Table Of ContentQuantum Science and Technology
Maria Schuld · Francesco Petruccione
Supervised
Learning with
Quantum
Computers
Quantum Science and Technology
Series editors
Raymond Laflamme, Waterloo, ON, Canada
Gaby Lenhart, Sophia Antipolis, France
Daniel Lidar, Los Angeles, CA, USA
Arno Rauschenbeutel, Vienna University of Technology, Vienna, Austria
Renato Renner, Institut für Theoretische Physik, ETH Zürich, Zürich, Switzerland
MaximilianSchlosshauer,DepartmentofPhysics,UniversityofPortland,Portland,
OR, USA
Yaakov S. Weinstein, Quantum Information Science Group, The MITRE
Corporation, Princeton, NJ, USA
H. M. Wiseman, Brisbane, QLD, Australia
Aims and Scope
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
quantum effects in biology. Last but not least, the series will include books on the
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
intending to enter the field of quantum science and technology.
More information about this series at http://www.springer.com/series/10039
Maria Schuld Francesco Petruccione
(cid:129)
Supervised Learning
with Quantum Computers
123
Maria Schuld Francesco Petruccione
Schoolof Chemistry andPhysics, Schoolof Chemistry andPhysics
QuantumResearch Group University of KwaZulu-Natal
University of KwaZulu-Natal Durban,SouthAfrica
Durban,SouthAfrica
and
and
National Institute for Theoretical
National Institute for Theoretical Physics (NITheP)
Physics (NITheP) KwaZulu-Natal, SouthAfrica
KwaZulu-Natal, SouthAfrica
and
and
Schoolof Electrical Engineering
Xanadu QuantumComputing Inc Korea AdvancedInstitute of Science
Toronto, Canada andTechnology(KAIST)
Daejeon,Republic of Korea
ISSN 2364-9054 ISSN 2364-9062 (electronic)
QuantumScience andTechnology
ISBN978-3-319-96423-2 ISBN978-3-319-96424-9 (eBook)
https://doi.org/10.1007/978-3-319-96424-9
LibraryofCongressControlNumber:2018950807
©SpringerNatureSwitzerlandAG2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
For Chris and Monique
Preface
Quantum machine learning is a subject in the making, faced by huge expectations
due to its parent disciplines. On the one hand, there is a booming commercial
interest in quantum technologies, which are at the critical point of becoming
availablefortheimplementationofquantumalgorithms,andwhichhaveexceeded
therealmofapurelyacademicinterest.Ontheotherhand,machinelearningalong
with artificial intelligence is advertised as a central (if not the central) future
technology into which companies are bound to invest to avoid being left out.
Combining these two worlds invariably leads to an overwhelming interest in
quantum machine learning from the IT industry, an interest that is not always
matchedbythescientificchallengesthatresearchersareonlybeginningtoexplore.
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
them accessible to a broader audience in order to foster future research.
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. Having said that, quantum mechanics is a field based on advanced
mathematicaltheory(anditdoesbynomeanshelpwithasimplephysicalintuition
either), and these access barriers are difficult to circumvent even with the most
well-intended introduction to quantum mechanics. Not every section is therefore
easytounderstandforreaderswithoutexperienceinquantumcomputing.However,
we hope that the main concepts are within reach and try to give higher level
overviews wherever possible.
vii
viii Preface
We thank our editors Aldo Rampioni and Kirsten Theunissen for their support
and patience. Our thanks also go to a number of colleagues and friends who have
helped to discuss, inspire and proofread the book (in alphabetical order): Betony
Adams, Marcello Benedetti, Gian Giacomo Guerreschi, Vinayak Jagadish, Nathan
Killoran, Camille Lombard Latune, Andrea Skolik, Ryan Sweke, Peter Wittek and
Leonard Wossnig.
Durban, South Africa Maria Schuld
March 2018 Francesco Petruccione
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Merging Two Disciplines. . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 The Rise of Quantum Machine Learning. . . . . . . . . . . . . . 4
1.1.3 Four Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.4 Quantum Computing for Supervised Learning. . . . . . . . . . 6
1.2 How Quantum Computers Can Classify Data. . . . . . . . . . . . . . . . 7
1.2.1 The Squared-Distance Classifier . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Interference with the Hadamard Transformation. . . . . . . . . 9
1.2.3 Quantum Squared-Distance Classifier . . . . . . . . . . . . . . . . 13
1.2.4 Insights from the Toy Example . . . . . . . . . . . . . . . . . . . . 16
1.3 Organisation of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Machine Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 Prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1 Four Examples for Prediction Tasks . . . . . . . . . . . . . . . . . 23
2.1.2 Supervised Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 How Data Leads to a Predictive Model. . . . . . . . . . . . . . . 30
2.2.2 Estimating the Quality of a Model . . . . . . . . . . . . . . . . . . 32
2.2.3 Bayesian Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.4 Kernels and Feature Maps . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Training. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.1 Cost Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3.2 Stochastic Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Methods in Machine Learning. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1 Data Fitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.2 Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . 48
ix
x Contents
2.4.3 Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4.4 Kernel Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3 Quantum Information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1 Introduction to Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.1 What Is Quantum Theory?. . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.2 A First Taste. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.1.3 The Postulates of Quantum Mechanics . . . . . . . . . . . . . . . 84
3.2 Introduction to Quantum Computing . . . . . . . . . . . . . . . . . . . . . . 91
3.2.1 What Is Quantum Computing? . . . . . . . . . . . . . . . . . . . . . 91
3.2.2 Bits and Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.2.3 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2.4 Quantum Parallelism and Function Evaluation. . . . . . . . . . 101
3.3 An Example: The Deutsch-Josza Algorithm. . . . . . . . . . . . . . . . . 103
3.3.1 The Deutsch Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.3.2 The Deutsch-Josza Algorithm. . . . . . . . . . . . . . . . . . . . . . 104
3.3.3 Quantum Annealing and Other Computational Models. . . . 106
3.4 Strategies of Information Encoding . . . . . . . . . . . . . . . . . . . . . . . 108
3.4.1 Basis Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4.2 Amplitude Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.4.3 Qsample Encoding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.4.4 Dynamic Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.5 Important Quantum Routines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.5.1 Grover Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.5.2 Quantum Phase Estimation. . . . . . . . . . . . . . . . . . . . . . . . 117
3.5.3 Matrix Multiplication and Inversion . . . . . . . . . . . . . . . . . 119
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4 Quantum Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.1 Computational Complexity of Learning . . . . . . . . . . . . . . . . . . . . 127
4.2 Sample Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.2.1 Exact Learning from Membership Queries . . . . . . . . . . . . 133
4.2.2 PAC Learning from Examples . . . . . . . . . . . . . . . . . . . . . 134
4.2.3 Introducing Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.3 Model Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Information Encoding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.1 Basis Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 Preparing Superpositions of Inputs . . . . . . . . . . . . . . . . . . 142
5.1.2 Computing in Basis Encoding . . . . . . . . . . . . . . . . . . . . . 145
5.1.3 Sampling from a Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 146
