Deploying a quantum annealing processor to detect tree cover in aerial imagery of California

22 Pages·2017·2.56 MB·English

RESEARCHARTICLE
Deploying a quantum annealing processor to
detect tree cover in aerial imagery of
California
EdwardBoyda1,2*,SaikatBasu3,SangramGanguly2,4,AndrewMichaelis4,5,
SupratikMukhopadhyay3,RamakrishnaR.Nemani6
1DepartmentofPhysicsandAstronomy,SaintMary’sCollegeofCalifornia,Moraga,CA,UnitedStatesof
America,2BayAreaEnvironmentalResearchInstitute,MoffettField,CA,UnitedStatesofAmerica,
3DepartmentofComputerScience,LouisianaStateUniversity,BatonRouge,LA,UnitedStatesofAmerica,
a1111111111
4EarthScienceDivision,NASAAmesResearchCenter,MoffettField,CA,UnitedStatesofAmerica,
a1111111111
5UniversityCorporationatCSUMontereyBay,Seaside,CA,UnitedStatesofAmerica,6NASAAdvanced
a1111111111 SupercomputingDivision,NASAAmesResearchCenter,MoffettField,CA,UnitedStatesofAmerica
a1111111111
a1111111111 *[email protected]
Abstract
OPENACCESS Quantumannealingisanexperimentalandpotentiallybreakthroughcomputationaltechnol-
Citation:BoydaE,BasuS,GangulyS,MichaelisA, ogyforhandlinghardoptimizationproblems,includingproblemsofcomputervision.We
MukhopadhyayS,NemaniRR(2017)Deployinga presentacasestudyintrainingaproduction-scaleclassifieroftreecoverinremotesensing
quantumannealingprocessortodetecttreecover
imagery,usingearly-generationquantumannealinghardwarebuiltbyD-waveSystems,
inaerialimageryofCalifornia.PLoSONE12(2):
Inc.Beginningwithinaknownboostingframework,wetraindecisionstumpsontexturefea-
e0172505.doi:10.1371/journal.pone.0172505
turesandvegetationindicesextractedfromfour-band,one-meter-resolutionaerialimagery
Editor:ShijoJoseph,KeralaForestResearch
fromthestateofCalifornia.Wethenimposearegulatedquadratictrainingobjectiveto
Institute,INDIA
selectanoptimalvotingsubsetfromamongthesestumps.Thevotesofthesubsetdefine
Received:June29,2016
theclassifier.Foroptimization,thelogicalvariablesintheobjectivefunctionmaptoquantum
Accepted:February6,2017
bitsinthehardwaredevice,whilequadraticcouplingsencodeasthestrengthofphysical
Published:February27,2017 interactionsbetweenthequantumbits.Hardwaredesignlimitsthenumberofcouplings
Copyright:Thisisanopenaccessarticle,freeofall betweenthesebasicphysicalentitiestofiveorsix.Toaccountforthislimitationinmapping
copyright,andmaybefreelyreproduced, largeproblemstothehardwarearchitecture,weproposeatruncationandrescalingofthe
distributed,transmitted,modified,builtupon,or
trainingobjectivethroughatrainablemetaparameter.Theboostingprocessonourbasic
otherwiseusedbyanyoneforanylawfulpurpose.
108-and508-variableproblems,thusconstituted,returnsclassifiersthatincorporatea
TheworkismadeavailableundertheCreative
CommonsCC0publicdomaindedication. diverserangeofcolor-andtexture-basedmetricsanddiscriminatetreecoverwithaccura-
ciesashighas92%invalidationand90%onatestsceneencompassingtheopenspace
DataAvailabilityStatement:Dataareavailable
fromFigshare(DOI:10.6084/m9.figshare. preservesanddensesuburbanbuildofMillValley,CA.
4644535).
Funding:ThisworkwassupportedbytheNASA
EarthScienceDivisionandperformedusingthe
computingfacilitiesoftheNASAAdvanced
Supercomputing(NAS)divisionandNASAEarth
Exchange(NEX).Anyopinions,findings,and Introduction
conclusionsorrecommendationsexpressedinthis
Theproliferationofveryhighresolution(VHR)aerialandsatelliteimageryopensthewayto
materialarethoseoftheauthorsanddonot
significantimprovementsinremotesensingdataproducts.Itisnowpossibletoidentify
necessarilyreflectthatofNASAortheUnited
StatesGovernment.Thefundershadnorolein structuresatbetterthan1-meterresolution,downfrom30metersinexistingLandsat-based
PLOSONE|DOI:10.1371/journal.pone.0172505 February27,2017 1/22
DeployingaquantumannealingprocessortodetecttreecoverinaerialimageryofCalifornia
studydesign,datacollectionandanalysis,decision solutions.Objects—individualsheds,tractors,streams,islands,rocks,trees,vines,andfur-
topublish,orpreparationofthemanuscript. rows—comeintofocusfromoutofbroadswathsofforestorfield,allowingfordetailedsite-
Competinginterests:Theauthorshavedeclared specificstudiesaswellasmoreaccuratedelineationsoflandcoverinthelarge.VHRdatasets
thatnocompetinginterestsexist. arerichinpotentialities.Atthesametime,newlysophisticatedcomputeralgorithmsare
requiredtoparsethedata.
