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QuantumAlgorithmsforSystemsofLinearEquations
RolandoSommaTheoreticalDivisionLosAlamosNationalLaboratory
Jointworkwith
WorkshopattheIntersectionofMachineLearningandQuantumInformationUniversityofMaryland
September2018
RobinKothariMicrosoft
Yigit SubasiLosAlamos
AndrewChildsMaryland
DavideOrsucciVienna
QuantumAlgorithmsforSystemsofLinearEquations
References:
• “Quantumlinearsystemsalgorithmwithexponentiallyimproveddependenceonprecision”,A.M.Childs,R.Kothari,andR.D.Somma,SIAMJ.Comp.46,1920(2017).
• “Quantumalgorithmsforlinearsystemsinspiredbyadiabaticquantumcomputing”,Y.Subasi,R.D.Somma,andD.Orsucci,arXiv:1805.10549(2018).
Abriefhistoryofresultsinquantumcomputing
• SimulatingquantumsystemswasthemainmotivationbehindFeynman’sideaofaquantumcomputer(1982).
• Forexample,algorithmsforsimulatingthedynamicsofn spinsystemswithclassicalcomputershavecomplexitythatisexponentialinn.Quantumalgorithms,inprinciple,haveonlycomplexitythatispolynomialinn.
Abriefhistoryofresultsinquantumcomputing
• SimulatingquantumsystemswasthemainmotivationbehindFeynman’sideaofaquantumcomputer(1982).
• Forexample,algorithmsforsimulatingthedynamicsofn spinsystemswithclassicalcomputershavecomplexitythatisexponentialinn.Quantumalgorithms,inprinciple,haveonlycomplexitythatispolynomialinn.
• PeterShordiscoversaquantumalgorithmforefficientfactorizationofintegerswithimportantapplicationstocryptography.Shor’salgorithmresultsinasuperpolynomial quantumspeedup(1994).Hisresultwasamainmotivationforthediscoveryofotherquantumalgorithms.
Abriefhistoryofresultsinquantumcomputing
• SimulatingquantumsystemswasthemainmotivationbehindFeynman’sideaofaquantumcomputer(1982).
• Forexample,algorithmsforsimulatingthedynamicsofn spinsystemswithclassicalcomputershavecomplexitythatisexponentialinn.Quantumalgorithms,inprinciple,haveonlycomplexitythatispolynomialinn.
• PeterShordiscoversaquantumalgorithmforefficientfactorizationofintegerswithimportantapplicationstocryptography.Shor’salgorithmresultsinasuperpolynomial quantumspeedup(1994).Hisresultwasamainmotivationforthediscoveryofotherquantumalgorithms.
• L.Groverdiscoversaquantumalgorithmforunstructuredsearchresultinginapolynomial(quadratic)quantumspeedup(1997).AmainideainGrover’sresult(amplitudeamplification)hasbeenextensivelyusedinotherquantumalgorithmsforproblemssuchasoptimization,search,andmore.
Abriefhistoryofresultsinquantumcomputing
• SimulatingquantumsystemswasthemainmotivationbehindFeynman’sideaofaquantumcomputer(1982).
• Forexample,algorithmsforsimulatingthedynamicsofn spinsystemswithclassicalcomputershavecomplexitythatisexponentialinn.Quantumalgorithms,inprinciple,haveonlycomplexitythatispolynomialinn.
• PeterShordescubre unalgoritmo cuántico parafactorizar enterosconaplicaciones importantes acybersecurity,resultando en unareducción exponencial delacomplejidad clásica (1994).Sibiencomputadoras cuánticas degrantamaño noexisten,este fue elprincipalresultado quedisparó investigación en algoritmoscuánticos
• L.Groverdiscoversaquantumalgorithmforunstructuredsearchresultinginapolynomial(quadratic)quantumspeedup(1997).AmainideainGrover’sresult(amplitudeamplification)hasbeenextensivelyusedinotherquantumalgorithmsforproblemssuchasoptimization,search,andmore.
Otherquantumalgorithmsforlinearalgebraproblems?
Let’sconsidertheproblemofsolvingasystemoflinearequationsortherelatedproblemofinvertingamatrix:
LinearEquations:Animportantproblem
A.~x = ~
b ~x = A
�1~
b
Let’sconsidertheproblemofsolvingasystemoflinearequationsortherelatedproblemofinvertingamatrix:
LinearEquations:Animportantproblem
A.~x = ~
b ~x = A
�1~
b
NxN N-dimensional
Let’sconsidertheproblemofsolvingasystemoflinearequationsortherelatedproblemofinvertingamatrix:
LinearEquations:Animportantproblem
A.~x = ~
b ~x = A
�1~
b
• Thereisavarietyofclassicalalgorithmstosolvethisproblem.Nevertheless,evenwhenthematrixA andvectorƃ aresparse,thecomplexityof``exact’’classicalalgorithmsisatleastlinearinN.
Let’sconsidertheproblemofsolvingasystemoflinearequationsortherelatedproblemofinvertingamatrix:
LinearEquations:Animportantproblem
A.~x = ~
b ~x = A
�1~
b
• Thereisavarietyofclassicalalgorithmstosolvethisproblem.Nevertheless,evenwhenthematrixA andvectorƃ aresparse,thecomplexityof``exact’’classicalalgorithmsisatleastlinearinN.
