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Ubiquity,anACMpublication July2017

http://ubiquity.acm.org 1 ©2017AssociationforComputingMachinery

IsQuantumComputingforReal?

AnInterviewwithCatherineMcGeochofD-WaveSystems

byWalterTichy

Editor’s Introduction

In this interview, computer scientist Catherine McGeoch demystifies quantum computing and introduces us to a new world of computational thinking.



IsQuantumComputingforReal?

AnInterviewwithCatherineMcGeochofD-WaveSystems

byWalterTichy

Imaginetheworld'sfastestcomputerrequiringatotallynewwayofthinking:nosequencing,noconditions,noloops—indeed,noinstructions.Themachineisnotasetoflogicboxescontrolledby clock; it is instead a large number of cells each containing a quantum bit or "qubit,"connectedwitheachotherthroughweightedlinks.Changingjustonebittriggersacascadeofchanges inall theothers.Aqubit isnoordinarybitbecause ituses thequantumprincipleofsuperpositiontostorethetwobit-valuessimultaneously.A link isnoordinarywirebecauseitusesthequantumprincipleofentanglementtoavoidrelyingonelectriccurrents.Inamachinewith 2,000 qubits, all 22000possible combinations of bit-values are represented at the sametime.Whencooledtonearabsolutezero,andonceaprobleminstancehasbeen loaded intothequbits,themachinetakesonlyafewmicrosecondstosettleintoapatternthatencodestheproblem’ssolution.This,inshort,isthecomputationalmodelofquantumannealingandawaytobuildpracticalquantumcomputers.Quantumannealingmight replace theTuringandVonNeumannmodelswe’resousedtoandusherinanewkindofcomputationalthinking.

That quantumeffects such as superposition and entanglement could be used to build vastlymorepowerfulcomputerswasfirstsuggestedintheearly1980scourtesyofRichardFeynmanofCalTechandDavidDeutschoftheUniversityofOxford.In1994,PeterShor,thenatAT&TBellLabs, showed how a quantum computer could be used to factor large numbers. The firstquantum bits made of atoms, molecules, or photons appeared in the late 1990s and early2000s.Despite theprogress,physicists still struggle toconstructmore thana fewqubits thatcanremaininsuperpositionandentanglementforanyusefultimespan.

Then in 1999, the Canadian company D-Wave Systems embarked on a mission to makequantum computing practical. Over the years, the company has been fabricating quantumcomputerswitha steadilydoublingnumberofqubits. In January2017,D-Waveannouncedamachinewithanunprecedented2,000qubits.Withthismanybits, realisticproblemscan betackled.



In this interview,computer scientistCatherineMcGeoch,who leftAmherstCollege to joinD-Wave Systems and specializes in NP-hard problems, will demystify quantum computing andintroduceustoanewworldofcomputationalthinking.

WalterTichy:Wheneveraquantumcomputerinaresearchlabgoesfrom,say,8qubitsto10,this isbignews.NowD-Wavehasannouncedanewquantumcomputer, going from1,000qubitsto2,000!Whatisgoingonhere?Arewenearingpracticalquantumcomputing?

CatherineMcGeoch:Mostofthoseresearchlabsaretryingtobuildquantumcomputersthatbelong to the quantum gatemodel of computation (QGM). D-Wave implements a family ofquantumannealingalgorithmsinhardware,whichisthereal-worldcounterpartofadifferentabstractmodelcalledadiabaticquantumcomputation(AQC)[1].

Theengineeringchallengesthatarisewhentryingtobuildworkingquantumcomputersunderthesetwomodelsofcomputationareconsiderablydifferent.ForQGM,it isamajortechnicalchallengetominimizedecoherence,whichwreckstheprobabilityofasuccessfulcomputation(all quantum computation is probabilistic). That's why announcement of, say, a working 10-qubit device makes headlines—in the QG model it is a real achievement to managedecoherenceatthatscale.

The AQCmodel is more robust against decoherence (although coherence is still necessary).Annealing-basedquantumcomputinghasitsownsetofengineeringchallenges,butscalingupinqubitsizewhilemaintainingadequatelevelsofcoherencehasn'treallybeenonthetoptenlist.

Astowhetherwearenearingpracticalquantumcomputing,well,yes,that'stheidea.

WT:Canyoutelluswhich labsorcompanieshaveboughtD-Wavecomputingsystems,andwhattheyaredoingwiththem?

CM:LockheedMartinhasownedaD-Wavesystemsince2011,whichitshareswithUSC.TheiroriginalD-WaveOne(128qubits)wasupgradedtoaD-WaveTwo(500qubits),andthenaD-Wave 2X (1,000 qubits). Google, NASA, and Universities Space Research Association (USRA)havebeensharingaD-Wavesystemsince2013.TheirD-WaveTwowaslaterupgradedtoaD-Wave2X,andtheyrecentlypurchasedanupgradetotheD-Wave2000Q(2,000qubits)system.Los Alamos National Laboratory acquired a D-Wave 2X system in 2015. Temporal DefenseSystemspurchasedaD-Wave2000Qsysteminearly2017.