Duetohighvariabilitywithinclassesandinatmospheric,lighting,andphoto-geometric
conditions,land-coverclasscognitionatveryhighresolutionremainsadifficultchallenge.In
thisrealm,object-orientedtechniquesforintegratedsegmentationandclassificationhave
showngreatrecentpromise.Theyofferarichersemanticsandmoreaccurateclassification
whencomparedtoclusteringofspectralandtexturalprimitivesalone.(See,forexample,[1]in
thecontextofcomputervisionor[2]forareviewinthecontextofremotesensing.)Object-
orientedappraochesputsignificantdemandsoncomputationalinfrastructure.Themachine
learningalgorithmsleadtomemory-andprocessor-intensivetraining(optimization)prob-
lemsinwhichthousandsofparametersmustbedetermined,whiletherelevantVHRdatasets
themselvesextendtoterabytesinsize.Giventhesepressures,itisnaturaltoaskwhatsortsof
breakthroughs,algorithmicortechnological,maylieonthehorizon.
Quantumcomputing(see,e.g.,[3])isonesuchpossibility.Broadlydefined,quantumcom-
putingisanefforttoencodehardcomputationalproblemsinthedynamicsofquantumphysi-
calsystems.Thestatespaceofquantumsystemsisexponentiallylargeinthenumberofbasic
physicalvariables,andiftappedproperly,canyieldcomputationalresultsexponentiallyfaster
thanthebestavailableclassicalalternatives.Thisadvantagehasbeendemonstratedformally
forparticularproblems,integerfactorization[4]beingtheexamplemostoftencitedduetoits
roleinthewidely-usedRSApublic-keycryptographyscheme.Thecommunityisactively
workingtocharacterizethescalingadvantageswecanexpectforbroaderclassesofproblems.
Withinthequantumcomputingparadigm,quantumannealing[5–7]isacomputational
metaheuristicdesignedtosolveoptimizationproblems.Akintosimulatedannealing,quantum
annealingseekstheminimumofacostfunctioninacomplexconfigurationspace.Physically,
thecostfunctionencodesasthesystem’senergy.Thealgorithmproceedsbypreparingthesys-
teminaquantumsuperpostionofallpossibleconfigurationsinthesolutionspace,allequally
probable,thusinitiatingauniquelyquantumparallelprocessing.Thesystemthenisevolvedin
timeuntilthesoughtminimalenergyconfigurationisoverwhelminglyprobable.Inprinciple,
intheabsenceofthermalnoise,itcanbearrangedsothattheminimalenergyconfiguration
willbemeasuredonread-outwithprobabilityarbitrarilyclosetoone.Ratherthansampling,
physicalinteractionsbetweenquantumbitsdrivethesystemtotheenergyminimum.Aspart
ofthisprocess,thesystemhasthepossibilityofquantumtunnelingthroughtall,narrowbarri-
ersintheenergylandscapetoescapelocalminimainlessthanexponentialtime.
AquantumannealingprocessorbuiltbyD-waveSystems,Inc.,with1152quantumbits
(qubits)isnowoperatingatNASA’sAmesResearchCenter.ThedeploymentoftheD-wave
2Xfollowsearliertrialsof128-qubitand512-qubitprocessorsatLockheedMartinandat
Ames.Muchworkhasgonetocharacterizetheperformanceofthesemachines.Evidenceof
thepersistenceofquantumcoherenceduringcomputationhasbeenobservedinsubsystemsof
eightqubits[8–10].Ontheotherhand,theprocessorhashandilybeenbeatenforspeedby
desktopCPUsrunningoptimizedsimulatedannealingand/ormoretargetedsamplingalgo-
rithms[11–13].Inlate2015,afirstsetofproblemswerecraftedonwhichtheD-wavequantum
annealerrunssignificantlyfasterthanclassicalsimulatedannealing[14].
