Quantum Image Processing in Practice - Artyom M Grigoryan

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Résumé :

Preface xiii

Acknowledgments xvii

About the Companion Website xix

Part I Mathematical Foundation of Quantum Computation 1

1 Introduction 3

References 4

2 Basic Concepts of Qubits 5

2.1 Measurement of the Qubit 7

2.1.1 Operations on Qubits 10

2.1.2 Elementary Gates 10

References 14

3 Understanding of Two Qubit Systems 15

3.1 Measurement of 2-Qubits 16

3.1.1 Projection Operators 17

3.2 Operation of Kronecker Product 20

3.2.1 Tensor Product of Single Qubits 21

3.3 Operation of Kronecker Sum 22

3.3.1 Properties on Matrices 23

3.3.2 Orthogonality of Matrices 23

3.4 Permutations 24

3.4.1 Elementary Operations on 2-Qubits 25

References 36

4 Multi-qubit Superpositions and Operations 37

4.1 Elementary Operations on Multi-qubits 38

4.2 3-Qubit Operations with Local Gates 38

4.3 3-Qubit Operations with Control Bits 41

4.4 3-Qubit Operations with 2 Control Bits 43

4.5 Known 3-Qubit Gates 49

4.6 Projection Operators 51

References 52

5 Fast Transforms in Quantum Computation 53

5.1 Fast Discrete Paired Transform 53

5.2 The Quantum Circuits for the Paired Transform 57

5.3 The Inverse DPT 58

5.3.1 The First Circuit for the Inverse QPT 59

5.4 Fast Discrete Hadamard Transform 60

5.5 Quantum Fourier Transform 65

5.5.1 The Paired DFT 65

5.5.2 Algorithm of the 4-Qubit QFT 75

5.5.3 The Known Algorithm of the QFT 77

5.6 Method of 1D Quantum Convolution for Phase Filters 81

References 85

6 Quantum Signal-Induced Heap Transform 87

6.1 Definition 87

6.1.1 The Algorithm of the Strong DsiHT 89

6.1.2 Initialization of the Quantum State by the DsiHT 94

6.2 DsiHT-Based Factorization of Real Matrices 97

6.2.1 Quantum Circuits for DCT-II 98

6.2.2 Quantum Circuits for the DCT-IV 105

6.2.3 Quantum Circuits for the Discrete Hartley Transform 107

6.3 Complex DsiHT 110

References 111

Part II Applications in Image Processing 113

7 Quantum Image Representation with Examples 115

7.1 Models of Representation of Grayscale Images 116

7.1.1 Quantum Pixel Model (QPM) 116

7.1.2 Qubit Lattice Model (QLM) 122

7.1.3 Flexible Representation for Quantum Images 123

7.1.4 Representation of Amplitudes 125

7.1.5 Gradient and Sum Operators 128

7.1.6 Real Ket Model 130

7.1.7 General and Novel Enhanced Quantum Representations (GQIR and NEQR) 131

7.2 Color Image Quantum Representations 135

7.2.1 Quantum Color Pixel in the RGB Model 135

7.2.1.1 3-Color Quantum Qubit Model 136

7.2.2 NASS Representation 137

7.2.3 NASSTC Model 137

7.2.4 Novel Quantum Representation of Color Images (NCQI) 137

7.2.5 Multi-channel Representation of Images (MCRI) 139

7.2.6 Quantum Image Representation in HSI Model (QIRHSI) 141

7.2.7 Transformation 2 ? 2 Model for Color Images 142

References 145

8 Image Representation on the Unit Circle and MQFTR 147

8.1.1 Preparation for FTQR 147

8.1.2 Constant Signal and Global Phase 148

8.1.3 Inverse Transform 149

8.1.4 Property of Phase 150

8.2 Operations with Kronecker Product 150

8.3 FTQR Model for Grayscale Image 151

8.4 Color Image FTQR Models 151

8.5 The 2D Quantum Fourier Transform 153

8.5.1 Algorithm of the 2D QFT 153

8.5.2 Examples in Qiskit 157

References 159

9 New Operations of Qubits 161

9.1 Multiplication...

Sommaire:

