiedquantumly givena
quantumstateaswitness• k(n)-boundedQMAm(c,s)istheclassofpromiseproblemsL=(Lyes,Lno)sothatthereexistsaverifier{Vx}actingonO(m(|x|)+k(|x|))qubits:
• QMA isacentraltopicofstudyinquantumcomplexitytheory• “QuantumNP”• Manyconnectionstophysics(i.e.,estimatingthegroundstateenergyofaLocalHamiltonianisQMA-complete[Kitaev’02])
• Butsomeofthemostnaturalquestionsareembarrassinglyopen• Ourmainresultisamethodforspace-efficientQMA error-amplification ICALP2016
x 2 L
yes
) 9| i�h |⌦ h0k|
�V
†x
|1ih1|out
V
x
�| i ⌦ |0ki
�� c
x 2 L
no
) 8| i�h |⌦ h0k|
�V
†x
|1ih1|out
V
x
�| i ⌦ |0ki
� s
2.Pastwork:QMA erroramplification
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QMA error amplificationusingrepetition• “Repetition”[Kitaev ’02]
• AskMerlintosendmanycopiesoftheoriginalwitness• Verifierrepeatsoriginalprotocoloneachone,measuresandtakesmajorityvoteofoutcomes• UsingChernoff bound,toobtainerror2-p,needO(p/(c-s)2)repetitions• Problemwiththis:numberofwitnessandspacequbitsgrowwithimprovingerrorbounds• i.e.,foranygivenp:
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k�bounded QMAm(c, s) ✓ (k· p
(c� s)2)�bounded QMAm· p
(c�s)2(1�2
�p, 2�p)
“In-place”QMA amplification• “Amplificationwithoutdestroyingwitness”[MarriottandWatrous ’04]
• Definetwoprojectors: and• Noticethemax.acceptanceprobabilityofVx isthemaximaleigenvalueof• Verificationprocedure:
• InitializeastateconsistingofMerlin’switnesstensored withancilla qubitsinitializedtoall-zerostate• Alternatinglymeasure andmanytimes
• Usepostprocessingtoanalyzeresultsofmeasurements(rejectingiftwoconsecutivemeasurementoutcomesdiffertoomanytimes)
• Analysisrelieson“Jordan’slemma”• Giventwoprojectors,Hilbertspacedecomposesinto1and2-dimensionalsubspacesinvariantunderprojectors
• Basicallyallowsverifiertorepeateachmeasurementwithout“losing”Merlin’switness• Becauseapplicationoftheseprojectors“stays”inside2Dsubspaces
• Asaresult,wecanattainthesametypeoferrorreductionasinrepetition,withoutneedingadditionalwitnessqubits
• However,weneedadditionalspacetokeeptrackofmeasurementoutcomes
{⇧0, 1�⇧0} {⇧1, 1�⇧1}
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⇧0 = |0ih0|anc ⇧1 = V †x
|1ih1|out
Vx
k � bounded QMAm(c, s) ✓ (k +
p
(c� s)2)� bounded QMAm(1� 2
�p, 2�p)
⇧0⇧1⇧0
3.Ourresults
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Ourresults:Space-efficientQMA erroramplification
• Nagaj,Wocjan,andZhang[NWZ’11]improvementsonMarriott-Watrous:
• Noticetoachieveerror2-poly requirespolynomialextraancilla qubits!• MainTheorem:
• Asaconsequence,weshowthefirst“strong”erroramplificationprocedureforunitaryquantumlogspace protocols• Wegivethreeproofsofmaintheoremusingdifferentprocedures
• I’lltalkaboutthesimplestone• Othertwoproofsachievebetterparameters
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k�bounded QMAm(c, s) ✓ (k+p log1
c� s)�bounded QMAm(1�2
�p, 2�p)
k� bounded QMAm(c, s) ✓ (k+ log
p
c� s)� bounded QMAm(1� 2
�p, 2�p)
MainTheorem(Proofsketch1/3)• We’llusethephaseestimationalgorithm[Kitaev ‘95]
• Importantingredientinmanyquantumalgorithms• GivenquantumcircuitforimplementingunitaryUandeigenvector|𝜓>estimateseigenphase θ• Uptoprecisionj withfailureprobabilityα usingO(log(1/jα)) ancilla qubits
• Definereflections• Thesearethe“Grover”reflectionsthatapplyaphaseflipifnotintheprojectedsubspace
• UsingJordan’slemma:• Within2Dsubspaces,theproductR0R1 isarotationbyananglerelatedtoacceptanceprobabilityofverifierVx