Motivatedtoadvanceourremotesensingcapabilitiesandtobetterunderstandthepossibil-
itiesofquantumannealingvisionalgorithms,wesetouttotrainaproduction-scaleclassifier
ofaerialimageryontheD-waveprocessor.Webeginwithanimplementationofaboosting
algorithmknownasQBoost,developedspecificallyfortheD-wavearchitecture.Itwas
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DeployingaquantumannealingprocessortodetecttreecoverinaerialimageryofCalifornia
employedin2009toidentifycarsinphotographsofstreetscenes,havingbeentrainedona
processorwith52functioningqubits[15–17].Unfortunately,QBoost,alongwithproblems
fromacommongeneralclassofquadratictrainingobjectives,doesnotscalewellontheD-
wavearchitectureoronanyforeseeablequantumannealingprocessor.Bytruncatingand
rescalingcouplingsintheQBoosttrainingobjective,withtheintroductionofanadditional
trainablemetaparameter,weareabletomapproblemsofhundredsofvariablestotheD-wave
chipandtobuildthedesiredclassifieroftreecoverinaerialimagery.
Thisworkisanoffshootofaprototypestudy[18]plannedeventuallytodelivertreecover
estimatesforthecontinentalUnitedStatesvia1-meter-resolutionVHRdatafromtheNational
AgricultureImageryProgram(NAIP)[19].Theobject-orientedplatformproducespixelwise
probabilisticmapsfortreecoverastheoutputofaconditionalrandomfield,whichitselfinte-
gratesoutputsfromaregion-mergingsegmentationroutineandaneuralnetworkclassifier.In
theprototype,treecovermapsweregeneratedfor11,095inputNAIPtilescoveringthestateof
California,withcorrectdetectionratesof85%inregionsoffragmentedforestand70%for
urbanareas.Wehaveformulatedtheboostedclassifiersothatitcanworkinconcertwithor
standinfortheneuralnetworkinthelargerobject-orientedplatform.Althoughthisremains
workinprogress,weareaimingataviablescientificapplicationofD-waveoutputinthenear
term.Ourcontributionsincludethedemonstrationoftreecoverclassification,alongwitha
detailedanalysisoftrainingonourremotesensingdataandashortcutsolutiontoembedthis
classofproblemsintotheD-wavearchitecture.Interalia,wediscoveredsomesimple,classi-
callyfast-to-trainquadraticdecisionstumpsonderivedimagefeaturesthatthemselvespro-
ducesurprisinglygoodclassificationoftreecoverinCalifornia.Forpointofreference,
antecedentcasestudiesofpotentialD-waveapplicationsinclude[20–27],while[28]presentsa
broadcollectionofpotentialapplicationsofinteresttoNASA.
Thepaperwillproceedasfollows.Wefirstreviewthestructureofproblemsamenableto
solutionsontheD-wavequantumannealingprocessor.Mathematically,theyarequadratic
unconstrainedbinaryoptimization(QUBO)problems,andinphysics,theyaregeneralized
Isingmodelsofaspinglass.WediscussQBoostinthiscontext,theproblemofembedding
intotheD-wavearchitecture,andourproposedmodificationstoQBoost.Wepresentthe
detailsofourimplementationontheNAIPdataset,layingoutthetwoproblems,oneon108
qubits,anotheron508qubits,whicharethefocusofthisstudy,alongwithourresults.We
concludewithadiscussionofchallengesandpossibleimprovementstothisframework.
QuantumannealingontheD-waveprocessor
InquantummechanicstheenergyfunctionisknownastheHamiltonian,denotedH.It
encodesalldynamicsofasystemandwillvarywithtimetalongwithambientconditions.The
basicprocessofquantumannealingistointerpolatephysicallybetweenaninitialHamilitonian
H ,withaneasy-to-implementminimalenergyconfiguration(orgroundstate),andaproblem
0
HamiltonianH whoseminimalconfigurationissought.Forinstance,foralinearinterpola-
P
tionscheduleandcomputationtimeτ,
(cid:16) (cid:17)
t t
HðtÞ¼ 1(cid:0) H þ H : ð1Þ
t 0 t P
TheinterpolationiseffectedphysicallyontheD-wavechipbyadjustingcurrentsthatflow
toindividualqubits,eachofwhichisatinysuperconductingcircuit.Thesystembeginsinthe
groundstateofH andends,ideally,inthegroundstateofH .Forperfectlyisolatedquantum
0 P
systems,thegroundstateofH canbeattainedforsufficientlylargeτwithprobabilityarbi-
P
trarilyclosetoone.Inpractice,duetothermalnoiseandlossofquantumcoherence,optimal
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DeployingaquantumannealingprocessortodetecttreecoverinaerialimageryofCalifornia
computetimesintheD-wavedeviceareactuallylessthanitscurrentlyminimalallowable
time,τ=20μs.[11]Inthiscontext,itshouldbenotedthattheparameterτcapturesonlythe
actualannealingtimeanddoesnotincludetimesforcooling,initialization,andreadoutofthe
device.
