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AlexisDeVos: ReversibleComputing — 2010/8/5 — 13:36 — page I — le-tex
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AlexisDeVos
ReversibleComputing
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AlexisDeVos: ReversibleComputing — 2010/8/5 — 13:36 — page II — le-tex
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AlexisDeVos: ReversibleComputing — 2010/8/5 — 13:36 — page III — le-tex
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Alexis De Vos
Reversible Computing
Fundamentals, Quantum Computing, and Applications
WILEY-VCH Verlag GmbH & Co. KGaA
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TheAuthor AllbookspublishedbyWiley-VCHarecarefully
produced.Nevertheless,authors,editors,and
Prof.Dr.AlexisDeVos publisherdonotwarranttheinformation
UniversiteitGent containedinthesebooks,includingthisbook,to
elektronikaeninformatiesystemen befreeoferrors.Readersareadvisedtokeepin
SintPietersnieuwstraat41 mindthatstatements,data,illustrations,
9000Gent proceduraldetailsorotheritemsmay
Belgium inadvertentlybeinaccurate.
LibraryofCongressCardNo.:appliedfor
BritishLibraryCataloguing-in-PublicationData:
Acataloguerecordforthisbookisavailable
fromtheBritishLibrary.
Bibliographicinformationpublishedbythe
DeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhis
publicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableonthe
Internetathttp://dnb.d-nb.de.
©2010WILEY-VCHVerlagGmbH&Co.KGaA,
Weinheim
Allrightsreserved(includingthoseoftranslation
intootherlanguages).Nopartofthisbookmay
bereproducedinanyform–byphotoprinting,
microfilm,oranyothermeans–nortransmitted
ortranslatedintoamachinelanguagewithout
writtenpermissionfromthepublishers.Regis-
terednames,trademarks,etc.usedinthisbook,
evenwhennotspecificallymarkedassuch,are
nottobeconsideredunprotectedbylaw.
Typesetting le-texpublishingservicesGmbH,
Leipzig
PrintingandBinding FabulousPrinters
PteLtd,Singapore
CoverDesign Formgeber,Eppelheim
PrintedinSingapore
Printedonacid-freepaper
ISBN 978-3-527-40992-1
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AlexisDeVos: ReversibleComputing — 2010/8/5 — 13:36 — page V — le-tex
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V
Contents
Preface IX
Introduction 1
1 BooleanAlgebra 5
1.1 BooleanFunctionsofOneVariable 5
1.2 BooleanFunctionsofTwoVariables 6
1.3 BooleanFunctionsofnVariables 8
1.3.1 TheMintermExpansion 9
1.3.2 TheMaxtermExpansion 9
1.3.3 TheReed–MullerExpansion 10
1.3.4 TheMinimalESOPExpansion 11
1.4 LinearFunctions 11
1.5 AffineLinearFunctions 12
1.6 MonotonicFunctions 13
1.7 BooleanDerivative 13
1.8 BooleanDecompositions 14
1.9 ExercisesforChapter1 15
2 GroupTheory 17
2.1 Introduction 17
2.2 PermutationGroups 19
2.3 MatrixGroups 21
2.4 GroupGenerators 22
2.5 Subgroups 23
2.6 YoungSubgroups 23
2.7 Sylowp-Subgroups 24
2.8 CyclicSubgroups 25
2.9 ClosingRemarks 25
2.10 ExercisesforChapter2 26
3 ReversibleComputing 29
3.1 Introduction 29
3.2 ReversibleCircuits 32
3.3 ConservativeCircuits 33
ReversibleComputing.AlexisDeVos
Copyright©2010WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:978-3-527-40992-1
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VI Contents
3.4 MonotonicCircuits 34
3.5 LinearCircuits 34