Therearealsoanumberofcustomerswhobuyblocksofquantumcomputationtimeoverthecloud,aservicethathasbeenavailablesince2010.

The best way to find out what they are doing with quantum computation is to visit theirwebsitesandlookatthepapersthatarebeingpublished.

WT:Ijustnoticedthefirstcarcompany,Volkswagen,hasannouncedacollaborationwithD-WaveSystemstooptimizetrafficflow.Obviously,aD-WavecomputerisnotmeantforyourHelloWorldprogramorforsurfingtheInternet. Instead,quantumcomputersaresupposedto tackle really tough problems. Before we go into concepts such as superposition andentanglement,whichproblemsmightreadersunderstandthathavebeensolvedonD-Wavesystems?

CM: The quantum processing unit (QPU) is designed to find good solutions to NP-hardoptimizationproblems,whicharedefined in termsofcertaincost functions tobeminimized.These problems are found, for example, in operations research, artificial intelligence, andespeciallymachinelearningapplications,whichrequirelargesamplesofsolutionsintheirinnerloopcomputations.

Inprinciple,anything inNP—the famousclass containinghundredsof challengingapplicationproblems—canbeformulatedforsolutionontheQPU.Ofcourse,someinputsaretoobigtobesolveddirectlyontheQPU,inwhichcaseproblemdecompositionorahybridclassical/quantumqueryapproachmaybeused.

Notetherearenotheoreticalguaranteesavailableonhowclosetooptimalthesolutionwillbe,or on how much time it would take to find an optimal solution. In this sense, quantumannealingiscomparabletoaclassicaloptimizationheuristiclikesimulatedannealing.

Theproblems thathavebeenactually implementedonD-Wave systemsare toomany to listhere.Threerecentpapersdescribeproblemsinconstraintsatisfactionforcircuitfaultanalysis[2],multiplequeryoptimization[3]indatabases,andconstructingSATfilters(similartoBloomfilters)[4].

WT:Canitcrackcodes,i.e.,factorlargenumbers?

CM: Itcanfactor,butitdoesnotuseShor'salgorithm,itusesadifferentquantumannealing-basedalgorithm[5].AtcurrentQPUsizes (whichcanhandlenumbersof sizearound10bits)



classicalmethods canbeat certain lower-bound times imposedby theQPUcontrol software.Therearenotheoremsknownabouttheefficiencyofthisalgorithm,andwedon’tknowifthequantumalgorithmwillbefasterthanclassicalmethodswhenqubitcountsgrowlargeenoughtosolvemorechallenginginputs.

WT:Whatdoestheprogramminglanguagelooklike?

CM: The QPU can compute any logic circuit, which technically means that is has the samecapabilitiesasaclassicalALU.Butthereisn'treallyamachine-levelinstructionset,thereisjustone(parameterized)instructionthatbasicallysays“solveitviaquantumannealing.”

Insteadoffiguringouthowtowriteaprogramtosolveyourfavoriteproblem,theintellectualchallenge is how to formulate your problem so that it can be solved by this instruction. Forexample,thefactoringproblemissolvedbyformulatingitasamultiplicationcircuitwithinputsX,YandoutputN,togetherwithacostfunctionthatisminimizedwhen𝑁 = 𝑋 ⋅ 𝑌.YousetNtothenumbertobefactored,andthenasktheQPUtofindXandYtominimizethecostfunction.

Thistypeofproblemtranslationisawell-understoodconceptinNP-completenesstheory,andcookbook translationmethodsareknownfor theoptimizationproblemsweare interested insolving.Thereisalsoarapidlygrowingbodyofsoftwaretoolsavailable(C,Python,Fortran)tohelptheuserformulateproblems.

WT:Howfastisit?

CM: Current-generation D-Wave systems can solve a 2000-variable problem with an annealtime of fivemicroseconds. This is one of theminimum time bounds imposed by the controlsystem:Onsomeinputsthequantumalgorithmcouldprobablyworkfinewithannealtimesintensofnanoseconds,andonotherinputslongerannealtimesgivebetterresults.Theminimumelapsed time, includingoverhead forsetupandreadout tomovetheproblemonandoff theQPU,isabout10milliseconds.Sothequantumpartofthecomputationisjustasmallfractionofelapsedtime.

For optimization heuristics the question of “how fast” is usually evaluated in terms of thetradeoffbetweencomputationtimeandsolutionquality:Howfastcanitreturnasolutionthatmeets a given quality threshold X? Or, for sampling applications: How fast can it returnNdistinctsolutionsofqualityY?



I don't have short answers to thesequestions, because they arehighly input-dependent andhard to summarize. Serious performance evaluation of quantum annealing has only beenpossibleinthelastcoupleyears.Bycomparison,scoresofresearchgroupshavespentdecadesstudyingthe(classical)simulatedannealingheuristic,yetthereisstillnosimplewaytodescribeitsperformance.