Preface xiii

Acknowledgments xvii

About the Companion Website xix

Part I Mathematical Foundation of Quantum Computation 1

1 Introduction 3

2 Basic Concepts of Qubits 5

2.1 Measurement of the Qubit 7

3 Understanding of Two Qubit Systems 15

3.1 Measurement of 2-Qubits 16

3.2 Operation of Kronecker Product 20

3.3 Operation of Kronecker Sum 22

3.4 Permutations 24

4 Multi-qubit Superpositions and Operations 37

4.1 Elementary Operations on Multi-qubits 38

4.2 3-Qubit Operations with Local Gates 38

4.3 3-Qubit Operations with Control Bits 41

4.4 3-Qubit Operations with 2 Control Bits 43

4.5 Known 3-Qubit Gates 49

4.6 Projection Operators 51

5 Fast Transforms in Quantum Computation 53

5.1 Fast Discrete Paired Transform 53

5.2 The Quantum Circuits for the Paired Transform 57

5.3 The Inverse DPT 58

5.4 Fast Discrete Hadamard Transform 60

5.5 Quantum Fourier Transform 65

5.6 Method of 1D Quantum Convolution for Phase Filters 81

6 Quantum Signal-Induced Heap Transform 87

6.1 Definition 87

6.2 DsiHT-Based Factorization of Real Matrices 97

6.3 Complex DsiHT 110

Part II Applications in Image Processing 113

7 Quantum Image Representation with Examples 115

7.1 Models of Representation of Grayscale Images 116

7.2 Color Image Quantum Representations 135

8 Image Representation on the Unit Circle and MQFTR 147

8.1 Preparation for FTQR 147

8.2 Operations with Kronecker Product 150

8.3 FTQR Model for Grayscale Image 151

8.4 Color Image FTQR Models 151

8.5 The 2D Quantum Fourier Transform 153

9 New Operations of Qubits 161

9.1 Multiplication 161

9.2 Quantum Fourier Transform Representation 169

9.3 Linear Filter (Low-Pass Filtration) 170

10 Quaternion-Based Arithmetic in Quantum Image Processing 177

10.1 Noncommutative Quaternion Arithmetic 178

10.2 Commutative Quaternion Arithmetic 180

10.3 Geometry of the Quaternions 182

10.4 Multiplicative Group on 2-Qubits 184

11 Quantum Schemes for Multiplication of 2-Qubits 195

11.1 Schemes for the 4?4 Gate Aq1 196

11.2 The 4?4 Gate with 4 Rotations 202

11.3 Examples of 12 Hadamard Matrices 205

11.4 The General Case: 4?4 Gate with 5 Rotations 210

11.5 Division of 2-Qubits 213

11.6 Multiplication Circuit by 2nd 2-Qubit (Aq2 214

12 Quaternion Qubit Image Representation (QQIR) 219

12.1 Model 2 for Quaternion Images 220

12.2 Examples in Color Image Processing 224

12.3 Quantum Quaternion Fourier Transform 227

12.4 Ideal Filters on QQIR 228

12.5 Cyclic Convolution of 2-Qubit Superpositions 230

12.6 Windowed Convolution 230

12.7 Convolution Quantum Representation 238

12.8 Other Gradient Operators 244

12.9 Gradient and Smooth Operators by Multiplication 246

13 Quantum Neural Networks: Harnessing Quantum Mechanics for Machine Learning 251

13.1 Introduction in Quantum Neural Networks: A New Frontier in Machine Learning 251

13.2 McCulloch-Pitts Processing Element 254

13.3 Building Blocks: Layers and Architectures 258

13.4 Artificial Neural Network Architectures: From Simple to Complex 259

13.5 Key Properties and Operations of Artificial Neural Networks 261

13.6 Quantum Neural Networks: A Computational Model Inspired by Quantum Mechanics 263

13.7 The Main Difference Between QNNs and CNNs 271

13.8 Applications of Q...