• Aresult[HHL08]:Quantumcomputerscanprepareaquantumstateproportionaltothesolutionofthesystemintimethatispolynomialintheconditionnumber,inverseofprecision,andthelogarithmofthedimension(undersomeassumptions).
Let’sconsidertheproblemofsolvingasystemoflinearequationsortherelatedproblemofinvertingamatrix:
LinearEquations:Animportantproblem
A.~x = ~
b ~x = A
�1~
b
• Thereisavarietyofclassicalalgorithmstosolvethisproblem.Nevertheless,evenwhenthematrixA andvectorƃ aresparse,thecomplexityof``exact’’classicalalgorithmsisatleastlinearinN.
• Aresult[HHL08]:Quantumcomputerscanprepareaquantumstateproportionaltothesolutionofthesystemintimethatispolynomialintheconditionnumber,inverseofprecision,andthelogarithmofthedimension(undersomeassumptions).
• Note:Thisisasomewhatdifferentproblem(QLSP)andclassicalalgorithmsmaydobetterinthiscase.However,theQLSPisBQP-Complete.
QuantumLinearSystemProblem(QLSP)
A.~x = ~
b
Assumptions
• A isHermitianofdimensionNxN• A iss-sparse• A isinvertibleanditsconditionnumberis𝜅<∞• ThespectralnormofA isboundedby1
QuantumLinearSystemProblem(QLSP)
A.~x = ~
b
Assumptions
• A isHermitianofdimensionNxN• A iss-sparse• A isinvertibleanditsconditionnumberis𝜅<∞• ThespectralnormofA isboundedby1
QuantumLinearSystemProblem(QLSP)
A.~x = ~
b
Assumptions
• A isHermitianofdimensionNxN• A iss-sparse• A isinvertibleanditsconditionnumberis𝜅<∞• ThespectralnormofA isboundedby1
QuantumLinearSystemProblem(QLSP)
A.~x = ~
b
Assumptions
• A isHermitianofdimensionNxN• A iss-sparse• A isinvertibleanditsconditionnumberis𝜅<∞• ThespectralnormofA isboundedby1
QuantumLinearSystemProblem(QLSP)
A.~x = ~
b
Assumptions
• A isHermitianofdimensionNxN• A iss-sparse• A isinvertibleanditsconditionnumberis𝜅<∞• ThespectralnormofA isboundedby1
Let CA(t, ✏) be the cost of simulating e�iAtwith precision ✏
QuantumLinearSystemProblem(QLSP)
A.~x = ~
b
Assumptions
• A isHermitianofdimensionNxN• A iss-sparse• A isinvertibleanditsconditionnumberis𝜅<∞• ThespectralnormofA isboundedby1
Let CA(t, ✏) be the cost of simulating e�iAtwith precision ✏
Hamiltoniansimulation
Note:RecentadvancesinHamiltoniansimulationresultedin
CA(t, ✏) = ˜O(tsTA log(t/✏))
• Complexityalmostlinearintheevolutiontime• Complexityispolylogarithmic intheinverseofaprecisionparameter
D.Berry,A.Childs,R.Cleve,R.Kothari,andRDS,PRL114,090502(2015)D.Berry,A.Childs,andR.Kothari,FOCS2015,792(2015)G.H.LowandI.Chuang,PRL118,010501(2017)
QuantumLinearSystemProblem(QLSP)Someapplications:
• Inphysics,wherethegoalistocomputetheexpectationvalueoftheinverseofamatrix.Thisideawasusedin[1]forobtainingtheresistanceofanetwork.
• Instatmech,where,e.g.,estimatingthehittingtimeofaMarkovchainalsoreducestocomputingtheexpectationvalueoftheinverseofamatrix[2]
• InML,forsolvingproblemsrelatedtoleast-squaresestimation[3],byapplyingthepseudoinverse:
• Forsolvingcertainlineardifferentialequations[4]:
[1]G.Wang,arXiv:1311.1851(2013).[2]A.ChowdhuryandR.Somma,QIC17,0041(2017)[3]N.Wiebe,D.Braun,andS.Lloyd,PRL109,050505(2012).[4]D.Berry,A.Childs,A.Ostrander,andG.Wang,CMP356,1057(2017)
QuantumLinearSystemProblem(QLSP)
Anote: Evenforthoseapplications,anumberofassumptionsmustbemadeinordertoobtainquantumspeedups.Theseassumptionsincludeefficientpreparationofcertainstates(ofexp manyamplitudes),nicescalingoftheconditionnumber,andsolvingcertainproblemslikecomputingexpectationvalues.Forthesereasons,shownquantumspeedupsaretypicallypolynomial.
