Quantum Image Processing in Practice
Description
Comprehensive resource addressing the need for a quantum image processing machine learning model that can outperform classical neural networks
Quantum Image Processing in Practice explores the transformative potential of quantum color image processing across various domains, including biomedicine, entertainment, economics, and industry. The rapid growth of image data, especially in facial recognition and autonomous vehicles, demands more efficient processing techniques. Quantum computing promises to accelerate digital image processing (DIP) to meet this demand.
This book covers the role of quantum image processing (QIP) in quantum information processing, including mathematical foundations, quantum operations, image processing using quantum filters, quantum image representation, and quantum neural networks. It aims to inspire practical applications and foster innovation in this promising field.
Topics include:
- Qubits and Quantum Logic Gates: Introduces qubits, the fundamental data unit in quantum computing, and their manipulation using quantum logic gates like Pauli matrices, rotations, the CNOT gate, and Hadamard matrices. The concept of entanglement, where qubits become interconnected, is also explored, highlighting its importance for applications like quantum teleportation and cryptography.
- Two and Multiple Qubit Systems: Demonstrates the importance of using two qubits to process color images, enabling image enhancement, noise reduction, edge detection, and feature extraction. Covers the tensor product, Kronecker sum, SWAP gate, and local and controlled gates. Extends to multi-qubit superpositions, exploring local and control gates for three qubits, such as the Toffoli and Fredkin gates, and describes the measurement of superpositions using projection operators.
- Transforms and Quantum Image Representations: Covers the Hadamard, Fourier, and Heap transforms and their circuits in quantum computation, highlighting their applications in signal and image processing. Introduces the quantum signal-induced heap transform for image enhancement, classification, compression, and filtration. Explores quantum representations and operations for images using the RGB, XYZ, CMY, HSI, and HSV color models, providing numerous examples.
- Fourier Transform Qubit Representation: Introduces a new model of quantum image representation, the Fourier transform qubit representation. Describes the algorithm and circuit for calculating the 2-D quantum Fourier transform, enabling advancements in quantum imaging techniques.
- New Operations and Hypercomplex Algebra: Presents new operations on qubits and quantum representations, including multiplication, division, and inverse operations. Explores hypercomplex algebra, specifically quaternion algebra, for its potential in color image processing.
- Quantum Neural Networks (QNNs): Discusses QNNs and their circuit implementation as advancements in machine learning driven by quantum mechanics. Summarizes various applications of QNNs and current trends and future developments in this rapidly evolving field.
The book also addresses challenges and opportunities in QIP research, aiming to inspire practical applications and innovation. It is a valuable resource for researchers, students, and professionals interested in the intersection of quantum computing and color image processing applications, as well as those in visual communications, multimedia systems, computer vision, entertainment, and biomedical applications.
Artyom M. Grigoryan is an Associate Professor with the Department of Electrical Engineering at the University of Texas at San Antonio. He is a Senior Member of IEEE and the Editor of the International Journal of Applied Control and Electrical and Electronics Engineering.
Sos S. Agaian is a Distinguished Professor of Computer Science at the Graduate Center/CSI at CUNY. He is an Associate Editor for IEEE journals, a Fellow of IS&T, SPIE, AAAS, IEEE, and AAIA, a Member of Academia Europaea, and a Foreign Member of the Armenian National Academy.
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Content
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
Preface
The modern world has witnessed remarkable applications in the dynamic field of image processing, where operations transform an image to enhance it or extract vital information. It is a vibrant and diverse field encompassing various applications, such as facial recognition, image segmentation and compression, noise reduction, and more. These applications require sophisticated techniques to transform, enhance, and extract image information. However, these techniques also demand substantial computational resources for image storage and processing, which pose significant challenges for scalability and efficiency. Therefore, there is a critical need for more advanced and innovative methods to handle visual information. On the other hand, quantum computing defines a probabilistic approach to represent classical information using methods from quantum theory. Quantum computing offers a probabilistic and parallel approach to computation, which differs fundamentally from the deterministic and sequential approach of classical computing. The basic unit of quantum information, the qubit, can exist in a superposition of two states until measured, which enables quantum parallelism and entanglement. These quantum phenomena can provide exponential speedups and enhanced security for specific computational tasks, such as factoring large numbers, searching unsorted databases, simulating quantum systems, and solving linear systems of equations.
Quantum image processing (QIP) is a research branch of quantum information and computing that aims to exploit the advantages of quantum computing for image processing. QIP studies how to encode and process images using various quantum image representations and operations in a quantum computer. QIP has the potential to outperform classical image processing in terms of computing speed, security, and minimum storage requirements. However, QIP also faces many challenges and open questions, such as quantum superiority, reading the classical data, measurement, noise and error correction, scalability and compatibility, and the practical implementation of QIP algorithms and circuits.
In this book, we provide a comprehensive introduction to QIP, covering the theoretical foundations, methodological developments, quaternion color imaging, and practical QIP applications. We describe the existing quantum image representations and their operations, such as geometric transformations, color transformations, filtering, and enhancement. We also explore the emerging topics and applications of QIP, such as quantum image filtration in the frequency domain, convolution, and fast unitary transforms. We discuss the current state of QIP research, addressing the controversies and opportunities, as well as the challenges and future directions of QIP. We illustrate the QIP algorithms and circuits with detailed examples, diagrams, and code snippets using the Qiskit framework. We also provide exercises and references for further learning and research.
