Quantum Machine Learning and Optimisation in Finance: On the Road to Quantum Advantage

443 Pages·2022·4.367 MB·English

by  Antoine Jacquier, Oleksiy Kondratyev

Quantum Machine Learning
and Optimisation in Finance
On the Road to Quantum Advantage
Antoine Jacquier
Oleksiy Kondratyev
BIRMINGHAM—MUMBAI
Quantum Machine Learning and Optimisation in
Finance
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Foreword
Asfarascomputationalprospects(includinghardwareandsoftware)andrelateddevelop-
mentsareconcerned,weliveinthemostexcitingtimes.Everydaybringsnewachievements,
newpromises,and,onoccasion,newdisappointments. Presently,someoftheideasdis-
cussedfordecadesarereachingtheusablestage,themostexcitingdevelopmentsbeing
thermonuclearfusion,all-purposeartificialintelligence,distributedledgertechnology,and
quantum computing. However, not surprisingly, breakthroughs in these fields are very
hardtoachieve,so,despitestrenuousgovernmentandprivateeffortsandlavishfunding,
theirstatedgoalshavenotbeenreachedyet.
JacquierandKondratyev,twoofthestrongestquantsoftheirgeneration,havewrittena
remarkablebookaboutquantumcomputersandtheirapplicationsinfinance,emphasizing
practical aspects of quantum machine learning and optimisation. To put the subject of
thisbookinapropercontext,letusbrieflytouchuponthehistoryofcomputingdevices.
Withoutexaggeration,onecanclaimthatthehistoryofhumanprogressiscloselyrelated
to the history of computing devices. Original simple but beneficial instruments, such
as abacuses, were invented at the dawn of civilization. Eventually, computing devices
developedintopotenttools,suchassupercomputers,thatdefineourday-to-dayexistence
andfutureasspecies.
Tostartwith,itisworthmentioningsomemilestones. AbacuseswereusedinBabylonia
as early as c. 2700–2300 BC; eventually, they spread worldwide and became known as
Romanabacuses. Computationaldeviceswereusedtogetherwithmemorydevices,such
astallysticksandclaytablets. Inclassicalantiquity,analogcomputers,suchasthefabled
Antikythera,probablyintendedforastronomicalcalculations,wereactivelyused. Subse-
quently, medieval Muslim astronomers and engineers brought such devices to the next
level. Asaresult,theydevelopedremarkableobjects,suchassundials,planispheres,and
astrolabes.
Eventually, the center of progress moved to Europe and later to North America, where
variouscalculatingtools,suchastheslideruleandmechanicalcalculator,wereintroduced
inearlymoderntimes.However,therealbreakthroughwasachievedin1804whenJacquard
createdaloomprogrammableviapunchedcards. Babbage,“thefatherofthecomputer”,
invented the first mechanical computer programmable with punch cards in the United
Kingdom. Unfortunately,hisDifferenceEngineandAnalyticalEnginewerenevercom-
pletedbecausetheirexecutioninvolvedseverepracticalissues,includinginsurmountable
engineeringobstacles,lackoffunding,andgeneralridicule(Babbage’sstoryshouldserve
asafairwarningtoalltheintrepidinventorswhocomewithideascenturiesaheadoftheir
time). Severaldecadeslater,theAmericaninventorHollerithwasmuchmoresuccessful.
Heusedpunchedcardstostoredatareadablebyatabulatormachine;IBMcantraceits
originstothesehumblecards.
Inthelate1800stoearly1900s,analogcomputers,suchastide-predictingmachines,using
physicalphenomenatomodeltheproblemtheywerebuilttosolve,becamealltherage.
Specifically,ananalogcomputer’sdevelopersmustfindphysicalprocessesgovernedbythe
sameorsimilarequationsastheproblemofactualinterest.Since,bydesign,suchcomputers
usenoisycontinuousvaluessubjecttovariouserrors,theyproduceapproximatesolutions.