Becauseofthefacilityofphysicalcontrolattainablewithbinaryqubitsandpairwiseinterac-
tionsbetweenthem,theproblemHamiltoniantakestheform:
X X
H ¼(cid:0) hs (cid:0) J ss:
P i i ij i j ð2Þ
i2V fi;jg2E
InphysicsthisHamiltonianwasfirststudiedastheIsingmodelofamagnet.Thebinary
variabless 2{−1,+1}arethuscalledspins,fixedinalatticegraphGwithverticesandedges
i
ðV;EÞ.Theprogrammableelementsarethelocalmagneticfields,h,andthecouplings
i
betweenspins,J .Bothareinprinciplecontinuumrealvariablesbutareinpracticelimitedto
ij
adiscretumbynoiseinthedevice.TheoptimizationseekstheminimumofH overallconfig-
P
urationsofthespins{s}.
i
Theintuitionfortheoptimizationisasfollows:Thenegativesigninthefirsttermindicates
thattheenergyislowerwhenaspins alignswith(hasthesamesignas)themagneticfieldh at
i i
latticesitei;thisimperativecompeteswiththedemandthats alignoranti-alignwithneighbor-
i
ingspinss,accordingtothesignofthecouplingJ .IfJ >0,thecouplingbetweenspinsisfer-
j ij ij
romagnetic,drivingthemtoalign.IfJ <0,thecouplingisantiferromagnetic,drivingthemto
ij
anti-align.TheproblemofminimizingtheIsingenergyfunctionwithantiferromagneticcou-
plingsisknowntobeNP-hard,meaningthatthecomputationaleffortrequiredforthehardest
instancesscalesexponentiallywithproblemsizeforallknownclassicalalgorithms[29,30].
ComputationontheD-waveisfirstaprocessofmappingtheproblemtotheIsingstruc-
ture,binaryandquadratic,thenembeddingitintotheavailablequbitlattice.OntheD-wave
thequbitsarearrangedaccordingtoachimeragraph,asillustratedinFig1.Eachqubitcouples
tofiveorsixothers,exceptwheretherearedefectsduetofaultyqubits.Iftheproblemdoesn’t
Fig1.ChimerastructureofqubitconnectivityontheD-wave2Xprocessor.Thefull1152-qubitgraph
extendstoa12x12latticeofgroupsofeightqubits.Withintheillustratedsubset,currentlyinoperablequbits
aremarkedinred.
doi:10.1371/journal.pone.0172505.g001
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DeployingaquantumannealingprocessortodetecttreecoverinaerialimageryofCalifornia
embeddirectly,auxiliaryqubitscanbeintroducedtoaugmenttheavailablecouplings,atasig-
nificantcostinqubits.Bothmappingandembeddingimplyrestrictionsonthetypesofprob-
lemsthatcanprofitablybetackledwiththeD-waveprocessor.Wewillinvestigatetheseissues
inthecontextoftheQBoostalgorithm.Forathoroughrecentstudy,andformoredetailson
quantumannealingintheD-waveprocessor,see[22,28].
Boosting
Boostingisthetacticofbuildingastrongclassifierasanoptimallyweightedcombinationof
weakclassifiers,eachofwhichmayclassifyonlymoderatelybetterthanrandomguessingon
itsown.Iftheweakclassifiersarelinearintheinputfeatures,theboostedclassifiercarvesouta
piecewise-planardecisionsurfacethatis,ifnottothesamedegreeasthatexpressedbyaneural
network,effectivelynonlinear.In2008Neven,Denchev,Rose,andMacreadyproposeda
boostedclassifier,christenedQBoost,thatcouldbetrainedonaD-waveprocessor[15].Given
Nbinaryweakclassifiersc,i=1...N,eachofwhichclassifiesadatasampletaccordingto
i
c(t)2{−1,+1},theysoughtastrongclassifierofform
i
!
XN
CðtÞ¼sign wcðtÞ : ð3Þ
i i
i¼1
Theauthorsachievedtheirbesttestresultswithbinaryweights,w 2{0,1},inwhichcase
i
thestrongclassifierissimplyanoptimalvotingsubsetofweakclassifiers.Thenaturalcost
functiontomatewiththeD-wavearchitectureisaregulatedquadraticloss.ForasetToftrain-
ingsamples,witheachelementthavingbeenassignedatraininglabely(t)2{±1},atraining
problemcanbeposedasfollows:
( ! )
X XN 2 XN
Find : min wcðtÞ(cid:0) yðtÞ þl w : ð4Þ
fwi;lg i i i
t2T i¼1 i¼1
Theregularizationtermgovernedbytheparameterλisintendedtoimprovegeneralization
andspeedinexecutionbykeepingthefinalclassifiercompact.Thenormalizationoftheweak
classifiersisthenadjustedsoasnottoundulypenalizelargepositivemarginsfromthedecision
hypersurface,
XN
cðtÞ2f(cid:0) 1=N;þ1=Ng $ (cid:0) 1(cid:20) wcðtÞ(cid:20)1: ð5Þ
i i i
i¼1
Thetrainingproblemthusformulatedisoneofquadraticunconstrainedbinaryoptimiza-
tion(QUBO).Intheirinitialtestsofthealgorithm,Nevenetal.optimizedtheQUBOproblem
directlyusingclassicalheuristicsolvers.ComparingwithAdaboost,theyfoundmodest
improvementsinclassificationaccuracyandsignificantimprovment(oforder50%)incom-
pactnessoftheboostedclassifiers.