1. UsephaseestimationonR0R1withMerlin’sstateandancillias setto0,toamplifyerrortoinversepolynomial(relatedtoapproachof[NWZ’11])• Acceptifphaseisaboveacertainthreshold,rejectotherwise• DothiswithprecisionO(c-s) andfailureprobabilityα=1/(8p)• Completenessis1-1/(8p),Soundnessis1/(8p)• UsesspaceO(log(p/(c-s)))
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R0 = 2⇧0 � I, R1 = 2⇧1 � I
MainTheorem(Proofsketch2/3)1. Vx
(1) runsmildphaseestimationtoachievecompleteness1-1/(8p(n))andsoundness1/(8p(n))
2. Take“AND”ofN1=O(p(n)) iterationsofVx(1)
• LetVx(2) bethequantumcircuitrepeatsthefollowingN1 times:
• AppliesVx(1) andincrementsacounteriftheoutputstateisreject
• Applies(Vx(1))†
• Acceptiff counterisstillsetto0• Completenessis1-N1/8p(n)≥1/2, Soundnessis(1/(8p(n)))N1≤2-p(n)
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MainTheorem(Proofsketch3/3)1. Vx
(1) runsmildphaseestimationtoachievecompleteness1-1/(8p(n))andsoundness1/(8p(n))
2. Vx(2) takes“AND”ofN1iterationsofVx
(1)toachieveconstantcompleteness,andexponentiallysmallsoundnesserror
3. Take“OR”ofN2=O(p(n)) iterationsofVx(2)
• RepeatsthefollowingN2 times:• AppliesVx
(2) andincrementsacounterby1iftheoutputstateisaccept• Applies(Vx
(2))†• Acceptiff counterisatleast1• Completenessisatleast1-2-p(n), Soundnessisatmostp(n)2-p(n)
• Keypoint:ThespaceusedinthenewverificationprocedureisO(log(p/(c-s)))+log(N1)+log(N2)=O(log(p/(c-s)))• Otherproofsachievesimilaramplificationresultswithoutphaseestimation
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ApplicationsofMainTheorem• Strongerrorreductionforunitaryquantumlogspace
• i.e.,foranya-b≥1/poly,QSPACE[log(n)](a,b)⊆ QSPACE[log(n)](1-2-poly,2-poly)• Uselessnessofquantumwitnessesinlog(n)-boundedQMA
• Idea:Logspace algorithmwithnowitnesscanchooserandomlog(n) bitbasisstateaswitness,andthenerroramplify
• i.e.,log(n)-boundedQMAlog(n)(2/3,1/3)=BQSPACE[log(n)]• QMA withexponentiallysmallcompleteness-soundnessgapiscontainedinPSPACE• i.e., PreciseQMA⊆PSPACE• ProofsuseMaintheoremand“uselessnessofquantumwitnessinlog(n)-boundedQMA”• HasseveralapplicationstophysicsviaLocalHamiltonianproblem
• StrongerroramplificationofMatchgate computations• Physicallymotivatedmodel(relatedtoquantumcomputationwithnoninteracting fermions)• Knowntobeclassicallysimulable [Valiant‘02]• Usesequivalenceoflogspace quantumcomputationandmatchgate computation[Jozsa et.al.‘10]
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ACompleteCharacterizationofUnitaryQuantumSpace[F.,Lin‘16]• Whyareweinterestedinunitary quantumspacecomplexity?• Motivatedbyrecentresult[F.,Lin‘16]
• Givestwonaturalcompleteproblemsfor(unitary)BQSPACE[k(n)]• Underclassicalk(n)-spacepoly(n)-timereductions• Givenasuccinctlyspecified2k(n)x2k(n) PSDmatrixA:
1. EstimateagivenentryofA-1 (assumingAiswell-conditioned)2. EstimatingminimumeigenvalueofAtoinverseexponentialprecision
• Interestingly,theMatrixinversionproblemwithdifferentparametersettingiscompleteforBQTIME[t(n)]aswell
• Asacorollary,wecanshowthelowerbound PSPACE⊆preciseQMA• Andsotogetherwithupperbound fromtoday’sresults,preciseQMA=PSPACE
• FormoredetailsseearXiv:1604.01384
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OpenQuestions
• Canwefindspace-efficientmethodsforin-placeamplificationofQMAwith intermediatemeasurements?• NotethatMarriott-Watrous projectionoperatorsexplicitlyusetheinverseoftheverificationprocedure
• WhatisthepowerofQMA withdoubly-exponentiallysmallgap?• CanshowthatthisisstillequaltoPSPACE ifprotocolhasperfectcompleteness
• Canweusethisupperbound onpreciseQMA toshowupperboundsforothercomplexityclasses?
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Thanks!
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