WT:Oneortwoconcreteexamples,withsolutionqualitiesandruntimes,wouldhelp.

CM: Here is an example from a recent paper: Trummer and Koch compared a D-Wave 2Xsystem (1,000qubits) to five classical algorithmson inputs formultiplequeryoptimization,adatabaseapplication[3].Theylookedatthetrade-offbetweencomputationtimeandsolutionquality.GenerallyspeakingtheQPUconvergedveryquicklytoverygoodsolutions,whereastheclassicalsolversconvergedmoreslowlybutcouldfindcomparable(sometimesbetter)solutionsifgivenlongenoughtime.Withtimes(notcountingsetup)fortheD-Wavesystemrangingfromabout onemillisecond to one second, the software solvers took up to 1,000 times longer to“catchup”andreturnsolutionsofcomparablequality.AseparatetestshowedsolutionsfoundbytheQPUinonesecondweretypicallywithin0.4percentofoptimal.

Atoneendoftheperformancespectrum,Denchevetal.foundaD-Wave2XQPUcanoptimallysolve a class of synthetic inputs 100 million times faster than some (but not all) classicalalgorithms[6].Thisisadifferenceonthescaleofonesecondversusthreeyearsofcomputationon a single core. At the other end, we know classical algorithms with nanosecond-scaleinstruction sets can solve small and/or easy inputs faster than the above-mentioned lowerbounds.Thatisthecurrentrangeofpossibilitythatwearedealingwith.

I should point out whatever numbers I give here—including those lower bounds—will likelyimproveasthetechnologycontinuestogetbetteralongwithincreasingqubitcounts.Denchevetal.,forexample,notedtheirD-Wave2XQPUsolvedthebiggestproblems10,000timesfasterthantheirpredictionsbasedonperformanceofthepreviousyear'sD-WaveTwoQPU[6].

WT:Butcanitbeatamassivelyparallelcomputer?

CM: Here is another question with no simple answers. First let me say the hoopla aboutquantumcomputationofferingenormousspeedupsoverclassicalcomputersisbasedonback-of-theenvelopecalculations fromtheoreticalargumentsabout theworst-casecostof solving



NP-hardproblems.Thatis,givenaninputwith2000variables,aclassicalalgorithmwouldhavetosearchaspaceof22000bitstringstocertifyithasfoundanoptimalsolution.

There have been about 280 nanoseconds since the Big Bang, so solving a problem in fivemicroseconds does indeed represent an unimaginably large speedup over that worst-casescenario.Parallelcomputationwouldnothelpunlessyouhaveanunimaginablylargenumberof cores on hand (the number of atoms in the known universe is about 2240). And in thistheoreticalcontext,youareallowedtohaveasmanyquantumcomputersasyouhaveclassicalcomputers. In other words, parallel models of computation don't change the fundamentalargumentsaboutproblemcomplexity.

However, the relevant theoretical questions are open and nobody knows if this line ofreasoning iscorrect.Fastsolutiontimesforagivensetofproblemsdonotconstituteaproofthatallquantumcomputationsarefast.Also,a fasterclassicalalgorithmmightbediscoveredsomeday,therebysettlingthefamous𝑃 = 𝑁𝑃questionintheaffirmative.

Nomatterhowthetheoryturnsout, it isnotreallyrelevanttopractice.Forperhapsobviousreasons, nobody uses the worst-case approach to solve real application problems. Instead,classicalheuristicsareused,whichcanfindprettygoodsolutions,prettyquickly,onsomesetsofinputs.

Havingworkingquantumcomputersinhandmakesitpossibletocompareperformanceagainstclassicalheuristics.Theruntimeofanysuchheuristiconaninputofsize2,000canbebetweena fewmicroseconds (best case),up toBigBang timescales (worst case). So thepotential forenormous speedup is still there, butwedon’t know in advancewhich inputswill be easy orhardforwhichheuristics.Thiscreatesthe“decadesofresearch”issueImentionedearlier.

The same issues apply when we talk about comparisons to massively parallel computingplatforms: A back of the envelope calculation about theoretical speedup doesn't capturereality,andrealityishardtosummarize.Thatbeingsaid,acoupleoftheheuristicswecompareagainsthaveshownmoderateruntimeimprovementsfromparallelization.Inothercases,thereisnospeedup,andsometimes theparallel code is slower.Verygenerally speaking, itdoesn'tappear that parallel computation will be a huge difference-maker in this arena (a D-Wavetechnicalreportdiscussesthistopicinmoredetail[7]).

WT:What’sinsidethelayersofcoolingequipmentofaD-Wavesystem?Inotherwords,whatarethebitsandhowaretheylinked?