However, such solutions are highly beneficial in many applications, including warfare,
navigation,andeconomics,tomentionbutafew. Oneofthemoreremarkableexamples
ofanalogcomputersisthePhillipsHydraulicComputer,usingsimpletoolstocontrolthe
water’sflowtomodelthenationaleconomyoftheUnitedKingdomwitha2%accuracy.
Othersarethebattleship’sfire-controlsystemsandbombsights. Shortlybeforeandduring
WorldWarII,digitalcomputersevolvedintoformidablecompetitorsofanalogcomputers
and,bythe1950s,replacedthemaltogether,exceptforsomehighlyspecializedapplications.
StemmingfromtheoreticalinsightsduetoTuring,whoinventedthecelebratedeponymous
Turing Machine, and several other pioneers and practical engineering inventions, such
asvacuumtubesandtransistors,digitalcomputersconqueredtheworld. Initially,digital
computers,suchastheAnanasov-Berrycomputer,theColossus,andtheENIAC,tomention
butafew,wereenormousmachinesbuilttoservemilitaryneeds,includingnuclearweapons
design and cryptography. Eventually, computers found a wide plethora of commercial
applications.
Moderndigitalcomputersstorethenecessarydatainamagneticmemoryintheformof
0-1bits;theyoperateonthisdatavialogicalgates. Thefactthatthedataisdigitisedhas
profoundimplications. Ontheonehand,whileundeniablyhumongous,theamountofdata
acomputercanstoreandthespeedatwhichitcanprocessthisdataarelimited. Onthe
otherhand,inevitableerrorsinherentinanyphysicaldevicearerelativelyeasytocontrol.
Quantum computers were independently conceived in the early 1980s by Benioff, who
proposed a quantum version of the Turing Machine, Feynman, and Manin. Feynman,
withhisusualeloquence,stated: “Natureisn’tclassical,dammit,andifyouwanttomake
a simulation of nature, you’d better make it quantum mechanical, and by golly, it’s a
wonderfulproblembecauseitdoesn’tlooksoeasy”. Quantumcomputersusequantum
mechanical phenomena to solve computational problems that conventional computers
cannot solve efficiently (or at all). To build a working quantum computer, one must
overcometremendousdifficultiesandachieveconditionssuchthatquantumeffectsbecome
dominant or, at least, noticeable. For example, such a computer must operate at low
temperaturesclosetoabsolutezero. Theprimarydistinctionbetweenquantumcomputers
and their classical brethren is how they store and operate on the data. Instead of the
0-1bitsusedbydigitalcomputers,quantumcomputersusetheso-calledqubits,capable
ofstoringcontinuousinformationrepresentedbyapointontheso-calledBlochsphere;
however, once measured, the data collapse to the classical 0-1 state. In theory, qubits
shouldbeentangledsothattheirquantumstatesdependonthestateofallotherqubits.
Thedata,storedbyaquantumcomputer,isprocessedviaquantumgates. Sincethedata
is continuous rather than discrete, the storage problem is not an issue per se; however,
the continuity of the data makes quantum computers prone to inevitable errors. As a
result,fault-tolerantquantumcomputingwithsuitablymanyqubitsremainsaverydistant
possibility. Furthermore,buildinganactualqubitisverydifficult. Differentpossibilities
havebeentried,suchasJosephson’sjunctions,trappedions,andmanyothers.
