Foundations of Quantum Programming

352 Pages·2016·5.58 MB·English

by  Mingsheng Ying

Foundations of
Quantum
Programming
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Foundations of
Quantum
Programming
Mingsheng Ying
UniversityofTechnologySydneyandTsinghuaUniversity
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Preface
“Perhapsthequantumcomputerwillchangeoureverydaylivesinthiscentury
inthesameradicalwayastheclassicalcomputerdidinthelastcentury.”
—excerptfrompressrelease,NobelPrizeinPhysics2012.
Quantum computers promise dramatic advantages over current computers.
Governments and industries around the globe are now investing large amounts
of money with the expectation of building practical quantum computers. Recent
rapid physical experimental progress has made people widely expect that large-
scalableandfunctionalquantumcomputerhardwarewillbebuiltwithin10–20years.
However, to realize the super-power of quantum computing, quantum hardware is
obviouslynotenough,andquantumsoftwaremustalsoplayakeyrole.Thesoftware
development techniques used today cannot be applied to quantum computers.
Essentialdifferencesbetweenthenatureoftheclassicalworldandthatofthequan-
tumworldmeanthatnewtechnologiesarerequiredtoprogramquantumcomputers.
Research on quantum programming started as early as 1996, and rich results
have been presented at various conferences or reported in various journals in the
last20years.Ontheotherhand,quantumprogrammingisstillaprematuresubject,
with its knowledge base being highly fragmentary and disconnected. This book is
intendedto providea systematic and detailed expositionof the subjectof quantum
programming.
Sincequantumprogrammingisstillanareaunderdevelopment,thebookdoesnot
focus on specific quantum programming languages or techniques, which I believe
will undergo major changes in the future. Instead, the emphasis is placed on the
foundational concepts, methods and mathematical tools that can be widely used
forvariouslanguagesandtechniques.Startingfroma basic knowledgeof quantum
mechanicsandquantumcomputation,thebookcarefullyintroducesvariousquantum
programconstructsandachainofquantumprogrammingmodelsthatcaneffectively
exploit the unique power of quantum computers. Furthermore, semantics, logics,
and verification and analysis techniques of quantum programs are systematically
discussed.
Withthehugeinvestmentandrapidprogressinquantumcomputingtechnology,
Ibelievethatwithin10yearsmoreandmoreresearcherswillentertheexcitingfield
of quantumprogramming.Theywill need a referencebookas the starting pointof
theirresearch.Also,a courseonquantumprogrammingwillbe taughtatmoreand
moreuniversities.Teachersandstudentswillneedatextbook.So,Idecidedtowrite
thisbookwiththetwo-foldaim:
(i) providingabasisforfurtherresearchinthearea;and
(ii) servingasatextbookforagraduateoradvancedundergraduatelevelcourse.
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x Preface
Quantum programming is a highly interdisciplinary subject. A newcomer and, in
particular, a student is usually frustrated with the requisite knowledge from many
different subjects. I have tried to keep the book as self-contained as possible,
with details being explicitly presented so that it is accessible to the programming
languagescommunity.
Writing this book gave me an opportunity to systemize my views on quantum
programming. On the other hand, topics included in this book were selected and
thematerialswereorganizedaccordingtomyownunderstandingofthissubject,and
severalimportanttopicswereomittedinthemainbodyofthebookduetomylimited
knowledgeaboutthem. As a remedy,some briefdiscussionsaboutthese topicsare
providedintheprospectschapterattheendofthebook.
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Acknowledgments
This book has been developed through my research in the last 15 years at the
QuantumComputationandQuantumInformationGroupoftheStateKeyLaboratory
of Intelligent Technology and Systems, Tsinghua University and the Quantum
Computation Laboratory of the Centre for Quantum Computation and Intelligent
Systems,UniversityofTechnologySydney.Ihaveenjoyedverymuchcollaborations
and discussions with my colleagues and students there. I would like to thank
allofthem.
