About the Author x

Preface xi

Acknowledgments xiv

1 Introduction 1

1.1 The New Quantum Age and the Second Quantum Revolution 1

1.2 Distributed Quantum Computing and the Rise of Quantum Internet Computing 4

1.2.1 Distributed Quantum Computing 5

1.2.2 Quantum Internet Computing 8

1.3 Aim and Scope of the Book 9

1.4 Outline of this Book 10

1.5 Related Books and Resources 11

2 Preliminaries 17

2.1 Qubit and Qubit States 17

2.2 Quantum Gates and Quantum Circuits 23

2.2.1 Single Qubit Gates 24

2.2.2 Measurement Operators 26

2.2.2.1 Measurement Postulate (General Measurement) 27

2.2.2.2 POVM 27

2.2.2.3 Projective Measurements 28

2.2.3 Multiple Qubit Gates 29

2.2.4 Quantum Circuits 29

2.2.5 Universal Quantum Computer and Gate Sets 32

2.3 Entanglement 33

2.4 Teleportation and Superdense Coding 39

2.5 Summary 42

2.6 Further Reading and Resources on Quantum Computing 42

3 Distributed Quantum Computing?Classical and Quantum 45

3.1 The Power of Entanglement for Distributed Computing 46

3.1.1 Enabling Distributed Computations with Fewer Bits of Communication 46

3.1.1.1 The Distributed Three-Party Product Problem 47

3.1.1.2 The Distributed Deutsch?Jozsa Promise Problem 49

3.1.1.3 The Distributed Intersection Problem 52

3.1.1.4 Discussion 56

3.1.2 Enabling Distributed Computations Not Possible Classically 56

3.1.2.1 Greenberger?Horne?Zeilinger and Mermin (GHZ&M) Game 56

3.1.2.2 Clauser?Horne?Shimony?Holt (CHSH) 60

3.1.2.3 Discussion 67

3.2 Other Quantum Protocols 67

3.2.1 Quantum Coin Flipping 67

3.2.1.1 Classical Coin Flipping 67

3.2.1.2 Quantum Coin Flipping 74

3.2.2 Quantum Leader Election 77

3.2.3 Quantum Key Distribution (QKD) 79

3.2.4 Quantum Anonymous Broadcasting 90

3.2.5 Quantum Voting 97

3.2.6 Quantum Byzantine Generals Solution 99

3.2.7 Quantum Secret Sharing 104

3.2.8 Quantum Oblivious Transfer (OT) 105

3.2.9 Discussion 107

3.3 Summary 107

4 Distributed Quantum Computing ? Distributed Control of Quantum Gates 113

4.1 Performing a Distributed CNOT 113

4.1.1 Using Teleportation 113

4.1.2 A More Efficient MethodWith Cat-Like States 114

4.2 Beyond the Distributed CNOT 118

4.2.1 Same Control Qubit for Multiple Target Qubits on Different Machines 118

4.2.2 Multiple Control Qubits for the Same Target Qubit on a Different Machine 123

4.2.3 Cat-Entangler and Cat-Disentangler Modules 126

4.3 Distributing Quantum Circuits and Compilation for Distributed Quantum Programs 127

4.4 Control and Scheduling for Distributed Quantum Computers 129

4.5 Distributed Quantum Computing Without Internode Entanglement 132

4.6 Summary 133

5 Delegating Quantum Computations 137

5.1 Delegating Private Quantum Computations 137

5.2 How to Verify Delegated Private Quantum Computations 149

5.2.1 X Gate Gadget 150

5.2.2 Z Gate Gadget 151

5.2.3 CNOT Gate Gadget 151

5.2.4 R Gate Gadget 151

5.2.5 H Gate Gadget 155

5.3 Quantum Computing-as-a-Service 158

5.4 Summary 158

6 The Quantum Internet 161

6.1 Entanglement Over Longer Distances 161

6.1.1 Bell States and Bell State Measurement 162

6.1.2 Entanglement Swapping 163

6.1.3 Transmission of Qubits Using Tree-Cluster States 165

6.2 Entanglement with Higher Fidelity 172

6.2.1 Fidelity 172

6.2.2 Twirling Map 174

6.2.3 Quality of Distributed Entanglement and Entanglement Purification 177

6.3 Distributed Quantum Computation Over the Quantum Internet ? Challenges 183

6.4 Summary 187

References 187

7 Conclusion 191

References 192

Index 193