Download - Quantum Algorithms for Systems of Linear Equationsqml2018.umiacs.io/slides/Somma.pdf · Let’s consider the problem of solving a system of linear equations or the related problem

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.




• PeterShordiscoversaquantumalgorithmforefficientfactorizationofintegerswithimportantapplicationstocryptography.Shor’salgorithmresultsinasuperpolynomial quantumspeedup(1994).Hisresultwasamainmotivationforthediscoveryofotherquantumalgorithms.




• PeterShordiscoversaquantumalgorithmforefficientfactorizationofintegerswithimportantapplicationstocryptography.Shor’salgorithmresultsinasuperpolynomial quantumspeedup(1994).Hisresultwasamainmotivationforthediscoveryofotherquantumalgorithms.

• L.Groverdiscoversaquantumalgorithmforunstructuredsearchresultinginapolynomial(quadratic)quantumspeedup(1997).AmainideainGrover’sresult(amplitudeamplification)hasbeenextensivelyusedinotherquantumalgorithmsforproblemssuchasoptimization,search,andmore.




• 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



A.~x = ~

b ~x = A

�1~

b

NxN N-dimensional



A.~x = ~

b ~x = A

�1~

b

• Thereisavarietyofclassicalalgorithmstosolvethisproblem.Nevertheless,evenwhenthematrixA andvectorƃ aresparse,thecomplexityof``exact’’classicalalgorithmsisatleastlinearinN.



A.~x = ~

b ~x = A

�1~

b


• Aresult[HHL08]:Quantumcomputerscanprepareaquantumstateproportionaltothesolutionofthesystemintimethatispolynomialintheconditionnumber,inverseofprecision,andthelogarithmofthedimension(undersomeassumptions).



A.~x = ~

b ~x = A

�1~

b


• 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


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)


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]


ConsideringthatmanyHamiltonianscanbesimulatedefficientlyonquantumcomputers,thecomplexitydependenceonthedimensionissmall(e.g.,logarithmic)


O [(Tb + CA(/✏, ✏/))]

ImprovementsoftheHHLAlgorithm[6]

[5]Harrow,Hassidim,Lloyd,PRL103,150502(2009)[6]A.Ambainis,STACS14,636(2012)


FurtherimprovementsbyAmbainis (VariableTimeAmplitudeAmplificationorVTAA):

[6]ThereexistsaquantumalgorithmthatsolvestheQLSPwithcomplexity

O⇥Tb + CA(/✏

3, ✏)⇤


O [(Tb + CA(/✏, ✏/))]

ImprovementsoftheHHLAlgorithm[6]



FurtherimprovementsbyAmbainis (VariableTimeAmplitudeAmplificationorVTAA):


O⇥Tb + CA(/✏

3, ✏)⇤

• NotethatthebestHamiltoniansimulationmethodshavequeryandgatecomplexitiesalmostlinearinevolutiontimeandlogarithmicinprecision

Almostlinearin𝜅!


O [(Tb + CA(/✏, ✏/))]

AquickviewoftheHHLalgorithmandVTAA




|bi !N�1X

j=0

cj |vjiI |�jiE



|bi !N�1X

j=0

cj |vjiI |�jiE

Thisregistercontainstheeigenvalueestimate(superposition):

• Itsufficestohavetheestimatewithrelativeprecision𝜖• Orderlog(𝜅/𝜖)ancillaryqubits



Roughly,thescalingoftheHHLalgorithmcanbeanalyzedfromtheworstcase:

|bi = (1/)|v1/i+p

1� 1/2|v1i




|bi = (1/)|v1/i+p

1� 1/2|v1i

Theactionof1/ A willroughlycreatetheequalsuperpositionstate,sobothareequallyimportant




|bi = (1/)|v1/i+p

1� 1/2|v1i


ForthedesiredprecisionweneedtoevolvewithA fortimeoforder𝜅/𝜖




|bi = (1/)|v1/i+p

1� 1/2|v1i



Theactionof1/ (𝜅 A )onthestatereducesitsamplitudebyorder1/ 𝜅 andorder𝜅amplitudeamplificationroundsareneeded




|bi = (1/)|v1/i+p

1� 1/2|v1i



Theactionof1/ (𝜅 A )onthestatereducesitsamplitudebyorder1/ 𝜅 andorder𝜅amplitudeamplificationroundsareneeded

FromhereweseethatweneedtoevolvewithA fortimethatis,atleast,order𝜅2/𝜖


Howcanweimprovethistimecomplexitytosomethingthatisalmostlinearintheconditionnumber?