CM:It’snotjustcoolingequipment.LikeSchrödinger’scat,theQPUmustdoitsworkinaclosedbox,isolatedasmuchaspossiblefromambientenergythatcouldderailthequantumprocess.Soitoperatesinahighlyshieldedenvironmentwherethetemperatureisbelow.015kelvin,themagneticfieldis50thousandtimeslessthanEarth’smagneticfield,andatmosphericpressureis10billiontimeslessthanoutside.

Thequbitsaremadeofmicroscopicloopsofniobium,ametalthatbecomesasuperconductorattemperaturesbelow9kelvin.Qubitsarelinkedtogetherbycouplers,alsomadeofniobium.Figure 1 shows a photograph of a QPU chip inside its mounting apparatus—it is the blacksquareinthecenter,aboutthesizeofathumbnail.

Figure1.Aquantumprocessingunit(QPU)initsmountingapparatus.



Figure2showsan8x8gridofso-calledChimeracells(onasmallerpredecessorchip)containingthe qubits and couplers that perform the actual computation. Each cell has eight qubitsconsistingof four thin loopsacrossand four thin loopsdown,connectedbycouplersat theirintersections.Therearealsocouplersbetweenqubitsinonecellandthoseinitsneighborcells.Othercomponentsonthechipincludequbitreadoutcircuitryandaframeworkofdigital-analogconverterstocontrolqubitandcouplerstates.ApaperbyBunyketal.describessomeoftheconsiderations that went into selecting this topology [8]. Basically, it offered the bestcombinationoffeatureswithincertaindesignconstraintsimposedbyphysics.(Othertopologiesareexpectedforuseinfuturegenerations.)

Figure 2. An 8x8 Chimera graph, with one Chimera cell magnified. The cell contains anarrangementofeightqubits,longthinloopsofniobiummarkedinred,whichareconnectedat their intersections by couplers. The interstices are filled with qubit read and controldevices.

Figure3 illustrateshowthisqubitarrangementcorrespondstoaChimeragraph.Theleftsideshowseightqubits(redloops)and16couplers(blueells).Alsoshownareeightcouplers(bluecircles) connecting to neighbor qubits in the north and east cells (couplers to the south andwestarenot shownhere). The right side showsagraphical viewwhere rednodes representqubitsandblueedgesrepresentcouplers.Eachcellformsa4x4completebipartitegraph.



Figure 3. Left: A single Chimera cell contains eight qubits laid out in thin loops (red), fouracross and four down.Qubits are connected by couplers (blue cells) at their intersections.Alsoshownareeightof16possiblecouplings(bluecircles)connectingtoqubits inadjacentcells.Right: Qubitandcouplerconnectivity isdescribedbyacompletebipartitegraphwitheightnodes (red, representingqubits) and16edges (blue, representing couplers). Edges toqubitsinneighborcellslocatednorthandeastarealsoshown.

The2000Qprocessordesign isbasedona16x16Chimeragraphcontaining2,048nodes.Theworking graph for any given QPU is a subgraph of a Chimera graph, because an imperfectfabricationprocessleavesasmallpercentageofqubitsandcouplersunusable.

WT: ThefundamentalproblemtheD-WaveprocessorsolvesistheIsingspinmodel.Pleaseexplainwhatthatis.

CM: You are given a graph 𝐺 = (𝑉,𝐸) with weights ℎ = {ℎ!} on the nodes and weights𝐽 = {𝐽!"} on the edges. The problem is to assign spin values 𝑠! = ± 1 to the nodes so as tominimizetheenergyfunctiondefinedby

𝐸 𝑠 = ℎ!𝑠!!∈𝒱

+ 𝐽!"𝑠!𝑠!(!,!)∈!

. (1)



Figure 4 shows an example problem defined on a 4-node graph. The right side shows onepossible spin assignment,which has energy𝐸 𝑠 = −8. This happens to be an optimal spinassignmentforthisinput.

Figure 4. An Ising model problem defined on a 4-node graph. The energy function isminimizedat𝑬(𝒔) = −𝟖 bythespinassignmentshownhere.

Ising spinmodel can be trivially transformed to quadratic unconstrained binary optimization(QUBO), defined on binary values 𝑥 = {0, 1} instead of spins, or maximum weighted two-satisfiability,definedonBooleanvalues𝑏 = {𝐹,𝑇},whichmaybemorefamiliartocomputerscientists.

WT:Nowforthehardpart:HowdoestheD-Wavecomputersolvethisproblem?Apparently,themachineitselfisagraph.Edgeandnodeweightswillhavetobeloadedsomehowintothegraph, and thenodesmust choose +1or−1 so that𝑬 𝒔 isminimized.Buthowdoes thishappen?

CM: First, the Ising model input (ℎ!, 𝐽!) defined on a general graph 𝐺 is mapped to anequivalentproblem(ℎ, 𝐽)ontheworkinggraph𝐶,usingaprocesscalledminor-embedding(forexamples,see[2,3,5,9]).Roughlyspeaking,minor-embeddingmapseachnodeofdegree𝑑 in𝐺 into𝑑/4nodesofdegree4 in𝐶,plusadditionaledgestochainthosenodestogether.Thisprocess expands the total graph size somewhat, but the mapping is always sufficient topreservetheinformationin(ℎ′, 𝐽′).