TheHHLAlgorithmfortheQLSP[5]
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
[HHL08]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
O [(Tb + CA(/✏, ✏/))]
TheHHLAlgorithmfortheQLSP[5]
[HHL08]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
ConsideringthatmanyHamiltonianscanbesimulatedefficientlyonquantumcomputers,thecomplexitydependenceonthedimensionissmall(e.g.,logarithmic)
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
O [(Tb + CA(/✏, ✏/))]
ImprovementsoftheHHLAlgorithm[6]
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)[6]A.Ambainis,STACS14,636(2012)
[HHL08]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
FurtherimprovementsbyAmbainis (VariableTimeAmplitudeAmplificationorVTAA):
[6]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
O⇥Tb + CA(/✏
3, ✏)⇤
ConsideringthatmanyHamiltonianscanbesimulatedefficientlyonquantumcomputers,thecomplexitydependenceonthedimensionissmall(e.g.,logarithmic)
O [(Tb + CA(/✏, ✏/))]
ImprovementsoftheHHLAlgorithm[6]
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)[6]A.Ambainis,STACS14,636(2012)
[HHL08]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
FurtherimprovementsbyAmbainis (VariableTimeAmplitudeAmplificationorVTAA):
[6]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
O⇥Tb + CA(/✏
3, ✏)⇤
• NotethatthebestHamiltoniansimulationmethodshavequeryandgatecomplexitiesalmostlinearinevolutiontimeandlogarithmicinprecision
Almostlinearin𝜅!
ConsideringthatmanyHamiltonianscanbesimulatedefficientlyonquantumcomputers,thecomplexitydependenceonthedimensionissmall(e.g.,logarithmic)
O [(Tb + CA(/✏, ✏/))]
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)[6]A.Ambainis,STACS14,636(2012)
|bi !N�1X
j=0
cj |vjiI |�jiE
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
|bi !N�1X
j=0
cj |vjiI |�jiE
Thisregistercontainstheeigenvalueestimate(superposition):
• Itsufficestohavetheestimatewithrelativeprecision𝜖• Orderlog(𝜅/𝜖)ancillaryqubits
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
|bi !N�1X
j=0
cj |vjiI |�jiE
Then we implement the conditional rotation: |˜�jiE ! |˜�jiE
1
˜�j
|0iO +
s1� 1
2˜�2j
|1iO
!
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
|bi !N�1X
j=0
cj |vjiI |�jiE
Then we implement the conditional rotation: |˜�jiE ! |˜�jiE
1
˜�j
|0iO +
s1� 1
2˜�2j
|1iO
!
Undophaseestimation
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
|bi !N�1X
j=0
cj |vjiI |�jiE
Then we implement the conditional rotation: |˜�jiE ! |˜�jiE
1
˜�j
|0iO +
s1� 1
2˜�2j
|1iO
!
Undophaseestimation
Amplitude amplification for amplifying the amplitude of the |0iO state
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
Roughly,thescalingoftheHHLalgorithmcanbeanalyzedfromtheworstcase:
|bi = (1/)|v1/i+p
1� 1/2|v1i
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
Roughly,thescalingoftheHHLalgorithmcanbeanalyzedfromtheworstcase:
|bi = (1/)|v1/i+p
1� 1/2|v1i
Theactionof1/ A willroughlycreatetheequalsuperpositionstate,sobothareequallyimportant
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
Roughly,thescalingoftheHHLalgorithmcanbeanalyzedfromtheworstcase:
|bi = (1/)|v1/i+p
1� 1/2|v1i
Theactionof1/ A willroughlycreatetheequalsuperpositionstate,sobothareequallyimportant
ForthedesiredprecisionweneedtoevolvewithA fortimeoforder𝜅/𝜖
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
Roughly,thescalingoftheHHLalgorithmcanbeanalyzedfromtheworstcase:
|bi = (1/)|v1/i+p
1� 1/2|v1i
Theactionof1/ A willroughlycreatetheequalsuperpositionstate,sobothareequallyimportant
ForthedesiredprecisionweneedtoevolvewithA fortimeoforder𝜅/𝜖
Theactionof1/ (𝜅 A )onthestatereducesitsamplitudebyorder1/ 𝜅 andorder𝜅amplitudeamplificationroundsareneeded
AquickviewoftheHHLalgorithmandVTAA
[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)
Roughly,thescalingoftheHHLalgorithmcanbeanalyzedfromtheworstcase:
|bi = (1/)|v1/i+p
1� 1/2|v1i
Theactionof1/ A willroughlycreatetheequalsuperpositionstate,sobothareequallyimportant
ForthedesiredprecisionweneedtoevolvewithA fortimeoforder𝜅/𝜖
Theactionof1/ (𝜅 A )onthestatereducesitsamplitudebyorder1/ 𝜅 andorder𝜅amplitudeamplificationroundsareneeded
FromhereweseethatweneedtoevolvewithA fortimethatis,atleast,order𝜅2/𝜖
AquickviewoftheHHLalgorithmandVTAA
Howcanweimprovethistimecomplexitytosomethingthatisalmostlinearintheconditionnumber?
