Quantum image processing in practice : a mathematical toolbox

• Main Authors: Grigoryan, Artyom (Author), Agaian, S. S. (Author)
• Format: Electronic eBook
• Language: English
• Published: Hoboken, New Jersey : John Wiley & Sons, Inc., [2025]
• Subjects: Image processing > Digital techniques.; Quantum computing.; elektronické knihy; electronic books
• ISBN: 9781394265183; 9781394265176; 9781394265169; 9781394265152
• Physical Description: 1 online zdroj (xix, 295 stran) : ilustrace

"Image processing represents a critical use of artificial intelligence in various applications, including biomedicine, entertainment, economics, and industry. For example, image processing is extensively used in fast-growing markets like facial recognition and autonomous vehicles. Recently, the...

Table of Contents:

• 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 161
• 9.1.1 Conjugate Qubit 162
• 9.1.2 Inverse Qubit 162
• 9.1.3 Division of Qubits 163
• 9.1.4 Operations on Qubits with Relative Phases 163
• 9.1.5 Quadratic Qubit Equations 164
• 9.1.6 Multiplication of n-Qubit Superpositions 165
• 9.1.7 Conjugate Superposition 167
• 9.1.8 Division of Multi-qubit Superpositions 167
• 9.1.9 Operations on Left-Sided Superpositions 167
• 9.1.10 Quantum Sum of Signals 168
• 9.2 Quantum Fourier Transform Representation 169
• 9.3 Linear Filter (Low-Pass Filtration) 170
• 9.3.1 General Method of Filtration by Ideal Filters 173
• 9.3.2 Application: Linear Convolution of Signals 174
• References 176
• 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
• 10.4.1 2-Qubit to the Power 188
• 10.4.2 Second Model of Quaternion and 2-Qubits 190
• References 193
• 11 Quantum Schemes for Multiplication of 2-Qubits 195
• 11.1 Schemes for the 4×4 Gate A q 1 196
• 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
• References 218
• 12 Quaternion Qubit Image Representation (QQIR) 219
• 12.1 Model 2 for Quaternion Images 220
• 12.1.1 Comments: Abstract Models with Quaternion Exponential Function 221
• 12.1.2 Multiplication of Colors 222
• 12.1.3 2-Qubit Superposition of Quaternion Images 222
• 12.2 Examples in Color Image Processing 224
• 12.2.1 Grayscale-2-Quaternion Image Model 224
• 12.3 Quantum Quaternion Fourier Transform 227
• 12.4 Ideal Filters on QQIR 228
• 12.4.1 Algorithm of Filtration G p = Y p F p by Ideal Filters 229
• 12.5 Cyclic Convolution of 2-Qubit Superpositions 230
• 12.6 Windowed Convolution 230
• 12.6.1 Edges and Contours of Images 235
• 12.6.2 Gradients and Thresholding 235
• 12.7 Convolution Quantum Representation 238
• 12.7.1 Gradient Operators and Numerical Simulations 241
• 12.8 Other Gradient Operators 244
• 12.9 Gradient and Smooth Operators by Multiplication 246
• 12.9.1 Challenges 248
• References 248
• 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
• 13.5.1 Reinforcement Learning: Learning Through Trial and Reward 262
• 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 QNN in Image Processing 276
• 13.9 The Current and Future Trends and Developments in Quantum Neural Networks 281
• References 282
• 14 Conclusion and Opportunities and Challenges of Quantum Image Processing 285
• References 288
• Index 291.