Organization of the Book
This book is organized into 14 chapters, as follows:
Chapter 1: Introduction
This chapter provides an overview of the main concepts and motivations of quantum computing and image processing. It outlines the structure and objectives of the book.
Chapter 2: Basic Concepts of Qubits
This chapter delves into the core concepts and principles, such as computational qubit states, superposition, operations on qubits, permutations, elementary gates, and qubit measurement. It presents the operations and gates in matrix and graphical notations and illustrates them with examples; 3-D model of qubits is presented together with the known Bloch sphere.
Chapter 3: Understanding 2-Qubit Systems
This chapter focuses on 2-qubit systems, which are the building blocks of multi-qubit systems. It discusses the mathematical tools and techniques for manipulating 2-qubit systems, such as projection operators, Kronecker product and sum, qubit entanglement, orthogonality, and unitary transformations. It also describes the elementary operations and main gates for 2-qubit systems, such as CNOT, SWAP, local and controlled gates, and explores their properties and applications with practical examples.
Chapter 4: Multi-Qubit Superpositions and Operations
This chapter extends the concepts and methods of 2-qubit systems to multi-qubit systems, which are essential for quantum image processing. It examines multi-qubit superpositions of different types. Many 3-qubit gates with 1 and 2 control bits are described with matrices and circuit elements. It also highlights the key 3-qubit gates, such as Toffoli and Fredkin, bit SWAP, and Hadamard gates, and shows how they can be used to implement classical logic functions and reversible circuits.
Chapter 5: Fast Transforms in Quantum Computation
This chapter introduces the quantum analogs of the classical fast transforms, such as the discrete paired, Fourier, and Hadamard transforms which are widely used in image processing. It provides detailed descriptions of the algorithms and implementations of these quantum-fast transforms, supported by examples and circuit designs. It also compares the advantages and disadvantages of these quantum fast transforms concerning their classical counterparts. Examples and circuits of these transforms and their inverses on 2-, 3-, and 4-qubits are presented. The paired transform is the core of the Fourier and Hadamard transforms. Therefore, the quantum paired transform is described in detail. The 1-D quantum circular convolution for phase filters with circuits is also presented with examples.
Chapter 6: Quantum Signal-Induced Heap Transform
This chapter presents a novel concept of quantum fast transform, which refers to the so-called discrete signal-induced heap transform (DsiHT), which can generate a unique unitary and fast transform for any given signal. It explains the theory and algorithm of quantum signal-induced heap transform (QsiHT) and demonstrates its applications in quantum cosine and Hartley transforms with quantum circuits. It also shows how DsiHT can be used to factorize and decompose any transform in a set of rotated gates and permutation, as well to initiate any quantum superposition from the basis state 0.
Chapter 7: Quantum Image Representation with Examples
This chapter describes the various quantum image representations proposed in the literature and compares their features and limitations. It covers the models for both grayscale and color images, such as QLM, NEQR, FRQI, RKL, GQIP, QIRHSI, and MQFTR. It also presents the 2 × 2 model of color image respresentation as a single grayscale image in quantum computation. It explores the quantum representation of different color models, such as RGB, CMY, XYZ, HSV, and HSI, and discusses the challenges and opportunities of quantum color image processing.
Chapter 8: Image Representation on the Unit Circle and MQFTR
This chapter focuses on a specific type of quantum image representation, called the multi-qubit Fourier transform representation (MQFTR), which encodes the image information on the unit circle using the Kronecker product of qubits. It explains the advantages and disadvantages of MQFTR. It presents some extensions and variations of MQFTR, such as MQFTR with phase shift and MQFTR with amplitude modulation. It also describes the quantum schemes for 2-D quantum Fourier transform with examples for 4 × 4 and 8 × 8 images.
Chapter 9: New Arithmetic on Qubits
This chapter introduces some novel concepts and methods of arithmetic operations on qubits, such as multiplication, division, conjugate, and inverse. It extends these operations to multi-qubit superpositions and discusses their properties and applications. It also shows how these operations can be implemented using quantum circuits and algorithms in image summation, linear convolution and filtration with examples.
Chapter 10: Quaternion-Based Arithmetic in Quantum Image Processing
This chapter explores the non-commutative quaternion arithmetic in quantum color image processing, which can offer some advantages over conventional complex arithmetic. It differentiates between the traditional and commutative quaternion algebras and discusses their properties and applications. It presents a new concept of the multiplicative group of 2-qubits and describes the main properties of the multiplication of 2-qubits and 2-qubit-based superpositions. It also describes the graphical representation of 2-qubits.
Chapter 11: Quantum Schemes for Multiplication of 2-Qubits
This chapter presents detailed quantum 2-qubit multiplication circuits that underlie many quantum arithmetic operations. It showcases a few circuits composed using the QsiHT concept and compares their performance and complexity. It shows how to design the quantum 4 × 4-gates of multiplication with 4, 5, and 6 rotations. It also describes 12 Hadamard matrices as multiplication gates.
Chapter 12: Quaternion Qubit Image Representation (QQIR)
This chapter introduces a new quantum image representation, called the quaternion qubit image representation (QQIR), which combines the features of 4-D quaternion arithmetic and MQFTR. It covers some basic operations in...