3.6 AffineLinearCircuits 36
3.7 ExchangeGates 37
3.8 SWAPGates 39
3.9 AffineExchangeGates 39
3.10 ControlGates 42
3.11 SylowCircuits 47
3.12 GateCostandLogicDepth 49
3.13 MethodsofSynthesis 51
3.14 Cosets 52
3.15 DoubleCosets 55
3.16 TheSynthesisAlgorithm 57
3.16.1 BasicIdeaoftheAlgorithm 58
3.16.2 WorkingOuttheAlgorithm 60
3.16.3 Results 61
3.17 VariableOrdering 64
3.18 Templates 65
3.19 TheLinearSynthesisAlgorithm 69
3.20 PresetBitsandGarbageBits 72
3.20.1 DuplicatingCircuit 72
3.20.2 ControlledNOT 73
3.20.3 AnApplication:theMajorityCircuit 74
3.21 AnotherApplication:theFullAdder 77
3.22 AvoidingGarbage 80
3.23 ExercisesforChapter3 82
4 Low-powerComputing 85
4.1 Entropy 85
4.2 PermutationMatrices 88
4.3 Landauer’sTheorem 89
4.4 Thermodynamics 90
4.5 AnApplication:PrototypeChips 93
4.6 SwitchCost 101
4.7 Moore’sLaw 103
4.8 Quasi-adiabaticAddressing 105
4.9 ExercisesforChapter4 110
5 AnalogComputing 113
5.1 ComputingwithRealNumbers 113
5.2 Synthesis 115
5.3 AnApplication:theColorTransform 117
5.4 AboutDeterminants 119
5.5 LIFTGatesversusSCALEGates 120
5.6 Conclusion 122
5.7 ComputationswithComplexNumbers 123
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Contents VII
5.8 AnApplication:theFourierTransform 124
5.9 NonlinearComputations 126
5.10 ExercisesforChapter5 130
6 ComputinginModulo2b 131
6.1 AdditioninModulo2b 131
6.2 MultiplicationbyaConstant 135
6.3 Scalingby(cid:1)1 136
6.4 Scalingby1/2orby2 137
6.5 Lifting 139
6.6 ExercisesforChapter6 141
7 QuantumComputing 143
7.1 DoublyStochasticMatrices 143
7.2 UnitaryMatrices 144
7.3 EntropyintheQuantumWorld 147
7.4 Entanglement 148
7.5 ControlCircuitsandControlGates 149
7.6 Synthesis 151
7.7 Decomposition 152
7.8 Discussion 154
7.9 Bottom-UpandTop-Down 156
7.10 Bottom-UpApproach 157
7.10.1 One-(Qu)bitCalculations 157
7.10.2 Two-(Qu)bitCalculations 159
7.10.3 Three-andMulti-(Qu)bitCalculations 161
7.11 Top-DownApproach 162
7.12 AnApplication:theQuantumFourierTransform 163
7.13 NonlinearComputations 166
7.14 ExercisesforChapter7 167
8 ReversibleProgrammingLanguages 169
8.1 Theif-then-elseStructure 170
8.2 Thedo-untilStructure 175
8.3 ExercisesforChapter8 177
AppendixA TheNumberofLinearReversibleCircuits 181
AppendixB Boundsfortheq-Factorial 183
AppendixC ATheoremaboutUniversalReversibleGates 185
C.1 UniversalityinConventionalLogicCircuits 185
C.2 PreliminaryConcepts 186
C.3 No-Fan-OutTheoremandItsConsequences 187
C.4 FinalTheorem 190
C.5 Discussion 191
C.6 ExercisesforAppendixC 192
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VIII Contents
AppendixD SynthesisEfficiency 193
D.1 ExercisesforAppendixD 195
AppendixE Birkhoff’sTheorem 197
E.1 ExercisesforAppendixE 201
AppendixF MicroentropyandMacroentropy 205
F.1 ExercisesforAppendixF 207
AppendixG ComputingtheSuccessivePowersofaMatrix 209
G.1 ExercisesforAppendixG 210
PostScriptum 211
References 213
SolutionstotheExercises 223
Index 247
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Description:Written by one of the few top internationally recognized experts in the field, this book concentrates on those topics that will remain fundamental, such as low power computing, reversible programming languages, and applications in thermodynamics. It describes reversible computing from various points
