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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