The ℎ and 𝐽 weights represent electromagnetic forces that push qubits towards certainmagneticpolarities(𝑢𝑝 = +1,𝑑𝑜𝑤𝑛 = −1).Forexample,intheenergyfunction(1),termℎ!𝑠! isminimized(i.e.,negative)whenℎ! and𝑠! haveoppositesigns,andterm𝐽!" 𝑠! 𝑠! isminimized

when 𝐽!" isnegativeand𝑠! , 𝑠! havematchingsigns,or 𝐽!" ispositiveand𝑠! , 𝑠! haveoppositesigns.

These forces and polarities correspond to various levels ofmagnetic energy, and the qubitsnaturallyseektheirlowest-energystate(calledthegroundstate),justaswaterseeksthelowestpointinanaturallandscape.

A quantum annealing algorithm𝐻(𝑟) is analog, and runs over a time interval 𝑟: 0 → 1. It isspecifiedbyfourcomponents:(1)theproblemHamiltonian𝐻!,amatrixthatdescribessystemenergiesdeterminedbyℎ and𝐽; (2)an initialHamiltonian𝐻! thatdescribes initialconditions;(3)apairoffunctions𝐴(𝑟) (decreasing intime) and𝐵(𝑟) (increasing intime) thatcontrolatransitionfrom𝐻! to𝐻!;and(4)thetotalannealingtime𝑡!forthetransition,so𝑟 = 𝑡 ∕ 𝑡!.Likethis:

𝐻 𝑟 = 𝐴 𝑟 𝐻! + 𝐵 𝑟 𝐻! as 𝑟 = 0 → 1. (2)

(SeeWikipediaonquantumannealingforadescriptionoftheunderlyingphysics.)Attheendof the process, the qubit spin states are read and interpreted as a solution to the problem.Differentquantumannealingalgorithmscorrespondtodifferentchoicesforthesecomponents.Inthecurrent2000Qsystem,𝐻!issettoadefaultvalue,𝐻!and𝑡!areparameterssetbytheuser,and𝐴 𝑠 ,𝐵(𝑠)arefromafamilyoffunctionsthatcanbepartiallyspecifiedbytheuser.

WT: Letme try to describe this process inmy ownwords, since this is crucial. The initialHamiltoniansets thequbitweights toneutral,meaningall thequbitscansettlewithequalprobability on +𝟏 or−𝟏. The edgeweights are also set to neutral, so the energy function𝑬(𝒔)isthesameforallpossiblecombinationsof+𝟏and−𝟏ofthequbits.NowwegraduallytransitiontotheproblemHamiltonianbyturningtheweightsupordown.Theresultisthat𝑬(𝒔)isnolongerevenlydistributed—wegetanenergysurfacewithpeaksandvalleys.Whilewedo this, theannealingmagichappens: thewaterpools in thevalleys,meaning that theprobabilitiesforsettlingin+1or−𝟏shiftinsuchawayastopreferaminimumenergystate.Thishappensoveratimeperiodinwhichthepeaksandvalleysbecomemorepronounced.Isthispicturecorrect?



CM: Yes, that's about right, although I don't think “settingweights to neutral” is exactly therightwaytoputit.Wehavetothinkaboutitintermsoftwoquantumproperties,superpositionandentanglement.

Superposition. A classical bit can be in either state 0 or 1, and a register of 𝑁 bits can onlyrepresent one of 2!possible states at a time. But a qubit can be in superposition—simultaneously in both states at the same time, according to a certain probability.Superpositioncannotbeobserved—likethatfamouscat,whichisbothaliveanddeadwhiletheboxisclosed,but“snaps”toonestateortheotherwhenyouopentheboxandlookatit.

The initialHamiltonianpushesqubits intoanequal superpositionof−1 and+1. For the fullqubitsystem,thismeansthatallpossibleoutputsareequallylikely.

At initialization time, 𝐴(𝑠) puts the initial Hamiltonian at full strength and 𝐵(𝑠) puts theproblemHamiltonian at zero strength. The transition gradually dials down𝐴(𝑠) anddials up𝐵(𝑠), so that theℎ and 𝐽 forces are applied, first weakly and gradually becoming stronger.Whentheprocessfinishes,𝐻!isatzerostrengthand𝐻!isatfullstrength.Attheend,thequbitstatesarereadandinterpretedasasolution.

Appealing again to our water analogy, thewater represents superposition probabilities. Thesuperposition state can be everywhere on the surface at any time, covering all possiblesolutions,butittendstopoolinlow-lyingareas:Adeeperpoolatagivenspot𝑠correspondstohigherprobabilityofsbeingreadwhenthealgorithmfinishes.

