Table Of ContentQuantum Science and Technology
Maria Schuld
Francesco Petruccione
Machine
Learning
with Quantum
Computers
Second Edition
Quantum Science and Technology
SeriesEditors
RaymondLaflamme,Waterloo,ON,Canada
DanielLidar,LosAngeles,CA,USA
ArnoRauschenbeutel,ViennaUniversityofTechnology,Vienna,Austria
RenatoRenner,InstitutfürTheoretischePhysik,ETHZürich,Zürich,Switzerland
JingboWang,DepartmentofPhysics,UniversityofWesternAustralia,Crawley,
WA,Australia
YaakovS.Weinstein,QuantumInformationScienceGroup,TheMITRE
Corporation,Princeton,NJ,USA
H.M.Wiseman,Brisbane,QLD,Australia
SectionEditor
MaximilianSchlosshauer,DepartmentofPhysics,UniversityofPortland,Portland,
OR,USA
The book series Quantum Science and Technology is dedicated to one of today’s
mostactiveandrapidlyexpandingfieldsofresearchanddevelopment.Inparticular,
the series will be a showcase for the growing number of experimental implemen-
tations and practical applications of quantum systems. These will include, but are
not restricted to: quantum information processing, quantum computing, and
quantum simulation; quantum communication and quantum cryptography; entan-
glement and other quantum resources; quantum interfaces and hybrid quantum
systems; quantum memories and quantum repeaters; measurement-based quantum
control and quantum feedback; quantum nanomechanics, quantum optomechanics
and quantum transducers; quantum sensing and quantum metrology; as well as
quantumeffectsinbiology.Lastbutnotleast,theserieswillincludebooksonthe
theoretical and mathematical questions relevant to designing and understanding
these systems and devices, as well as foundational issues concerning the quantum
phenomena themselves. Written and edited by leading experts, the treatments will
be designed for graduate students and other researchers already working in, or
intendingtoenterthefieldofquantumscienceandtechnology.
Moreinformationaboutthisseriesathttp://www.springer.com/series/10039
·
Maria Schuld Francesco Petruccione
Machine Learning
with Quantum Computers
Second Edition
MariaSchuld FrancescoPetruccione
XanaduQuantumComputingInc. SchoolofChemistryandPhysics
Toronto,ON,Canada UniversityofKwaZulu-Natal
Durban,SouthAfrica
ISSN2364-9054 ISSN2364-9062 (electronic)
QuantumScienceandTechnology
ISBN978-3-030-83097-7 ISBN978-3-030-83098-4 (eBook)
https://doi.org/10.1007/978-3-030-83098-4
Originallypublishedwiththetitle:SupervisedLearningwithQuantumComputers
1stedition:©SpringerNatureSwitzerlandAG2018
2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SwitzerlandAG2021
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
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DedicatedtoPeterWittek,whowassupposed
tobeaco-authorofthisedition.
Preface to the Second Edition
Much has happened in the 3 years since the first edition of this book. Quantum
machinelearninghaswitnessedatremendoussurgeinpopularity,anditsvocabulary
is becoming increasingly known in mainstream quantum computing. Variational
circuits, machine learning models derived from quantum circuits that depend on
adaptable classical“control” parameters, have become acentralfocus ofresearch.
The training of such “quantum models” is facilitated by software libraries such
asPennyLane,TensorFlowQuantum andYao,whichprovidesimulatorsandcloud
accesstoquantumhardware.Inotherwords,quantummachinelearningisoneofthe
firstfieldsthatisnotonlydoneonpaperbutalsotestedonrealdevices.Finally,alot
ofprogresshasbeenmadeinourtheoreticalunderstandingofwhathappenswhen
quantumcomputerslearnfromdata—includingquestionsoftheirtrainabilitywith
thegradientdescent-typealgorithmsthatareubiquitousinclassicalmachinelearning,
theirproximitytokernelmethodsthatlearninhigh-dimensionalfeaturespacesand
thenatureoftheseparationbetweenclassicalandquantummachinelearning.This
editionthereforepresentsalotofnewmaterial,whiledroppingsectionsthatdidnot
stand the test of time. It also includes perspectives on unsupervised learning and
generativemodels,whichexplainsthechangeofthetitle.
We want to thank Amira Abbas, Betony Adams and Daniel Park for proof-
reading, and our readers for their kind feedback—in particular, Pranav Gokhale
forhisunwaveringsupportinspottingtypos.