ToconverttheQUBOtoIsingform,onemakesthetransformations =2w −1.Thenew
i i
variabless takevaluess =±1.Expandingthequadratic,thecostfunctionbecomes
i i
! !
X X X X
l(cid:0) 2 cðtÞyðtÞ w þ cðtÞcðtÞ ww þconst
i i i j i j
i t2T i;j t2T
! ! ð6Þ
X l X 1X 1X X
! (cid:0) cðtÞyðtÞþ cðtÞcðtÞ s þ cðtÞcðtÞ ss þconst0:
2 i 2 i j i 2 i j i j
i t2T j;t2T i>j t2T
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Inthelatterequation,anextrafactoroftwointhequadratictermcompensatesforrewrit-
ingthesumtopassoverallindexpairs(i,j)onceonly.Wecanthenidentifythemagneticfields
andcouplingsoftheIsingframeHamiltonian(Eq(2)),
l X 1X
h ¼(cid:0) þ cðtÞyðtÞ(cid:0) cðtÞcðtÞ ð7Þ
i 2 i 2 i j
t2T j;t2T
1X
J ¼(cid:0) cðtÞcðtÞ ð8Þ
ij 2 i j
t2T
TheconstantsdroppedfromEq(6)donotaffecttheoptimization.Onecanreadilyinter-
prethowvarioustermsinfluencetheconstructionofthestrongclassifier.Thecontribution
∑ c(t)y(t)toh describeshowwelltheoutputc(t)ofaweakclassifiercorrelatestothe
t2T i i i
traininglabelsy(t)overthetrainingsetT.Iftheycorrelatewell,theygiveastrongpositive
contributiontothemagneticfield,drivingthespintobepositive.Apositivespinindicates
thatthecorrespondingweightisequaltoone:Theweakclassifier’svoteistabulatedinthe
P
finalstrongclassifier.ThecouplingJ ¼(cid:0) 1 cðtÞcðtÞlikewisedescribesthecorrelation
ij 2 t2T i j
ofweakclassifiersc andc overthetrainingset.Ifthetwoweakclassifierscorrelatewell,
i j
J <0.Thespinss ands tendtooppositevalues,meaningoneandnottheotherwouldbe
ij i j
includedinthefinalstrongclassifier.Thisisasitshouldbe.Towhateverextenttheycorre-
late,theysupplyredundantinformationonthedata.
Embeddingintothechimeragraph
TheQBoostprocedureresultsinafully-connectedIsingproblem,witheachs coupledtoevery
i
others bya(generically)non-zeroJ .TorunontheD-waveprocessortheproblemneedsto
j ij
beembeddedintothechimeragraph.Themaximaldegreeofthechimeragraphissix.The
fullyconnectedIsingproblemonNspinsconstitutesagraphofdegreeN−1.Nonethelessthe
lattercanbeembeddedintotheformerbymappingeachspinnottoanindividualqubitbutto
aconnectedsubgraphofqubits,suchthateverysubgraph(correspondingtoans)isconnected
i
byatleastonechimeragraphedgetoeveryothersubgraph(correspondingtoans)[31].The
j
graphedgesbewteensubgraphscanbeassignedtheproblemcouplingsJ .Withinasubgraph,
ij
internalgraphedgescanbeassignedlarge,ferromagneticcouplingsJ toimposethecondition
F
thatallqubitsassociatedtoagivenspinalign,encodingoneandthesamespinstate.
Thisembeddingcomesatahighcostinqubits.Sinceeachauxiliaryqubitinachimerasub-
graphcouplestoatmostdotherqubits,thesubgraphsizemustscalewithNtoprovidesuffi-
cientcouplingstoothersubgraphs.AstherearenecessarilyNsubgraphs,theembedding
overheadinqubitsscalesquadraticallywiththenumberofspinsN.Fortheexplicitexamples
studiedrecentlyin[32],N=30wasthelargestfully-connectedproblemembedableina
512-qubitchimeragraph.Muchrecentwork[22,24,33–35]hasgoneintothisandrelated
embeddingschemes,examiningmappingsoflogicalqubitstophysicalqubitsubgraphs,opti-
malsettingsfortheinternalcouplingsJ ,thedistributionofproblemcouplingsJ among
F ij
graphedges,andmoregenerallyseekingproblemsthatarelessthanfullyconnectedandthere-
foremoreamenabletoembeddinginthechimeragraph.Improvingtheconnectivityofhard-
waregraphswillbecriticaltobroadeningthescopeofproblemssolvableonfuturequantum
annealers.