Thisalgorithmcreatesasurfacethatisinitiallyflat,withwaterspreadevenlyacrossit.Thenitgraduallyevolvesfromtheflatsurfacetoonethatmatches𝐸(𝑠),andtheprobabilitiesforeachqubitsettleoneither−1or+1.Ifeverythinggoeswell(thisistheprobabilisticheuristicpart),bythetimethealgorithmfinishes,thewaterhascollectedinonespotthatcorrespondstoanoptimalsolution.

WT:Wheredoesentanglementhappen?Isupposeyouneeditwhensplittingaproblemnodeacross severalphysicalnodes, inwhichcase theymustmaintain thesamestate.But if youlinkthesplitnodeswithastrongnegativeweight,thentheforceswillmakesuretheyhavethe same state. Is this entanglement? Some physicists claim there is no large-scaleentanglementonD-Wavemachines.HowmanynodescanaD-Wavesystemkeepentangled?

CM:Agroupofentangledqubitscanchangetheir (superposition)statessimultaneously inanindivisibleway,ratherthanindividually.Theedgeweightscreatemagneticcouplingsbetween



qubits: these couplings create entanglement in the ground state of the qubit system as itevolves.

Considerasmall2-qubitsystemwithfourpossiblespinconfigurations,withweightsℎ!, ℎ! =0 and 𝐽!" = +1. Theclassicalminimum-energy spin statesare (+1,−1) and(−1,+1).Duringtheanneal,thegroundstateforthesystemisasuperpositionofbothstates.Sotheindividualqubits are neither +1 nor−1, they are in equal superposition, pointing up and down at thesametime.Duetoentanglement,theirsuperpositionstatesarelinked,sothat,ifyoureadoneand then theother theywouldhaveopposite signs. Thismysterious instantaneous linkage isnotthesameasamagneticsignal,whichwouldapplytoclassicalstates(upordownonly),andwouldattenuateoverdistance.

In the context of quantum annealing, superposition and entanglement together create aphenomenon known as “tunneling.” In our water metaphor, this means under certaincircumstancesthelandscapeiseffectivelyporous,andthequbitstatescangodirectlythroughhills rather than climbing over them. Classical algorithms (andmagnets) cannot tunnel: Theymustchangestatebyclimbingoverhillsandmovingaroundthelandscapestepbystep.

Tunneling is themagicalproperty that, inan idealscenario,allowsthequantumalgorithmtoslidedirectlytoagroundstateeveniftherearehillsalongtheway,aslongastheannealtime𝑡!isaboveacertainthreshold(unknowninpractice).Noreal-worldquantumcomputationcanachieve the theoretically ideal situation, though. Despite all the shielding and coldtemperatures, there is always some amount of noise from the ambient environment thatinterfereswiththecomputation.Wehaveaquantumannealingalgorithmthatusestunnelinginlimitedways,andweareworkingtounderstandthisphenomenon.

Asfarasclaimsthatthereisnolarge-scaleentanglement,Ithinkitwouldbemoreaccuratetosay large-scale entanglement has not beendemonstrated. Indeed, it is a significant scientificchallengetoperformanexperimenttodetectentanglement—whichonlytakesplacewhenthequbitsareshieldedfromambientenergy—sinceenergyfromtheexperimentalapparatuscandestroy the very effect it is trying to detect. The physics experiments demonstratingentanglement in a D-Wave QPU were only designed to detect entanglement in an 8-qubitsystem[9].

Alaternon-intrusiveexperimentbasedonanalyzingoutputsfromcarefully-constructedinputs,showedtunnelingdoesoccur(whichcouldonlyhappenifentanglementispresent)onsystemsof 945 qubits [6]. But I don't think it is possible to infer from that result howmany qubitsactuallydidtunnel,orhowmanywereentangled.



WT:Youmentionedthatquantumannealingisanalog.Isupposetheedgeandnodeweightsandtheforcesareanalog.What is theresolutionoftheseweights, i.e.,howmanypossiblevalues are there? And how does the analog computation affect the problems that can besolvedandtheanswersgenerated?

CM: Right, thetransitionof forces from𝐻! to𝐻! isanalog.Theweightscanbespecifiedasregularfloating-pointnumbers.Analogcontrolerrorscreatesmallperturbationsonℎand𝐽,sotheenergyfunctionthattheQPUsolvesisslightlydifferentfromtheonespecifiedbytheuser.If the difference is big enough that states swap ranks (i.e., the best solution in the originalproblem is only second-best in the perturbed problem), then the algorithm has lowerprobabilityoffinishingingroundstate,andhigherprobabilityoffinishinginthenext-beststate.

Ican'tgiveanexactnumberfortherequiredweightresolutiononinputsofinterestingsize:itdependsonhowfarapart theenergiesof the low-energystatesare.SinceweworkwithNP-hardproblemswedon't typicallyhaveknowledgeof theoptimalandnear-optimal states, sothisquantityishardtoanalyzeorpredictinpracticalscenarios.