Durban,SouthAfrica MariaSchuld
April2021 FrancescoPetruccione
vii
Preface to the First Edition
Quantum machine learning is a subject in the making, faced by huge expectations
duetoitsparentdisciplines.Ontheonehandthereisaboomingcommercialinterest
in quantum technologies, which are at the critical point of becoming available for
theimplementationofquantumalgorithms,andwhichhaveexceededtherealmofa
purelyacademicinterest.Ontheotherhand,machinelearningalongwithartificial
intelligenceisadvertisedasacentral(ifnotthecentral)futuretechnologyintowhich
companiesareboundtoinvesttoavoidbeingleftout.Combiningthesetwoworlds
invariablyleadstoanoverwhelminginterestinquantummachinelearningfromthe
ITindustry,aninterestthatisnotalwaysmatchedbythescientificchallengesthat
researchersareonlybeginningtoexplore.
To find out what quantum machine learning has to offer, its numerous possible
avenues first have to be explored by an interdisciplinary community of scientists.
Weintendthisbooktobeapossiblestartingpointforthisjourney,asitintroduces
somekeyconcepts,ideasandalgorithmsthataretheresultofthefirstfewyearsof
quantum machine learning research. Given the young nature of the discipline, we
expectalotofnewanglestobeaddedtothiscollectioninduetime.Ouraimisnot
to provide a comprehensive literature review, but rather to summarise themes that
repeatedlyappearinquantummachinelearning,toputthemintocontextandmake
themaccessibletoabroaderaudienceinordertofosterfutureresearch.
On the highest level, we target readers with a background in either physics or
computer science that have a sound understanding of linear algebra and computer
algorithms.Havingsaidthat,quantummechanicsisafieldbasedonadvancedmath-
ematicaltheory(anditdoesbynomeanshelpwithasimplephysicalintuitioneither),
andtheseaccessbarriersaredifficulttocircumventevenwiththemostwell-intended
introductiontoquantummechanics.Noteverysectionisthereforeeasytounderstand
for readers without experience in quantum computing. However, we hope that the
main concepts are within reach and try to give higher level overviews where ever
possible.
ix
x PrefacetotheFirstEdition
WethankoureditorsAldoRampioniandKirstenTheunissenfortheirsupportand
patience.Ourthanksalsogotoanumberofcolleaguesandfriendswhohavehelped
to discuss, inspire and proofread the book (in alphabetical order): Betony Adams,
MarcelloBenedetti,GianGiacomoGuerreschi,VinayakJagadish,NathanKilloran,
Camille Lombard Latune, Andrea Skolik, Ryan Sweke, Peter Wittek and Leonard
Wossnig.
Durban,SouthAfrica MariaSchuld
March2018 FrancescoPetruccione
Contents
1 Introduction ................................................... 1
1.1 Background ............................................... 2
1.1.1 MergingTwoDisciplines .............................. 2
1.1.2 TheRiseofQuantumMachineLearning ................ 5
1.1.3 FourIntersections .................................... 6
1.1.4 Fault-TolerantVersusNear-TermApproaches ............ 7
1.2 AToyExampleofaQuantumAlgorithmforClassification ....... 9
1.2.1 TheSquared-DistanceClassifier ........................ 10
1.2.2 InterferencewiththeHadamardTransformation .......... 11
1.2.3 QuantumSquared-DistanceClassifier ................... 15
1.2.4 InsightsfromtheToyExample ......................... 18
1.2.5 OrganisationoftheBook .............................. 19
References ..................................................... 20
2 MachineLearning .............................................. 23
2.1 ExamplesofTypicalMachineLearningProblems ............... 24
2.2 TheThreeIngredientsofaLearningProblem ................... 27
2.2.1 Data ............................................... 29
2.2.2 Model .............................................. 32
2.2.3 Loss ............................................... 36
2.3 RiskMinimisationinSupervisedLearning ..................... 37
2.3.1 MinimisingtheEmpiricalRiskasaProxy ............... 38
2.3.2 QuantifyingGeneralisation ............................ 40
2.3.3 Optimisation ........................................ 42
2.4 TraininginUnsupervisedLearning ............................ 44
2.5 MethodsinMachineLearning ................................ 47
2.5.1 LinearModels ....................................... 47
2.5.2 NeuralNetworks ..................................... 51
2.5.3 GraphicalModels .................................... 62
2.5.4 KernelMethods ...................................... 66
References ..................................................... 76
xi