[6]A.Ambainis,STACS14,636(2012)
AquickviewoftheHHLalgorithmandVTAA
Howcanweimprovethistimecomplexitytosomethingthatisalmostlinearintheconditionnumber?OneanswerisviaVariableTimeAmplitudeAmplification(VTAA)[6]
[6]A.Ambainis,STACS14,636(2012)
AquickviewoftheHHLalgorithmandVTAA
[6]A.Ambainis,STACS14,636(2012)
Howcanweimprovethistimecomplexitytosomethingthatisalmostlinearintheconditionnumber?OneanswerisviaVariableTimeAmplitudeAmplification(VTAA)[6]Theroughideaisasfollows(again,consideringtheworstcase):
• Firstwedoabad-precisionphaseestimationtodistinguishlargefromsmalleigenvalues.ThismaybedoneevolvingwithA fortimeindependentof𝜅
• Thenweimplementaroughapproximationof1/𝜅 A toeigenstatesoflargeeigenvalue
• Weneedorder𝜅 amplitudeamplificationsteps• Weimplementanaccurateapproximationof1/ 𝜅 A toeigenstatesofsmall
eigenvalue• Amplitudeamplificationfororder1steps• UndophaseestimationorapplytheFouriertransform
|bi = (1/)|v1/i+p
1� 1/2|v1i
AquickviewoftheHHLalgorithmandVTAA
[6]A.Ambainis,STACS14,636(2012)
Howcanweimprovethistimecomplexitytosomethingthatisalmostlinearintheconditionnumber?OneanswerisviaVariableTimeAmplitudeAmplification(VTAA)[6]Theroughideaisasfollows(again,consideringtheworstcase):
• Firstwedoabad-precisionphaseestimationtodistinguishlargefromsmalleigenvalues.ThismaybedoneevolvingwithA fortimeindependentof𝜅
• Thenweimplementaroughapproximationof1/𝜅 A toeigenstatesoflargeeigenvalue
• Weneedorder𝜅 amplitudeamplificationsteps• Weimplementanaccurateapproximationof1/ 𝜅 A toeigenstatesofsmall
eigenvalue• Amplitudeamplificationfororder1steps• UndophaseestimationorapplytheFouriertransform
ThecomplexityofVTAAintermsofprecisionisworsethanthatofHHL
|bi = (1/)|v1/i+p
1� 1/2|v1i
Thistalk:twoquantumalgorithmsfortheQSLP
• IwillpresenttwoquantumalgorithmsfortheQLSPthatimprovepreviousresultsindifferentways:
[7]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
[7]A.Childs,R.Kothari,RDS,SIAMJ.Comp.46,1920(2017).
˜O [(Tb + CA( log(/✏, ✏/))]
Thistalk:twoquantumalgorithmsfortheQSLP
• IwillpresenttwoquantumalgorithmsfortheQLSPthatimprovepreviousresultsindifferentways:
[7]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
• Thisresultsinanexponentialimprovementontheprecisionparameter
[7]A.Childs,R.Kothari,RDS,SIAMJ.Comp.46,1920(2017).
˜O [(Tb + CA( log(/✏, ✏/))]
Thistalk:twoquantumalgorithmsfortheQSLP
• IwillpresenttwoquantumalgorithmsfortheQLSPthatimprovepreviousresultsindifferentways:
[7]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
• Thisresultsinanexponentialimprovementontheprecisionparameter• ItcanbeimprovedusingaversionofVTAAto:
[7]A.Childs,R.Kothari,RDS,SIAMJ.Comp.46,1920(2017).
[7]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity
˜O [(Tb + CA( log(/✏, ✏/))]
˜O [Tb + CA( log(/✏, ✏))]
Thistalk:twoquantumalgorithmsfortheQSLP
• ThepreviousresultallowedustoproveapolynomialquantumspeedupforhittingtimeestimationintermsofthespectralgapofaMarkovchainandprecision(A.Chowdhury,R.D.Somma,QIC17,0041(2017)).
• Havingasmallcomplexitydependenceonprecisionisimportantfor,e.g.,computingexpectationvaluesofobservablesatthequantummetrologylimit.
Whytheseimprovementsareimportant?
Thistalk:twoquantumalgorithmsfortheQSLP
• IwillpresenttwoquantumalgorithmsfortheQLSPthatimprovepreviousresultsindifferentways:
[8]ThereexistsaquantumalgorithmthatsolvestheQLSPbyevolvingwithHamiltoniansthatarelinearcombinationsof(productsof)A,theprojectorintheinitialstate,andPaulimatrices.Theoverallevolutiontimeis
[8]Y.Subasi,RDS,D.Orsucci,arXiv:1805.10549(2018).
O(/✏)
Thistalk:twoquantumalgorithmsfortheQSLP
• IwillpresenttwoquantumalgorithmsfortheQLSPthatimprovepreviousresultsindifferentways:
[8]ThereexistsaquantumalgorithmthatsolvestheQLSPbyevolvingwithHamiltoniansthatarelinearcombinationsof(productsof)A,theprojectorintheinitialstate,andPaulimatrices.Theoverallevolutiontimeis
UsingHamiltoniansimulation,thistransferstocomplexity
[8]Y.Subasi,RDS,D.Orsucci,arXiv:1805.10549(2018).
O(/✏)
O(Tb/✏+ CA(/✏, ✏))
Thistalk:twoquantumalgorithmsfortheQSLP
• IwillpresenttwoquantumalgorithmsfortheQLSPthatimprovepreviousresultsindifferentways:
[8]ThereexistsaquantumalgorithmthatsolvestheQLSPbyevolvingwithHamiltoniansthatarelinearcombinationsof(productsof)A,theprojectorintheinitialstate,andPaulimatrices.Theoverallevolutiontimeis
UsingHamiltoniansimulation,thistransferstocomplexity
O(/✏)
O(Tb/✏+ CA(/✏, ✏))
• Themethodisverydifferentandbasedonadiabaticevolutions.Itdoesnotrequireofcomplicatedsubroutinessuchasphaseestimationandvariabletimeamplitudeamplification,thereforereducingthenumberofancillaryqubitssubstantially.
[8]Y.Subasi,RDS,D.Orsucci,arXiv:1805.10549(2018).
