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.