IamparticularlyindebtedtoIchiroHasuo(UniversityofTokyo)andYuanFeng
(University of Technology Sydney) who patiently read the draft of this book and
kindly provided invaluable comments and suggestions. I am very grateful to the
anonymous reviewers for the book proposal; their suggestions were very helpful
for the structure of the book. I also would like to sincerely thank Steve Elliot,
Punithavathy Govindaradjane, Amy Invernizzi, and Lindsay Lawrence, my editors
andprojectmanagersatMorganKaufmann.
Special thanks go to the Centre for Quantum Computation and Intelligent Sys-
tems,FacultyofEngineeringandInformationTechnology,UniversityofTechnology
Sydneyforgivingmethefreedomtopursuemythoughts.
My research on quantum programming has been supported by the Australian
Research Council, the National Natural Science Foundation of China, and the
Overseas Team Program of the Academy of Mathematics and Systems Science,
ChineseAcademyofSciences.Allofthemaregratefullyacknowledged.
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CHAPTER
1
Introduction
“The challenge [of quantum software engineering] is to rework and extend the
wholeofclassicalsoftwareengineeringintothequantumdomainsothatprogram-
merscanmanipulatequantumprogramswiththesameeaseandconfidencethat
theymanipulatetoday’sclassicalprograms.”
excerptfromthe2004reportGrandChallenges
inComputingResearch[120].
Quantumprogrammingisthestudyofhowtoprogramfuturequantumcomput-
ers.Thissubjectmainlyaddressesthefollowingtwoproblems:
• Howcanprogrammingmethodologiesandtechnologiesdevelopedforcurrent
computersbeextendedforquantumcomputers?
(cid:129) Whatkindsofnewprogrammingmethodologiesandtechnologiescan
effectivelyexploittheuniquepowerofquantumcomputing?
Many technologiesthat have been very successful in traditional programmingwill
be broken when used to program a quantum computer, due to the weird nature of
quantumsystems(e.g.,nocloningofquantumdata,entanglementbetweenquantum
processes, and non-commutativity of observables which are all assertions about
program variables). Even more importantand difficult is to discover programming
paradigms, models and abstractions that can properly exploit the unique power of
quantumcomputing–quantumparallelism–butcannotbesourcedfromknowledge
oftraditionalprogramming.
1.1 BRIEF HISTORY OF QUANTUM PROGRAMMING
RESEARCH
The earliest proposalfor quantumprogrammingwas made by Knillin 1996[139].
HeintroducedtheQuantumRandomAccessMachine(QRAM)modelandproposed
a set of conventionsfor writing quantum pseudo-code. In the 20 years since then,
researchonquantumprogramminghasbeencontinuouslyconducted,mainlyinthe
followingdirections.
FoundationsofQuantumProgramming.http://dx.doi.org/10.1016/B978-0-12-802306-8.00001-X 3
Copyright©2016ElsevierInc.Allrightsreserved.
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4 CHAPTER 1 Introduction
1.1.1 DESIGN OF QUANTUM PROGRAMMING LANGUAGES
Early research on quantum programming focused on the design of quantum pro-
gramming languages. Several high-level quantum programming languages have
been defined in the later 1990s and early 2000s; for example, the first quantum
programminglanguage,QCL,wasdesignedbyÖmer[177],whoalsoimplemented
a simulator for this language. A quantum programming language in the style
of Dijkstra’s guarded-command language, qGCL, was proposed by Sanders and
Zuliani[191,241].AquantumextensionofC++wasproposedbyBettellietal.[39],
and implemented in the form of a C++ library. The first quantum language of the
functional programming paradigm, QPL, was defined by Selinger [194] based on
theideaofclassicalcontrolandquantumdata.Aquantumfunctionalprogramming
language QML with quantum control flows was introduced by Altenkirch and
Grattage [14]. Tafliovich and Hehner [208,209] defined a quantum extension of a
predicativeprogramminglanguagethatsupportstheprogramdevelopmenttechnique
inwhicheachprogrammingstepisprovencorrectwhenitismade.
Recently, two general-purpose, scalable quantum programming languages,
Quipper and Scaffold, with compilers, were developed by Green et al. [106] and
Abhariet al. [3], respectively.A domain-specificquantum programminglanguage,
QuaFL, was developed by Lapets et al. [150]. A quantum software architecture
LIQUi|>, together with a quantum programming language embedded in F#, was
designedandimplementedbyWeckerandSvore[215].
