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.