Elements of Quantum Computing: History, Theories and Engineering Applications

133 Pages·2014·2.689 MB·English

by  Seiki Akama

Seiki Akama
Elements
of
Quantum
Computing
HISTORY,
THEORIES
AND ENGINEERING
APPLICATIONS
123
Elements of Quantum Computing
Seiki Akama
Elements of Quantum
Computing
History, Theories and Engineering
Applications
ABC
SeikiAkama
Kawasaki
Japan
ISBN978-3-319-08283-7 ISBN978-3-319-08284-4 (eBook)
DOI10.1007/978-3-319-08284-4
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014942264
c SpringerInternationalPublishingSwitzerland2015
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Preface
Now,computertechnologyisrapidlydeveloped.Itappearsthatthecomputerworld
iscompletelyestablished.Certainly,thefactmaybetrueaccordingtocurrentthink-
ingbutthereseemstobearoomforfurtherdevelopments.Thisisbecausewehave
somepossibilitiestorealizeacomputerfromadifferentpointofview.
One of such possibilities is the so-called quantum computer. It is a computer
based on quantum mechanics, and it can do super speed computation. It is well
known that theoretical possibilities of quantum computer have been argued since
thebeginningofthe1980’s.
There has been a growinginterest in the 1990’s,and some quantumcomputers
attheexperimentallevelwererecentlyimplemented.However,mostpeopledonot
appeartobefamiliarwiththefieldofquantumcomputer.
Itisalsotrueforexpertsworkingoncomputerscience.Thereasonisthatwecan-
notunderstandquantumcomputerswithoutthebackgroundofquantummechanics.
Itcanbealsopointedoutthatthefoundationsforquantumcomputerarenotalways
obvious.
Webelievethatonemuststudythebasicsofquantumcomputingsystematically.
Unfortunately,itisdifficulttoreadmostpapers,includingsurveypapers,onquan-
tumcomputing.Ifthereisareadabletextbookonthesubject,manypeoplewillbe
abletounderstand(orstudy)quantumcomputing.
Thepurposeofthisbookistoprovideaquickintroductiontoquantumcomput-
ing for the readers who have no backgrounds of both theory of computation and
quantummechanics.
Buttounderstandquantumcomputing,weneedtheknowledgeofbothcomputer
scienceandquantummechanics.Onecannoteasilylearnrelevantmaterialsinboth
fields.Thus,wegiveanintuitiveexplanationofthematerial,andthenaddtheoretical
justification.
Thestructureofthebookisasfollows.
Chapter1givesaquickreviewofquantumcomputingandquantummechanics.
First, we outline a quantumcomputerto understandquantumcomputing.Second,
VI Preface
webrieflyexplaintheideasofquantummechanics.Wealsodescribethehistoryof
bothquantumcomputingandquantummechanics.
Chapter2surveyscurrentcomputermodelsincludingNeumann-typecomputer,
Turing Machine and Boolean algebra. Before looking at quantum computers, we
havetounderstandthesemodelsproperly.
Chapter 3 introduces quantum mechanics. After looking at the basic ideas of
quantummechanics,wavemechanics,matrixmechanics,andtheuncertaintyprin-
cipleareexplained.Sincethematerialsinthischapteraremathematicallyadvanced,
thereaderswhoarenotinterestedinthemcanskipandproceedtoChapter4.
Chapter4explainsquantumcomputerinsomedetails.Afterthehistoricalsurvey
of the origins of quantumcomputers, we introduce the quantum Turing Machine.
Qubit,quantumgatesandShor’salgorithmarealsodiscussed.
Chapter 5 outlines the representative applications of quantum computing. We
takeupquantumcodes,quantumcommunications,quantumteleportationandquan-
tumprogramming.
Chapter6concludesthebookwithdiscussingthefutureofquantumcomputing.
In particular,implementationsand importantproblemsof quantumcomputingare
discussed.
Wedonotassumethatthereaderhasanyspecialknowledgeabouttheoryofcom-
putationandquantummechanics.We hopethatthebookisofhelpforthereaders
whowanttounderstandquantumcomputingquickly.
SeikiAkama
May2014
Contents
1 Introduction.................................................. 1
1.1 WhatIsQuantumComputing? .............................. 1
1.1.1 IntuitiveDefinitionofQuantumComputers ............. 1
1.1.2 HistoryofQuantumComputers ....................... 2
1.2 WhatIsQuantumMechanics?............................... 5
1.2.1 ClassicalPhysics ................................... 5
1.2.2 QuantumTheory.................................... 6
2 ModelsofaComputer ......................................... 17
2.1 Neumann-TypeComputer .................................. 17
2.1.1 ENIACandEDVAC................................. 18
2.1.2 ArchitectureofNeumann-TypeComputers.............. 19
2.2 TuringMachine........................................... 20
2.2.1 StructureofTuringMachine.......................... 22
2.2.2 FormalDefinitionofTuringMachine .................. 23
2.3 BooleanAlgebra .......................................... 24
2.3.1 AxiomatizationofBooleanAlgebras................... 25
2.3.2 Truth-ValueTable................................... 27
2.3.3 BooleanAlgebraandHardware ....................... 29
3 QuantumMechanics .......................................... 33
3.1 BasicsofQuantumMechanics............................... 33
3.1.1 History............................................ 33
3.1.2 FromClassicalMechanicstoQuantumMechanics ....... 35
3.2 WaveMechanics .......................................... 35
3.2.1 WaveFunction ..................................... 37
3.2.2 Schro¨dingerEquation................................ 37