Quantumcomputinghasnumerousexcitingprospectivegoals,althoughitisunclearifall
(orany)ofthemcanbereached. Thebestknowniscryptography,moreconcretely,factoris-
inghugeintegersintoprimefactorsusingShor’salgorithm,thusbreakingconventional
asymmetricencryption. ItisworthmentioningthatShor’salgorithmisprobabilistic;hence,
itisparticularlywelladaptedtoquantumcomputers. Thepromiseofachievingthefactori-
sationbreakthroughgalvanisedgovernmentsandprivatecompaniesintopouringbillions
ofdollarsintoquantumcomputingandrelatedfields. Forinstance, theUSgovernment
launchedtheNationalQuantumInitiativetoexploreandpromoteQuantumInformation
Science. In addition, the National Institute of Standards and Technology evaluates and
standardizesquantum-resistantpublic-keycryptographicalgorithms. However,thebiggest
numberreliablyfactoredbyShor’salgorithmis35,factoredin2021usingacomputerwith
veryfewqubits. Anotherexcitingapplicationistosearchproblems,whichcanbeefficiently
handledwithGrover’salgorithm,althoughpracticalimplementationsarelacking. Several
researchgroups,suchastheGoogle-NASAcollaboration,activelystudymachinelearning
applications. Nevertheless,large-scaleapplicationsarestillfarfrombeingconquered.
ThissituationbringsusbacktotheJacquierandKondratyevbook. Giventherelatively
slowprogressachievedinquantumcomputation,itisnaturaltogobacktotheearlydaysof
moderncomputinghardwareandseeifonecanuseaquantum-likecomputerasamodern
analogcomputer;thisispreciselywhatJacquierandKondratyevdo. Specifically,rather
thanwaitingforquantumcomputerstoreachtheirtopform,theyadvocateusingquantum
annealingcomputersandsimilaranalogmachinestoperformtherequiredcalculations.
SuchcomputersstartwithasimpleHamiltonianwithaknowngroundstateandslowly(adi-
abatically)evolvetheoriginalHamiltonianintotheHamiltonianofinterestwhilealways
preservingthegroundstate. Thus,whentheprocessiscompleted,themeasurementallows
onetofindthegroundstatefortheactualproblemratherthanasimplifiedone. Jacquier
and Kondratyev use analog quantum computers to solve several exciting mathematical
financeproblemswithverveandpanache. Specifically,theydiscussquantumboostingand
demonstratehowtoapplyittopredictcreditcarddefaultsandsolveclassificationprob-
lems. Subsequently,JacquierandKondratyevturntheirattentiontoquantumBoltzmann
machinesandexplainhowtousethemfordistributionsampling.
Analog quantum computers can be used to solve many financial engineeringproblems.
For instance, a quantum annealer is a perfect tool for solving the multi-period integer
portfoliooptimisationproblem,whichisNP-Complete. MonteCarlosimulations,which
arefundamentalforderivativespricingandotherrelatedtasks,areparticularlynatural
toperformusingquantumcomputerssincetheycanbeviewedastruerandomnumber
generators.
Whenitcomestothedigitalgatemodelquantumcomputing, JacquierandKondratyev
discussquantumneuralnetworksandtheirapplicationstomachinelearning. Later,the
authorscoverQuantumCircuitBornMachine,VariationalQuantumEigensolver,andthe
QuantumApproximateOptimisationAlgorithm. Finally,JacquierandKondratyevdiscuss
newquantumalgorithms,suchasquantumkernels,BayesianQuantumCircuit,Quantum
FourierTransform,andQuantumMonteCarloSimulation.
JacquierandKondratyevareexcitedaboutquantumalgorithmsandtheirpotentialappli-
cations;however,theyarenotstarry-eyedandapproachquantumalgorithmscautiously.
Specifically,theyputmucheffortintoshowinghowclassicalalgorithmscansolveproblems
theyareinterestedinandwhenquantumalgorithmsoutperformtheirclassicalbrethren.
WhilebuildingquantumcomputersoriginallyenvisionedbyFeynmanandManinmight
stillbedecadesaway,morepracticalanalogquantumcomputersalreadyexist. Experience
suggests that finance is one of the fields where breakthroughs in computing tend to be
usedinreal-time. Quantswhowanttousequantumalgorithmsintheirday-to-daywork
couldscarcelydobetterthanstartingtheirjourneybystudyingthisbook.
Bonvoyage!
AlexanderLiptonandMarcosLópezdePrado,
AbuDhabiInvestmentAuthority