1.1.2 SEMANTICS OF QUANTUM PROGRAMMINGLANGUAGES
Formalsemanticsofaprogramminglanguagegivearigorousmathematicaldescrip-
tion of the meaning of this language, to enable a precise and deep understanding
of the essence of the language beneath its syntax. The operational or denotational
semantics of some quantum programming languages were already provided when
theyweredefined;forexample,qGCL,QPLandQML.
Two approachesto predicate transformer semantics of quantum programshave
been proposed. The first was adopted by Sanders and Zuliani [191] in designing
qGCL, where quantumcomputationis reducedto probabilistic computationby the
observation (measurement) procedure, and thus predicate transformer semantics
developed for probabilistic programs can be applied to quantum programs. The
secondwasintroducedbyD’HondtandPanangaden[70],whereaquantumpredicate
is defined to be a physical observable represented by a Hermitian operator with
eigenvalueswithin the unit interval. Quantum predicate transformersemantics was
further developed in [225] with a special class of quantum predicates, namely
projection operators. Focusing on projective predicates allows the use of rich
mathematical methods developed in Birkhoff-von Neumann quantum logic [42] to
establishvarioushealthinessconditionsofquantumprograms.
Semantic techniques for quantum computation have also been investigated in
some abstract, language-independent ways. Abramsky and Coeck [5] proposed a
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1.1 Brief history ofquantumprogramming research 5
category-theoreticformulationofthebasicpostulatesofquantummechanics,which
canbeusedtogiveanelegantdescriptionofquantumprogramsandcommunication
protocolssuchasteleportation.
Recentprogressincludes:HasuoandHoshino[115] founda semanticmodelof
afunctionalquantumprogramminglanguagewithrecursionviaGirard’sGeometry
of Interaction [101], categorically formulated by Abramsky, Haghverdi and Scott
[7]. Pagani, Selinger and Valiron [178] discovered a denotational semantics for
a functional quantum programming language with recursion and an infinite data
type using constructions from quantitative semantics of linear logic. Jacobs [123]
proposedacategoricalaxiomatizationofblockconstructsinquantumprogramming.
Staton [206] presented an algebraic semantic framework for equational reasoning
aboutquantumprograms.
1.1.3 VERIFICATION AND ANALYSIS OF QUANTUM PROGRAMS
Human intuition is much better adapted to the classical world than the quantum
world.Thisfactimpliesthatprogrammerswillcommitmanymorefaultsindesigning
programs for quantum computers than in programmingclassical computers. Thus,
it is crucial to develop verification techniques for quantum programs. Baltag and
Smets [30] presented a dynamic logic formalism of information flows in quantum
systems. Brunet and Jorrand [50] introduced a way of applying Birkhoff-von
Neumannquantumlogicinreasoningaboutquantumprograms.Chadha,Mateusand
Sernadas[52]proposedaproofsystemoftheFloyd-Hoarestyleforreasoningabout
imperativequantumprogramsin whichonly boundediterationsare allowed.Some
useful proof rules for reasoning about quantum programs were proposed by Feng
et al. [82] for purely quantum programs. A Floyd-Hoare logic for both partial and
total correctness of quantumprogramswith (relative) completenesswas developed
in[221].
Program analysis techniques are very useful in the implementation and opti-
mization of programs. Termination analysis of quantum programs was initiated in
[227], where a measurement-based quantum loop with a unitary transformation as
theloopbodywas considered.Terminationofa moregeneralquantumloopwith a
quantumoperationastheloopbodywasstudiedin[234]usingthesemanticmodel
of quantum Markov chains. It was also shown in [234] that the Sharir-Pnueli-Hart
method for proving properties of probabilistic programs [202] can be elegantly
generalizedtoquantumprogramsbyexploitingtheSchrödinger-Heisenbergduality
between quantum states and observables. This line of research has been continued
in [152,153,235,236,238] where termination of nondeterministic and concurrent
quantum programs was investigated based on reachability analysis of quantum
Markov decision processes. Another line of research in quantum program analysis
was initiatedby Jorrandand Perdrix[129] who showedhowabstractinterpretation
techniquescanbeusedinquantumprograms.
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