Intheir2009demonstrationofaQBoostclassifiertrainedtodetectcarsinstreetscenes
[17],Nevenetal.embeddedviaadifferentapproach.TheymappedeachIsingspintoasingle
qubitanddiscardedvaluesJ thatdidn’tembedintothechimeragraph.Tothispurposethey
ij
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designedagreedyheuristicthatassignsspinstoqubitsinsuccession,eachspintothequbit
whichwillmaximizetheedgeweightretained(thesumofthemagnitudesoftheembeddedJ )
ij
withrespecttothepreviouslyembeddedspins.Underthisschemetheyretained11%oftotal
edgeweightona52-qubitembedding.(Only52qubitswerefunctioningontheavailableD-
waveprocessor,andtheyiteratedtrainingstepstogrowalargerclassifier.)
Thisstrategydoesnotscale.Droppingtoohighaproportionofcouplingsleadstoasce-
narioinwhicheachspinvariablecanbeoptimizedindependentoftheothers.If,foragiven
spins ,themagneticfieldh isbiggerthanthesumofcouplingstootherspinsjretainedinthe
a a
embeddedlatticegraph,i.e.,
X
if jh j> jJ j;
a aj ð9Þ
fa;jg2E
thevalueofs intheoptimalsolutionisdeterminedsimplybythesignofh .Thiscanbeseen
a a
byconsideringthetotalcontributiontotheenergyduetospins ,namely,
a
X
E ¼(cid:0) h s (cid:0) J s s:
a a a aj a j ð10Þ
fa;jg2E
Asinthepreceedingequation,thesumhererunsoverthecoupledspinsj.Ifthespins is
a
anti-alignedwithitsmagneticfield,thefirsttermcontributes−h s =+|h |totheenergy.Flip-
a a a
pingthesignofs willdecreasethecontributionfromthattermby−2|h |.Atthesametime,
a a
thesecondtermisbounded,
X X X
(cid:0) jJ j(cid:20)(cid:0) J s s (cid:20) jJ j;
aj aj a j aj ð11Þ
fa;jg2E fa;jg2E fa;jg2E
andsoflippingthesignofs ,regardlessoftheconfigurationoftheotherspins{s},imposesan
Pa j
energycostofatmostþ2 jJ j.Whenthecondition(9)holds,flippingthespinleadsto
fa;jg2E aj
anetdecereaseofenergy,andsothespinnecessarilyalignswithitsmagneticfield.
Theconsequencesaretwo-fold.First,onecandeterminetheoptimalconfigurationofsuch
spinssimplybycheckingthesignsoftheirmagneticfields.Thisisnotataskthatcallsfora
quantumcomputer.Theimplicationfortheclassifieristhelossoffinebalancethatwastobe
achievedamongallpossibleweakclassifiers.Weseektoretainonlytheminimalsetofweak
classifiersthatcapturestheimportantfeaturesofthedata,butweakclassifierswhosespinssat-
isfycondition(9)willbeincludedorexcludedirrespectiveoftheinclusionofothers.
Unfortunately,thisscenarioistobeexpectedasthetotalnumberNofinputweakclassifiers
growslarge.Thebasemotivationforquantumcomputingisthehopethatruntimeswillscale
betterthanforclassicalalternativeswiththenumberofinputvariables.Theeffortonly
becomesjustifiedonproblemswiththousandsortensofthousandsofbinaryvariables.Atthe
sametime,thenumberofconnectionsbetweenqubits(fiveorsixinthecaseoftheD-wave
chimeragraph)islikelytoremainsmall,duetothechallengeofbuildingandcontrollinginter-
actionsbetweenmorethanafewbasicphysicalentities.Foraproblemwithaninitiallyfully
connectedgraph,asimpleone-variable-to-one-qubitembeddingwilldiscardthousandsor
tensofthousandsofcouplingsagainstsomesomesmallfinitenumberretained.Anycomputa-
tionalproblemthatbeginsbyimposingaquadraticlossfunctiononalinearcombinationof
binaryvariables,asinEq(4),resultsinafullyconnectedgraph.Whilesomecouplingsmay
turnouttobezero,genericallyeveryspincouplestoeveryotherspin.