Thistalk:twoquantumalgorithmsfortheQSLP
• PhaseestimationandVTAArequireseveralancillaryqubits(beyondthoseneededforHamiltoniansimulation)
• Withintwoweeksofpostingourresult,agroupimplementedouralgorithminNMR,claimingthatitisthelargestsimulatedinstancesofar(8x8)[9]
Whythisimprovementisimportant?
[9]J.Wen,et.al.,arXiv:1806.0329(2018)
Firstalgorithm:AFourierapproachforsolvingtheQSLP
Firstalgorithm:AFourierapproachforsolvingtheQSLP
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• 1/A isnotunitaryandweneedtofindaunitaryimplementationforit.Wethengothroughasequenceofapproximations:
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• 1/A isnotunitaryandweneedtofindaunitaryimplementationforit.Wethengothroughasequenceofapproximations:
wearegettingcloser:Linearcombinationofunitaries
Firstalgorithm:AFourierapproachforsolvingtheQSLP
Firstalgorithm:AFourierapproachforsolvingtheQSLP
�1.0 �0.5 0.5 1.0
�20
�10
10
20
Firstalgorithm:AFourierapproachforsolvingtheQSLP
�1.0 �0.5 0.5 1.0
�20
�10
10
20
Firstalgorithm:AFourierapproachforsolvingtheQSLP
�1.0 �0.5 0.5 1.0
�20
�10
10
20
Themaximum“evolutiontime”underA intheapproximationof1/A is
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• Sofarweapproximated1/A,withinthedesiredaccuracy,byafinitelinearcombinationofunitaries.EachunitarycorrespondstoevolvingwithA forcertaintime,andthemaxevolutiontimeisalmostlinearintheconditionnumber
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• Sofarweapproximated1/A,withinthedesiredaccuracy,byafinitelinearcombinationofunitaries.EachunitarycorrespondstoevolvingwithA forcertaintime,andthemaxevolutiontimeisalmostlinearintheconditionnumber
QLSP Hamiltoniansimulation
Implementingalinearcombinationofunitaries
Suppose we want to implement the operator �1U1 + �2U2 to some state | iwhere �i � 0, �1 + �2 = 1, and Ui unitary
Implementingalinearcombinationofunitaries
Suppose we want to implement the operator �1U1 + �2U2 to some state | iwhere �i � 0, �1 + �2 = 1, and Ui unitary
p�1|0i+
p�2|1i
| i Ui
V V †|0i
(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i
Implementingalinearcombinationofunitaries
Suppose we want to implement the operator �1U1 + �2U2 to some state | iwhere �i � 0, �1 + �2 = 1, and Ui unitary
p�1|0i+
p�2|1i
| i Ui
V V †|0i
(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i
Correctpart Incorrectpart(orthogonal)
Implementingalinearcombinationofunitaries
Suppose we want to implement the operator �1U1 + �2U2 to some state | iwhere �i � 0, �1 + �2 = 1, and Ui unitary
p�1|0i+
p�2|1i
| i Ui
V V †|0i
(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i
Correctpart Incorrectpart(orthogonal)
This idea can be generalized to the case where the goal is to implement
M�1X
i=0
�iUi
Implementingalinearcombinationofunitaries
Suppose we want to implement the operator �1U1 + �2U2 to some state | iwhere �i � 0, �1 + �2 = 1, and Ui unitary
p�1|0i+
p�2|1i
| i Ui
V V †|0i
(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i
Correctpart Incorrectpart(orthogonal)
This idea can be generalized to the case where the goal is to implement
M�1X
i=0
�iUi
1
�(M�1X
i=0
�iUi)| i|0 . . . 0i+ |⇠?i
Implementingalinearcombinationofunitaries
Suppose we want to implement the operator �1U1 + �2U2 to some state | iwhere �i � 0, �1 + �2 = 1, and Ui unitary
p�1|0i+
p�2|1i
| i Ui
V V †|0i
(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i
Correctpart Incorrectpart(orthogonal)
This idea can be generalized to the case where the goal is to implement
M�1X
i=0
�iUi
1
�(M�1X
i=0
�iUi)| i|0 . . . 0i+ |⇠?i Amplitudeamplificationtoobtainthecorrectpart
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• WeusetheLCUapproachtoimplementtheFourierapproximationof1/A
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• WeusetheLCUapproachtoimplementtheFourierapproximationof1/A
• Note:WeassumethatthegatecomplexityoftheoperationVissmallwithrespecttoothercomplexities
• Addingupallthecoefficientsinthelinearcombinationofunitaries,weobtain
� = O()
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• WeusetheLCUapproachtoimplementtheFourierapproximationof1/A
• Note:WeassumethatthegatecomplexityoftheoperationVissmallwithrespecttoothercomplexities
• Addingupallthecoefficientsinthelinearcombinationofunitaries,weobtain
• Thisisalsothenumberofamplitudeamplificationrounds
� = O()
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• WeusetheLCUapproachtoimplementtheFourierapproximationof1/A
• Note:WeassumethatthegatecomplexityoftheoperationVissmallwithrespecttoothercomplexities
• Addingupallthecoefficientsinthelinearcombinationofunitaries,weobtain
• Thisisalsothenumberofamplitudeamplificationrounds
• Then,theoverallcomplexityofthisapproachis
� = O()
˜O [(Tb + CA( log(/✏, ✏/))]
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• WeusetheLCUapproachtoimplementtheFourierapproximationof1/A
• Note:WeassumethatthegatecomplexityoftheoperationVissmallwithrespecttoothercomplexities
• Addingupallthecoefficientsinthelinearcombinationofunitaries,weobtain
• Thisisalsothenumberofamplitudeamplificationrounds
• Then,theoverallcomplexityofthisapproachis
� = O()
Thisisalmostquadraticintheconditionnumber.ToimproveittoalmostlinearweuseaversionofVTAAthatdoesn’truinthelogarithmicscalinginprecision
˜O [(Tb + CA( log(/✏, ✏/))]
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• VTAAforHHLreliesheavilyonphaseestimation,bringingaprohibitivecomplexitydependenceonprecision
• Butinourcaseweonlyneedtodistinguishtheregionsfortheeigenvalueswithhighconfidence,sothescalinginprecisionislogarithmic
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• VTAAforHHLreliesheavilyonphaseestimation,bringingaprohibitivecomplexitydependenceonprecision
• Butinourcaseweonlyneedtodistinguishtheregionsfortheeigenvalueswithhighconfidence,sothescalinginprecisionislogarithmic
• ThefinalalgorithmisVTAAappliedtoanotheralgorithmthatisbuiltuponasequenceofsteps.