There are various ways to compensate for analog errors; for example, a system utility isavailable that can average-out systematic errors, similar to the way randomized quicksortavoidstheworst-casescenario.Also,𝐻! canbere-formulated,ortechniquesofquantumerrorcorrectioncanbeapplied,toincreasetheseparationbetweengroundstatesandotherstatesin𝐸(𝑠),tomitigatetherank-swappingscenario.Also,asmentionedpreviously,newgenerationsofprocessorstendtoshowimprovedperformanceduetobettersuppressionofthesetypesoferrors.

WT:Doesquantumannealingrequirequantumentanglement?

CM: Quantum annealing exploits both quantum entanglement and superposition, but thoseproperties arenotnecessarily required to solveevery input. Forexample, if the landscape isshaped like a big cherry-bowl instead of having lots of hills and valleys, then the quantumalgorithmdoesnotneedtoinvokequantumtunneling:Itjustslidesdownthehilllikeaclassicalgreedyalgorithmwould.Entanglementwouldbepresentbutsimplemagneticforceswouldbeenoughtosolvethisproblem.

WT:Now thatwehavean ideaofhow themachineworks,pleaseexplainhowa concreteproblemismappedtotheIsingmodel,sothemachinecansolveit.



CM: In the circuit satisfiability problem, you are given a binary circuit and the question iswhetherthereisanassignmentofvaluestoinputsthatmakesthecircuitevaluateto1(assumethat0 = 𝐹and1 = 𝑇).Figure5showsasmallexamplecircuitwithinputs 𝑥!, 𝑥!, 𝑥!andtwogates,ANDaboveandORbelow. This circuit has value1, for example,when𝑥! = 1, 𝑥! = 1and𝑥! = 0.

Figure 5. Transformation from circuit satisfiability to Ising model. Each gate in the circuitcorrespondstoaBooleanclause,plusanextraclause(toprow)thatsaysthecircuitoutputis1.Eachclauseistranslatedtoanarithmeticexpressiondefinedonvariables𝒔𝒊 ∈ {−𝟏,+𝟏}.Thecircuitinputs𝒙𝟏,𝒙𝟐,𝒙𝟑(withvalues{𝟎,𝟏})makethecircuitevaluateto𝟏 (𝑻𝒓𝒖𝒆)exactlywhenthecorrespondingexpressioninputs𝒔𝟏, 𝒔𝟐, 𝒔𝟑(withvalues{−𝟏,+𝟏})minimizethesumofthesethreeexpressions.

Step one in the transformation is to write each gate as a Boolean clause of the form(𝑧 𝐸𝑄 (𝑥 𝑂𝑃 𝑦)),where𝑂𝑃isthegateoperationandtheclauseistruewhentheoutputwire𝑧equals the result of the operation applied to input wires 𝑥 and 𝑦. The two clauses for ourexample circuit are listedbelow it, togetherwithanextra clauseon top that says the circuitoutputisequalto1.Forinputs𝑥!, 𝑥!, 𝑥! thecircuitvalueis1exactlywhenallthreeclausesaresatisfied,thatis,allthreehavevalue1.

Thenextstepistotransformeachclausetoanarithmeticexpression.Assumethebinaryvalues(0,1)map tospinvalues(−1,+1), respectively.Since Isingmodel isaminimizationproblem,



the goal is to build an energy function𝐸(𝑠) defined on (−1,+1) that hasminimum energyexactlywhenthecorrespondingbinaryvalues(0,1)satisfyallthreeclauses.ThreeexpressionsmatchingthethreeclausesareshownatthebottomrightofFigure5.(Therearedifferentwaystosetuptheseexpressions;seereferencesformoreinformation[2,9].)

SomedetailsofthiscorrespondencefortheORclauseinthemiddlerowareshowninFigure6.Theleftsideofthetruthtableshowsallpossibleinputsandoutputsfortheclausedefinedon𝑥!, 𝑥!, 𝑥!: Theclausevalue is1 in rowswhere𝑥!equals theORof𝑥!and𝑥!.TherightsideshowsthecorrespondinginputsandoutputsfortheIsingmodeexpressiondefinedon𝑠!, 𝑠!, 𝑠!.Ineveryrowwheretheclauseissatisfied,theexpressionisminimizedwithvalue−3.

Figure6.Theclausetruthtable(left)andtheIsingmodelexpression(right)showallpossibleinputsandoutputsforthesetwoversionsoftheORcircuitofFigure5.Theclauseissatisfied(hasvalue𝟏 = 𝑻𝒓𝒖𝒆)inthesamerowsforwhichtheIMexpressionisminimized(hasvalue−𝟑).