WecanmaketheseconsiderationsmoreexplicitbyconsideringthescalingwithNofthe
varioustermsinIsingHamiltonian.Exceptinthecasethattheaccuracyofweakclassifiersis
tunedcloseto50%,thecorrelations∑ c(t)y(t)willbeO(|T|/N),with|T|thesizeofthe
t2T i
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DeployingaquantumannealingprocessortodetecttreecoverinaerialimageryofCalifornia
trainingset.Forinstance,inourimplementationfortreecoverclassification,theaveragetrain-
ingerrorofthelinearweakclassifiersis25%.Aweakclassifierwith25%trainingerrorwould
have
X
ciðtÞyðtÞ¼:25jTjð(cid:0) 1=NÞþ:75jTjðþ1=NÞ¼:5jTj=N: ð12Þ
t2T
TheNappearsherethroughthenormalizationgiveninEq(5).Thisleveloftrainingaccu-
racyimpliesalsothattheweakclassifiersarewellcorrelatedamongthemselves,withcorrela-
tionsthatscaleas
(cid:18) (cid:19)
X jTj
c ðtÞcðtÞ(cid:24)O : ð13Þ
a j N2
t2T
LettingkbethemaximumnumberofcouplingsbetweenqubitsinthegraphG ¼ðV;EÞ,
forlargeNwehavetheoverallscalingrules:
(cid:12) (cid:12)
(cid:12)(cid:12) l X 1X (cid:12)(cid:12) (cid:18)jTj(cid:19) l
jh j¼(cid:12)(cid:0) þ cðtÞyðtÞ(cid:0) cðtÞcðtÞ(cid:12)(cid:24)O (cid:6) ð14Þ
a (cid:12) 2 i 2 i j (cid:12) N 2
t2T j;t2T
X X(cid:12)(cid:12)(cid:12) 1X (cid:12)(cid:12)(cid:12) (cid:18)kjTj(cid:19)
jJ j¼ (cid:12)(cid:0) c ðtÞcðtÞ(cid:12)(cid:24)O : ð15Þ
aj (cid:12) 2 a j (cid:12) N2
fa;jg2E fa;jg2E t2T
Sincetheregulatorisfixedonceforallspinsandkisfinite,agenericspinwillsatisfythe
decouplingcondition(9),
X
jh j> jJ j;
a aj
fa;jg2E
asNgrowslarge.
Wecircumventedthesedifficulties,intheheuristicembeddingschemeofNevenetal.,by
rescalingtheretainedcouplingsJ tocompensateforthoselost.ThedynamicsofIsingferro-
ij
magnets,inwhichlong-rangeorderappearsinsystemswithonlylimited,localinteractions,
gaveusreasontohopethatasubsetoffiveorsixofN−1couplings,ifappropriatelyrescaled,
wouldbesufficienttomaintainthecharacteristicbalancesoughtbetweentheweakclassifiers.
Absentaprincipledwaytocomputearescalingonaspin-by-spinbasis,werescaledallcou-
plingsbyaconstantfactorαwhichwetreatedasanewvariationalmetaparameter.Intuitively,
αshouldworkouttobetheratiooflosttoretainedcouplings,α*N/5.(Thecurrentproces-
sorisconstructedonan1152-vertexchimeragraph,with55currentlyinoperablequbits,mak-
ingtheaveragenumberofviableedges5.6.Becausetheembeddingheuristicmaximizesthe
sumofmagnitudesofretainedcouplingsinpreferencetotheirnumber,theresultingembed-
dingsarenotmaximallydense.Ourembeddingstypicallyretainanaverageofbetweenfour
andfivecouplingsperqubit.)Aplotofvalidationerroragainstthemetaparametersofour
108-qubitproblem,definedbelowinthesection“TreeCoverClassification,”isshowninFig2.
pffiffiffi
Steppingαbyfactorsof 2fromN/64toN,wefindthesolutionofoveralllowestvalidation
pffiffiffi pffiffiffi
errorfora2fN 2=8;N=4;N 2=4g.Thismatcheswellwithourexpectationsforαandsitu-
atestheoptimalclassifierintheregimewherethecouplingsandmagneticfieldsshouldhave
comparable,competinginfluenceontheoptimization.Moreover,wecanseeinthereturned
classifierstheincreasinginfluenceofthecouplingswithincreasingα.Whenαisverysmall,
theoptimizationisgovernedbythemagneticfieldsandtheresultingclassifiersconsist
PLOSONE|DOI:10.1371/journal.pone.0172505 February27,2017 8/22
DeployingaquantumannealingprocessortodetecttreecoverinaerialimageryofCalifornia
Fig2.Validationerrorasafunctionofthecouplingrescalingfactorandregulatorforthe108-qubit
problem.Theregulatorisexpressedintermsofanewparameterf:λ=2f|T|/N.Foreachpair(α,f),the
problemwasoptimizedwith1000callstotheD-waveprocessorandtheclassifierofminimalvalidationerror
pffiffi pffiffi
recorded.Theoverallminimalerrorof9%,indeepestblue,isattainedfora2fN 2;N;N 2g.