[6]A.Ambainis,STACS14,636(2012)


Howcanweimprovethistimecomplexitytosomethingthatisalmostlinearintheconditionnumber?OneanswerisviaVariableTimeAmplitudeAmplification(VTAA)[6]




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



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]A.Childs,R.Kothari,RDS,SIAMJ.Comp.46,1920(2017).

˜O [(Tb + CA( log(/✏, ✏/))]




• Thisresultsinanexponentialimprovementontheprecisionparameter


˜O [(Tb + CA( log(/✏, ✏/))]




• Thisresultsinanexponentialimprovementontheprecisionparameter• ItcanbeimprovedusingaversionofVTAAto:



˜O [(Tb + CA( log(/✏, ✏/))]

˜O [Tb + CA( log(/✏, ✏))]


• ThepreviousresultallowedustoproveapolynomialquantumspeedupforhittingtimeestimationintermsofthespectralgapofaMarkovchainandprecision(A.Chowdhury,R.D.Somma,QIC17,0041(2017)).

• Havingasmallcomplexitydependenceonprecisionisimportantfor,e.g.,computingexpectationvaluesofobservablesatthequantummetrologylimit.

Whytheseimprovementsareimportant?



[8]ThereexistsaquantumalgorithmthatsolvestheQLSPbyevolvingwithHamiltoniansthatarelinearcombinationsof(productsof)A,theprojectorintheinitialstate,andPaulimatrices.Theoverallevolutiontimeis

[8]Y.Subasi,RDS,D.Orsucci,arXiv:1805.10549(2018).

O(/✏)




UsingHamiltoniansimulation,thistransferstocomplexity


O(/✏)

O(Tb/✏+ CA(/✏, ✏))




UsingHamiltoniansimulation,thistransferstocomplexity

O(/✏)

O(Tb/✏+ CA(/✏, ✏))

• Themethodisverydifferentandbasedonadiabaticevolutions.Itdoesnotrequireofcomplicatedsubroutinessuchasphaseestimationandvariabletimeamplitudeamplification,thereforereducingthenumberofancillaryqubitssubstantially.



• PhaseestimationandVTAArequireseveralancillaryqubits(beyondthoseneededforHamiltoniansimulation)

• Withintwoweeksofpostingourresult,agroupimplementedouralgorithminNMR,claimingthatitisthelargestsimulatedinstancesofar(8x8)[9]

Whythisimprovementisimportant?

[9]J.Wen,et.al.,arXiv:1806.0329(2018)

Firstalgorithm:AFourierapproachforsolvingtheQSLP


• 1/A isnotunitaryandweneedtofindaunitaryimplementationforit.Wethengothroughasequenceofapproximations:


• 1/A isnotunitaryandweneedtofindaunitaryimplementationforit.Wethengothroughasequenceofapproximations:

wearegettingcloser:Linearcombinationofunitaries


�1.0 �0.5 0.5 1.0

�20

�10

10

20


�1.0 �0.5 0.5 1.0

�20

�10

10

20

Themaximum“evolutiontime”underA intheapproximationof1/A is


• Sofarweapproximated1/A,withinthedesiredaccuracy,byafinitelinearcombinationofunitaries.EachunitarycorrespondstoevolvingwithA forcertaintime,andthemaxevolutiontimeisalmostlinearintheconditionnumber


• 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



p�1|0i+

p�2|1i

| i Ui

V V †|0i

(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i



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)



p�1|0i+

p�2|1i

| i Ui

V V †|0i

(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i


This idea can be generalized to the case where the goal is to implement

M�1X

i=0

�iUi



p�1|0i+

p�2|1i

| i Ui

V V †|0i

(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i



M�1X

i=0

�iUi

1

�(M�1X

i=0

�iUi)| i|0 . . . 0i+ |⇠?i



p�1|0i+

p�2|1i

| i Ui

V V †|0i

(�1U1 + �2U2)| i|0i+p�1�2(U1 � U2)| i|1i



M�1X

i=0

�iUi

1

�(M�1X

i=0

�iUi)| i|0 . . . 0i+ |⇠?i Amplitudeamplificationtoobtainthecorrectpart


• WeusetheLCUapproachtoimplementtheFourierapproximationof1/A



• Note:WeassumethatthegatecomplexityoftheoperationVissmallwithrespecttoothercomplexities

• Addingupallthecoefficientsinthelinearcombinationofunitaries,weobtain

� = O()





• Thisisalsothenumberofamplitudeamplificationrounds

� = O()