Therequirementthatallclausesmustbesatisfiedbecomesarequirementthatthesumofallexpressionsmustbeminimized.The laststep in the transformation is toaddtheexpressionstogetherandcollectterms(constantscanbediscarded).Theresult isan Isingmodel inputasshownon the leftofFigure7.Onepossibleway tomap (ℎ, 𝐽) toaChimeracell is touse thelabelingschemeshownontheright(theweightsofunusednodesandedgesaresetto0).



Figure7.Left:Isingmodelinputs(𝒉, 𝑱)forthecircuitofFigure4.Right:Amappingof(𝒉, 𝑱)ontoaChimeracell.Nodeswith𝒉 = 𝟎 arenotlabeled,edgeswith𝑱𝒊𝒋 = 𝟎arenotshown.The𝒉𝟐weightissplitacrosstwonodes,whichareconnectedbyachain(blackedge)havingastrongnegativeweight,inthiscase-10.

Noteℎ! issplitacrosstwonodesthatareconnectedbyaso-called“chain”(blackedge),whichis set to a strong negative value like−10 to ensure they end upwith the same spin in theoutput.Thissplitisnecessarybecausethetriangle𝐽!", 𝐽!", 𝐽!"can'tbeembeddeddirectlyontotheChimeratopology;thisdecisionispartoftheminor-embeddingprocessmentionedearlier.

SendthisproblemtotheQPU,anditwillreturnasampleofspinassignments𝑠!… 𝑠!,someofwhichcorrespondtogroundstatesof(ℎ, 𝐽)(therearefoursuchgroundstates).Inthosestates,theassignmentsofspinvaluesto𝑠!, 𝑠!, 𝑠!canbemappedbacktoassignmentsofbinaryvaluesto𝑥!, 𝑥!, 𝑥!forwhichthecircuitvalueis1,usingourconventionthatspin−1equalsbinary0andspin+1equalsbinary1.

WT: Thank you, Catherine, for this very informative interview. Obviously, a QPU is verydifferent froma traditional CPU. There is only a single solver hardwired into themachine,suchthatthequbitsalltugoneachotherinordertofindasolutiontotheIsingmodel.TheD-Wave systemseems tobea very cool (literally) andversatile accelerator for combinatorialproblems.

Thisinterviewhasbeencondensedandeditedforclarity.



Forthosewhowanttoseethemachine,a3-parttouroftheD-WavesystemcanbefoundonYouTube: look for “D-Wave lab tour.” Catherine'smonograph, aimed at a computer scienceaudience, discusses adiabatic quantum computation and quantum annealing in more detail[11].Catherinewould like to thankAndrewBerkleyandMark JohnsonofD-Waveforhelpingoutwiththephysicsdiscussion.

References

[1]Farhietal.AquantumadiabaticevolutionalgorithmappliedtorandominstancesofanNP-completeproblem.Science292,5516(2001).

[2]Z.Bianetal.Mappingconstrainedoptimizationproblemstoquantumannealingwithapplicationstofaultdiagnosis.FrontiersinICT3,14(2016).

[3]Trummer,I.andKoch.C.MultiplequeryoptimizationontheD-Wave2Xadiabaticquantumcomputer.ProceedingsoftheVLDBEndowment9,9(2016).

[4]A.Douglassetal.ConstructingSATFilterswithaQuantumAnnealer.InternationalConferenceonTheoryandApplicationsofSatisfiabilityTesting—SAT2015.SpringerLNCS9340,2015,104-120,.

[5]E.Andriyashetal.Boostingintegerfactoringperformanceviaquantumannealingoffsets.D-WaveTechnicalReport14-1002A-B.2016.Availableathttp://www.dwavesys.com/resources/publications.

[6]Denchevetal.Whatisthecomputationalvalueoffiniterangetunneling?,PhysicalReviewX6,3(2015.)

[7]LimitsonParallelSpeedupforClassicalIsingModelSolvers.D-WaveWhitePaper14-1004A-A.2017;https://www.dwavesys.com/sites/default/files/14-1004A_wp_Limits_on_Parallel_Speedup_for_Classical_Ising_Model_Solvers.pdf

[8]Bunyketal.Architecturalconsiderationsinthedesignofasuperconductingquantumannealingprocessor.IEEETransactionsonAppliedSuperconductivity24,4(August2014).

[9]Suetal.AquantumannealingapproachforBooleansatisfiabilityproblem.InProceedingsofthe53rdAnnualDesignAutomationConference(DAC'16).Article148.ACM,NewYork,2016.



[10]Lantingetal.Entanglementinaquantumannealingprocessor.PhysicsReviewX4,021041(May2014).

[11]McGeoch,C.C.AdiabaticQuantumComputationandQuantumAnnealing.Morgan&ClaypoolPublishers,2014.

AbouttheAuthorWalter Tichy has been professor of Computer Science at Karlsruhe Institute of Technology(formerly University Karlsruhe), Germany, since 1986. His major interests are softwareengineeringandparallelcomputing.Youcanreadmoreabouthimatwww.ipd.uka.de/Tichy.

DOI:10.1145/3084688

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