8 4 4
doi:10.1371/journal.pone.0172505.g002
predominantlyofthoseweakclassifierswhichindividuallyhavelowesttrainingerror.Towit,
theclassifiersreturnedatthefoursmallestvaluesofαshareincommonthetwelveweak
classifierswiththetwelvelowesttrainingerrors;whereas,theoptimalclassifierrealizedfor
pffiffiffi pffiffiffi
a2fN 2=8;N=4;N 2=4gincludesonlytwoofthosetwelve;andtheclassifieratα=N
includesoneofthetwelve.Whenwecometoourresults,wewillexploretheseeffectsandthe
propertiesoftheoptimalclassifierinmoredetail.
Incorporatingthenewrescalingfactor,theenergyfunctiontobeminimizedacrossvari-
ables{s,α,λ},becomes,finally,
i
X X
H ¼(cid:0) hs (cid:0) a J ss;
P i i ij i j
i2V fi;jg2E
l X 1X
h ¼(cid:0) þ cðtÞyðtÞ(cid:0) cðtÞcðtÞ ð16Þ
i 2 i 2 i j
t2T j;t2T
1X
J ¼(cid:0) cðtÞcðtÞ:
ij 2 i j
t2T
WewillrefertotheprocessoftruncationandrescalingoftheproblemHamiltonianasa
renormalization,anabuseofasuggestivetermfromstatisticalphysics.Inthinkingthrough
thisapproach,itisworthrememberingthatwehadalreadydeviatedfromthemostnatural
definitionofthetrainingproblematthepointofimposingaquadraticobjectivefunctionin
placeofthetotalnumberofmisclassifiedtrainingsamples(L vs.L norm).Wedeviatedagain
2 0
whenweregularizedthequadraticfunction.ThechoiceofL overL normismadehabitually
2 0
ongroundsofcomputationaltractabilityandjustifiedexpostfactobytheutilityofthesolu-
tionsthatresult.Likewisehere,welooktotheaccuracyoftheresultingclassifierstojustifythis
reformulationoftheoriginaloptimizationproblem.Themostaccurateclassifierfoundforour
108-qubitproblemusingtherenormalizedHamiltonianEq(16)hasavalidationerrorrateof
PLOSONE|DOI:10.1371/journal.pone.0172505 February27,2017 9/22
DeployingaquantumannealingprocessortodetecttreecoverinaerialimageryofCalifornia
9.00%.Thiscomparestoanerrorrateof10.13%forthebestsolutionfoundviasimulated
annealingontheoriginalQBoostcostfunction.Wehavefoundthatthefinalclassifiercanbe
improvedifselectedbyvalidationinpost-processingfromamongtheoutputsreturnedbythe
annealingprocess,andwedosoasmatterofcourse,althoughourresultsindicatethatthe
effectdiminishesforclassifiersoflargercardinality.
Twofinaldetailsoftheimplementationbearmentioninthecontextoftheembedding,for
bothofwhichwetakecuesfromtheoriginalreportonQBoost[15].Alongwiththerescaling
factorα,theregulatorλmustbedeterminedintraining.Beforesubmittingaproblemforopti-
mization,wespecifytheregulatorintermsofanewparameterf,
l fjTj
¼ : ð17Þ
2 N
Here,again,|T|isthenumberoftrainingsamplesandNthenumberofinputweakclassifi-
ers.Themetaparameters(α,f)arechosenbyactingtheoutputstrongclassifiersona
3000-samplevalidationset.(Thisstepiscoincidentwiththepost-validationstepmentionedin
thepreviousparagraph.)Ourpracticehasbeentodeterminethefractionfinitiallybyacoarse
parameterscanandthentoretestwithfinerstepsizesaroundtheminimuminf.Thecardinal-
ityofweakclassifiersinthestrongclassifieranditserrorratedependstronglyonf,asshown
inFig3withαfixedatN/4.TheeffectoftheregulatorforgeneralαcanbeseeninFig2.
Beyondenforcingcompactness,theregulatorevidentlyplaysanimportantroleinminimizing
classifiertrainingorvalidationerror.Withweakclassifiersnormalizedsothatc(t)2{−1/N,
i
+1/N},thequadraticloss,
!
XN 2
L¼ wcðtÞ(cid:0) yðtÞ ; ð18Þ
i i
i¼1
Fig3.Minimumvalidationerrorandweakclassifiersretainedasafunctionoftheregulatorλ=2f|T|/N.
Foreachftheproblemwasoptimizedwith500callstotheD-waveprocessor,andsubsequently(inset),with
1000callsinresolvingtheminimum,withαfixedatN/4.
doi:10.1371/journal.pone.0172505.g003
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