• Ateachstepwedothefollowing:i)Wedeterminetheregionoftheeigenvaluewithhighconfidence.ii)Weapply1/Awithinthenecessaryprecisionforthatregion(replacingtheconditionnumber)
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• VTAAforHHLreliesheavilyonphaseestimation,bringingaprohibitivecomplexitydependenceonprecision
• Butinourcaseweonlyneedtodistinguishtheregionsfortheeigenvalueswithhighconfidence,sothescalinginprecisionislogarithmic
• ThefinalalgorithmisVTAAappliedtoanotheralgorithmthatisbuiltuponasequenceofsteps.
• Ateachstepwedothefollowing:i)Wedeterminetheregionoftheeigenvaluewithhighconfidence.ii)Weapply1/Awithinthenecessaryprecisionforthatregion(replacingtheconditionnumber)
• Theoverallcomplexityofthisapproachis
˜O [Tb + CA( log(/✏, ✏))]
Firstalgorithm:AFourierapproachforsolvingtheQSLP
• VTAAforHHLreliesheavilyonphaseestimation,bringingaprohibitivecomplexitydependenceonprecision
• Butinourcaseweonlyneedtodistinguishtheregionsfortheeigenvalueswithhighconfidence,sothescalinginprecisionislogarithmic
• ThefinalalgorithmisVTAAappliedtoanotheralgorithmthatisbuiltuponasequenceofsteps.
• Ateachstepwedothefollowing:i)Wedeterminetheregionoftheeigenvaluewithhighconfidence.ii)Weapply1/Awithinthenecessaryprecisionforthatregion(replacingtheconditionnumber)
• Theoverallcomplexityofthisapproachis
UsingthebestHamiltoniansimulationmethods,thisisalmostlinearintheconditionnumberandpolylog ininverseofprecision
˜O [Tb + CA( log(/✏, ✏))]
Secondalgorithm:An“adiabatic”approachfortheQSLP
Secondalgorithm:An“adiabatic”approachfortheQSLP
• TheideahereistopreparetheeigenstateofaHamiltonianbypreparingasequencecontinuouslyrelatedeigenstatesofafamilyofHamiltonians
Secondalgorithm:An“adiabatic”approachfortheQSLP
• TheideahereistopreparetheeigenstateofaHamiltonianbypreparingasequencecontinuouslyrelatedeigenstatesofafamilyofHamiltonians
• Wewanttheeigenstatetobethedesiredquantumstate(aftertracingoutancillarysystems)
Secondalgorithm:An“adiabatic”approachfortheQSLP
• TheideahereistopreparetheeigenstateofaHamiltonianbypreparingasequencecontinuouslyrelatedeigenstatesofafamilyofHamiltonians
• Wewanttheeigenstatetobethedesiredquantumstate(aftertracingoutancillarysystems)
P
?b A.~x = P
?b~
b = 0
B
†B|xi = 0 , B = P
?b .A
Secondalgorithm:An“adiabatic”approachfortheQSLP
• TheideahereistopreparetheeigenstateofaHamiltonianbypreparingasequencecontinuouslyrelatedeigenstatesofafamilyofHamiltonians
• Wewanttheeigenstatetobethedesiredquantumstate(aftertracingoutancillarysystems)
Thefollowingpropertiescanbeproven:
• ThedesiredstateistheuniquegroundstateofH• Theeigenvaluegapisorder1/ 𝜅2
P
?b A.~x = P
?b~
b = 0
B
†B|xi = 0 , B = P
?b .A
H
Secondalgorithm:An“adiabatic”approachfortheQSLP
• TheideahereistopreparetheeigenstateofaHamiltonianbypreparingasequencecontinuouslyrelatedeigenstatesofafamilyofHamiltonians
• Wewanttheeigenstatetobethedesiredquantumstate(aftertracingoutancillarysystems)
Thefollowingpropertiescanbeproven:
• ThedesiredstateistheuniquegroundstateofH• Theeigenvaluegapisorder1/ 𝜅2
• WenowseekthefamilyofinterpolatingHamiltonians
P
?b A.~x = P
?b~
b = 0
B
†B|xi = 0 , B = P
?b .A
H
Secondalgorithm:An“adiabatic”approachfortheQSLP
• WeassumeforthemomentthatA>1/ 𝜅
Secondalgorithm:An“adiabatic”approachfortheQSLP
• WeassumeforthemomentthatA>1/ 𝜅
• Wedefinetheinterpolatingmatrix A(s) = (1� s)I + sA , 0 s 1
Secondalgorithm:An“adiabatic”approachfortheQSLP
• WeassumeforthemomentthatA>1/ 𝜅
• Wedefinetheinterpolatingmatrix
• Similarly,wedefine
A(s) = (1� s)I + sA , 0 s 1
H(s) = B†(s)B(s) , B(s) = P?b A(s)
Secondalgorithm:An“adiabatic”approachfortheQSLP
• WeassumeforthemomentthatA>1/ 𝜅
• Wedefinetheinterpolatingmatrix
• Similarly,wedefine
• Thisislikesolvinganincreasinglydifficultsystemoflinearequations!