• Then,theoverallcomplexityofthisapproachis

� = O()

˜O [(Tb + CA( log(/✏, ✏/))]






• Then,theoverallcomplexityofthisapproachis

� = O()

Thisisalmostquadraticintheconditionnumber.ToimproveittoalmostlinearweuseaversionofVTAAthatdoesn’truinthelogarithmicscalinginprecision

˜O [(Tb + CA( log(/✏, ✏/))]


• VTAAforHHLreliesheavilyonphaseestimation,bringingaprohibitivecomplexitydependenceonprecision

• Butinourcaseweonlyneedtodistinguishtheregionsfortheeigenvalueswithhighconfidence,sothescalinginprecisionislogarithmic




• ThefinalalgorithmisVTAAappliedtoanotheralgorithmthatisbuiltuponasequenceofsteps.

• Ateachstepwedothefollowing:i)Wedeterminetheregionoftheeigenvaluewithhighconfidence.ii)Weapply1/Awithinthenecessaryprecisionforthatregion(replacingtheconditionnumber)






• Theoverallcomplexityofthisapproachis

˜O [Tb + CA( log(/✏, ✏))]






• Theoverallcomplexityofthisapproachis

UsingthebestHamiltoniansimulationmethods,thisisalmostlinearintheconditionnumberandpolylog ininverseofprecision

˜O [Tb + CA( log(/✏, ✏))]

Secondalgorithm:An“adiabatic”approachfortheQSLP


• TheideahereistopreparetheeigenstateofaHamiltonianbypreparingasequencecontinuouslyrelatedeigenstatesofafamilyofHamiltonians



• Wewanttheeigenstatetobethedesiredquantumstate(aftertracingoutancillarysystems)




P

?b A.~x = P

?b~

b = 0

B

†B|xi = 0 , B = P

?b .A




Thefollowingpropertiescanbeproven:

• ThedesiredstateistheuniquegroundstateofH• Theeigenvaluegapisorder1/ 𝜅2

P

?b A.~x = P

?b~

b = 0

B

†B|xi = 0 , B = P

?b .A

H




Thefollowingpropertiescanbeproven:

• ThedesiredstateistheuniquegroundstateofH• Theeigenvaluegapisorder1/ 𝜅2

• WenowseekthefamilyofinterpolatingHamiltonians

P

?b A.~x = P

?b~

b = 0

B

†B|xi = 0 , B = P

?b .A

H


• WeassumeforthemomentthatA>1/ 𝜅



• Wedefinetheinterpolatingmatrix A(s) = (1� s)I + sA , 0 s 1



• Wedefinetheinterpolatingmatrix

• Similarly,wedefine

A(s) = (1� s)I + sA , 0 s 1

H(s) = B†(s)B(s) , B(s) = P?b A(s)





• Thisislikesolvinganincreasinglydifficultsystemoflinearequations!

A(s) = (1� s)I + sA , 0 s 1

H(s) = B†(s)B(s) , B(s) = P?b A(s)






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






• 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


s = 0H(0) = P?

b

H(1) = A.P?b .A


• EachmeasurementcanbesimulatedbyevolvingwiththecorrespondingHamiltonianforrandomtime.Thisreducescoherencesbetweeneigenstates

s1

s2

sq

|bi

|xi


s = 0H(0) = P?

b

H(1) = A.P?b .A



• TheexpectedevolutiontimewiththeHamiltoniansintherandomizationmethodsatisfies

s1

s2

sq

|bi

|xi

TRM = O

✓L2

✏�

◆


s = 0H(0) = P?

b

H(1) = A.P?b .A




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




s1

s2

sq

|bi

|xi

TRM = O

✓2

log

2()

✏

◆


• 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)



• 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)



• 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)


• UsingtherandomizationmethodwiththenewHamiltonian,theexpectedevolutiontimeis

TRM = O

✓ log2()

✏

◆


• 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.



If |bi is sparse and A is sparse, then H 0(s) is also sparse




WecanuseaHamiltoniansimulationmethodtobuildaquantumcircuitthatsimulatestheevolution.Thequantumcircuitwillusequeries.

• Thecomplexityintermsofqueriesforis

• ThecomplexityintermsofqueriesforA isalmostorder

|bihb| O(Tb/✏)

CA(/✏, ✏)




• 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?

Download - Quantum Algorithms for Systems of Linear Equationsqml2018.umiacs.io/slides/Somma.pdf · Let’s consider the problem of solving a system of linear equations or the related problem

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