3.3 MatrixMechanics ......................................... 38
3.3.1 LinearAlgebra ..................................... 39
3.3.2 Measurement....................................... 42
3.3.3 Bra-KetNotation ................................... 46
VIII Contents
3.4 UncertaintyPrinciple ...................................... 48
3.4.1 StartingPoint ...................................... 48
3.4.2 MathematicalFormulation ........................... 48
3.5 QuantumSystems ......................................... 50
3.5.1 PostulatesofQuantumSystems ....................... 50
3.5.2 SomeFoundationalProblems ......................... 52
4 QuantumComputers.......................................... 57
4.1 OriginsofQuantumComputers.............................. 57
4.1.1 Feynman’sIdeas.................................... 58
4.1.2 SimulatingQuantumMechanics....................... 60
4.2 QuantumTuringMachine .................................. 61
4.2.1 Church-TuringHypothesisandBeyond................. 62
4.2.2 UniversalQuantumComputer ........................ 64
4.2.3 FormalizationofQuantumTuringMachine ............. 66
4.3 Qubit.................................................... 68
4.3.1 Bitvs.Qubit ....................................... 68
4.3.2 PolarizationofPhotons .............................. 70
4.4 QuantumGates ........................................... 71
4.4.1 OutlineofQuantumGates............................ 72
4.4.2 UsefulQuantumGates............................... 73
4.4.3 No-CloningTheorem................................ 82
4.5 Shor’sAlgorithm.......................................... 83
4.5.1 RSACode ......................................... 84
4.5.2 DescriptionofShor’sAlgorithm....................... 84
4.5.3 DiscreteFourierTransformandQuantumAlgorithm...... 85
4.6 OtherQuantumAlgorithms ................................. 87
4.6.1 Grover’sAlgorithm ................................. 87
4.6.2 HSAlgorithm ...................................... 88
4.6.3 QuantumWalks .................................... 88
4.6.4 AdiabaticQuantumAlgorithm ........................ 89
5 ApplicationsofQuantumComputing............................ 91
5.1 QuantumCodes........................................... 91
5.1.1 QuantumKeyDistribution ........................... 91
5.1.2 ExampleofQuantumCode........................... 92
5.2 QuantumCommunications.................................. 93
5.2.1 HolevoBound...................................... 94
5.2.2 QuantumTeleportation .............................. 96
5.3 QuantumProgramming .................................... 97
5.3.1 QCL.............................................. 97
5.3.2 OtherQuantumProgrammingLanguages............... 99
Contents IX
6 FutureofQuantumComputing................................. 101
6.1 ImplementationsofQuantumComputers...................... 101
6.1.1 NMRQuantumComputer............................ 102
6.1.2 TrappedIonQuantumComputer ...................... 103
6.1.3 OtherApproaches................................... 104
6.1.4 DiVincenzo’sFiveCriteria ........................... 105
6.2 ProblemsofQuantumComputing............................ 108
6.2.1 TheoreticalProblems................................ 108
6.2.2 PracticalProblems .................................. 112
References........................................................ 115
Glossary ......................................................... 119
Index ............................................................ 123
Chapter 1
Introduction
Abstract.Chapter1givesaquickreviewofquantumcomputingandquantumme-
chanics.First, we outlinea quantumcomputerto understandquantumcomputing.
Second, we briefly explain the ideas of quantummechanics. We also describe the
historyofbothquantumcomputingandquantummechanics.
1.1 WhatIsQuantum Computing?
Quantumcomputerisacomputerbasedonacomputationalmodelwhichusesquan-
tum mechanics. As you know, quantum mechanics is a subfield of physics which
studies phenomena at the micro level. The area related to quantum computers is
generallycalledthequantumcomputing,whichisregardedasapromisingfieldin
computerscience.
1.1.1 IntuitiveDefinitionofQuantum Computers
Quantum computers do computation based on the theory of quantum mechan-
ics, whose computation is performed very different from traditional ones, e.g.,
Neumann-type computer. Perhaps, the reader may not be able to understand the
description. But, in fact, we now have such computers, although they are experi-
mentallybuilt.
It ishere necessaryforthe readerto have mathematicalbackgroundin orderto
graspthebasicsofquantumcomputers.Webelievethatitisvalidtodefinequantum
computersinthisway,andwewilldefergivingformalexposition.However,itdoes
notseemtobeagoodideaforbeginners.
Becausewecannoteasilylearnanabstracttheoryingeneral,itishelpfultoobtain
intuitiveimageofit.Thiscanbeappliedtothestudyofquantumcomputers.Here,
onewouldliketoimaginethefollowingpicture.
Namely,quantumcomputersrepresentinformationbymakinguseofphenomena
in atoms at the micro level and employ quantum mechanical systems with some
propertiestodocomputations.
S.Akama,ElementsofQuantumComputing, 1
DOI:10.1007/978-3-319-08284-4_1, c SpringerInternationalPublishingSwitzerland2015
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