A(s) = (1� s)I + sA , 0 s 1
H(s) = B†(s)B(s) , B(s) = P?b A(s)
Secondalgorithm:An“adiabatic”approachfortheQSLP
• WeassumeforthemomentthatA>1/ 𝜅
• Wedefinetheinterpolatingmatrix
• Similarly,wedefine
• Thisislikesolvinganincreasinglydifficultsystemoflinearequations!
A(s) = (1� s)I + sA , 0 s 1
H(s) = B†(s)B(s) , B(s) = P?b A(s)
s = 0
s = 1
|bi
|xi
H(0) = P?b
H(1) = A.P?b .A
Secondalgorithm:An“adiabatic”approachfortheQSLP
• WeassumeforthemomentthatA>1/ 𝜅
• Wedefinetheinterpolatingmatrix
• Similarly,wedefine
• Thisislikesolvinganincreasinglydifficultsystemoflinearequations!
• Theminimumeigenvaluegapisorder1/𝜅2 andthelengthofthepathL islog(𝜅)
A(s) = (1� s)I + sA , 0 s 1
H(s) = B†(s)B(s) , B(s) = P?b A(s)
s = 0
s = 1
|bi
|xi
H(0) = P?b
H(1) = A.P?b .A
L
Therandomizationmethodtoprepareeigenstates[]
s = 0|bi
|xi
H(0) = P?b
H(1) = A.P?b .A
• Byperformingasequenceofprojectivemeasurementsatsufficientlyclosepoints,wecanpreparetherelatedeigenstateswithhighprobability
s1
s2
sq
Therandomizationmethodtoprepareeigenstates[]
s = 0H(0) = P?
b
H(1) = A.P?b .A
• Byperformingasequenceofprojectivemeasurementsatsufficientlyclosepoints,wecanpreparetherelatedeigenstateswithhighprobability
• EachmeasurementcanbesimulatedbyevolvingwiththecorrespondingHamiltonianforrandomtime.Thisreducescoherencesbetweeneigenstates
s1
s2
sq
|bi
|xi
Therandomizationmethodtoprepareeigenstates[]
s = 0H(0) = P?
b
H(1) = A.P?b .A
• Byperformingasequenceofprojectivemeasurementsatsufficientlyclosepoints,wecanpreparetherelatedeigenstateswithhighprobability
• EachmeasurementcanbesimulatedbyevolvingwiththecorrespondingHamiltonianforrandomtime.Thisreducescoherencesbetweeneigenstates
• TheexpectedevolutiontimewiththeHamiltoniansintherandomizationmethodsatisfies
s1
s2
sq
|bi
|xi
TRM = O
✓L2
✏�
◆
Therandomizationmethodtoprepareeigenstates[]
s = 0H(0) = P?
b
H(1) = A.P?b .A
• Byperformingasequenceofprojectivemeasurementsatsufficientlyclosepoints,wecanpreparetherelatedeigenstateswithhighprobability
• EachmeasurementcanbesimulatedbyevolvingwiththecorrespondingHamiltonianforrandomtime.Thisreducescoherencesbetweeneigenstates
• TheexpectedevolutiontimewiththeHamiltoniansintherandomizationmethodsatisfies
s1
s2
sq
|bi
|xi
L isthepathlength𝛥 isthemingap𝜖 istheerror(tracenorm)
TRM = O
✓L2
✏�
◆
TherandomizationmethodfortheQLSP[]
s = 0H(0) = P?
b
H(1) = A.P?b .A
• Byperformingasequenceofprojectivemeasurementsatsufficientlyclosepoints,wecanpreparetherelatedeigenstateswithhighprobability
• EachmeasurementcanbesimulatedbyevolvingwiththecorrespondingHamiltonianforrandomtime.Thisreducescoherencesbetweeneigenstates
• TheexpectedevolutiontimewiththeHamiltoniansintherandomizationmethodsatisfies
s1
s2
sq
|bi
|xi
TRM = O
✓2
log
2()
✏
◆
TherandomizationmethodfortheQLSP[]
• ThestrongdependenceoftheevolutiontimewiththespectralgapsuggestsonetoconsiderotherHamiltoniansthathavethesameeigenstatebutalargereigenvaluegap
TherandomizationmethodfortheQLSP[]
• ThestrongdependenceoftheevolutiontimewiththespectralgapsuggestsonetoconsiderotherHamiltoniansthathavethesameeigenstatebutalargereigenvaluegap
• Forthisproblem,spectralgapamplification[10]isuseful:
H(s) ! H 0(s) = B†(s)⌦ �� +B(s)⌦ �+ =
✓0 B(s)
B†(s) 0
◆
[10]R.D.SommaandS.Boixo,SIAMJ.Comp.593(2013)
TherandomizationmethodfortheQLSP[]
• ThestrongdependenceoftheevolutiontimewiththespectralgapsuggestsonetoconsiderotherHamiltoniansthathavethesameeigenstatebutalargereigenvaluegap
• Forthisproblem,spectralgapamplification[10]isuseful:
• Someresults:
H(s) ! H 0(s) = B†(s)⌦ �� +B(s)⌦ �+ =
✓0 B(s)
B†(s) 0
◆
Let |x(s)i be the eigenstate of 0-eigenvalue of H(s).
Then, |x(s)i|1i is an eigenstate of 0-eigenvalue of H 0(s).
This eigenstate is separated from others by an eigenvalue gap
p�(s)
[10]R.D.SommaandS.Boixo,SIAMJ.Comp.593(2013)
TherandomizationmethodfortheQLSP[]
• ThestrongdependenceoftheevolutiontimewiththespectralgapsuggestsonetoconsiderotherHamiltoniansthathavethesameeigenstatebutalargereigenvaluegap
• Forthisproblem,spectralgapamplificationisuseful:
• Someresults:
• Notethatthepathlengthdidnotchange.TheonlychangefortheRMistheuseofadifferentHamiltonian.
H(s) ! H 0(s) = B†(s)⌦ �� +B(s)⌦ �+ =
✓0 B(s)
B†(s) 0
◆
Let |x(s)i be the eigenstate of 0-eigenvalue of H(s).
Then, |x(s)i|1i is an eigenstate of 0-eigenvalue of H 0(s).
This eigenstate is separated from others by an eigenvalue gap
p�(s)
TherandomizationmethodfortheQLSP[]
• UsingtherandomizationmethodwiththenewHamiltonian,theexpectedevolutiontimeis
TRM = O
✓ log2()
✏
◆
TherandomizationmethodfortheQLSP[]
• UsingtherandomizationmethodwiththenewHamiltonian,theexpectedevolutiontimeis
• Thecaseofnon-positivematrixA canbeanalyzedsimilarlyusing
TRM = O
✓ log2()
✏
◆
A(s) = (1� s)(�anc
z
⌦ I) + s(�anc
x
⌦A)
Therandomizationmethod,theQLSP,andthegatemodel
For A > 0, the Hamiltonian is H 0(s) = (I � |bihb|)((1� s)I + sA)�� +H.c.
Therandomizationmethod,theQLSP,andthegatemodel
For A > 0, the Hamiltonian is H 0(s) = (I � |bihb|)((1� s)I + sA)�� +H.c.
If |bi is sparse and A is sparse, then H 0(s) is also sparse
Therandomizationmethod,theQLSP,andthegatemodel
For A > 0, the Hamiltonian is H 0(s) = (I � |bihb|)((1� s)I + sA)�� +H.c.
If |bi is sparse and A is sparse, then H 0(s) is also sparse
WecanuseaHamiltoniansimulationmethodtobuildaquantumcircuitthatsimulatestheevolution.Thequantumcircuitwillusequeries.
• Thecomplexityintermsofqueriesforis
• ThecomplexityintermsofqueriesforA isalmostorder
|bihb| O(Tb/✏)
CA(/✏, ✏)
Therandomizationmethod,theQLSP,andthegatemodel
For A > 0, the Hamiltonian is H 0(s) = (I � |bihb|)((1� s)I + sA)�� +H.c.
If |bi is sparse and A is sparse, then H 0(s) is also sparse
• Thescalingintheprecisionparametercanbedonepolyligarithmic byusingfastermethodsforeigenpath traversal[11]
WecanuseaHamiltoniansimulationmethodtobuildaquantumcircuitthatsimulatestheevolution.Thequantumcircuitwillusequeries.
• Thecomplexityintermsofqueriesforis
• ThecomplexityintermsofqueriesforA isalmostorder
|bihb| O(Tb/✏)
CA(/✏, ✏)
[11]S.Boixo,E.Knill,andR.D.Somma,arXiv:1005.3034(2010)
Someconclusionsandobservations
• Quantumcomputingseemspromising.Severalquantumalgorithmsforproblemsinlinearalgebrawithsignificantspeedupsexist
• Ipresentedquantumalgorithmstosolvethequantumlinearsystemsproblem.Thetechniquescanbegeneralizedtoapplyotheroperators(otherthantheinverseofamatrix)toquantumstates.
• Theadvantagesofthefirstalgorithmareinthatthecomplexitydependenceonprecisionisonlypolylogarithmic,exponentiallyimprovingpreviousalgorithmsforthisproblem
• Theadvantagesofthesecondalgorithmareinthatitdoesn’trequiremanyancillaryqubitsandtheproblemreducestoasimpleHamiltoniansimulationproblem
• ItwouldbeimportanttounderstandtheapplicabilityofthisalgorithmtoscientificproblemsbeyondtheonesImentioned.Howimportantaretheproblemsandalgorithms?