Evolution and Applications of Quantum Computing
Description
The book is about the Quantum Model replacing traditional computing's classical model and gives a state-of-the-art technical overview of the current
efforts to develop quantum computing and applications for Industry 4.0.
A holistic approach to the revolutionary world of quantum computing is presented in this book, which reveals valuable insights into this rapidly emerging technology. The book reflects the dependence of quantum computing on the physical phenomenon of superposition, entanglement, teleportation, and interference to simplify difficult mathematical problems which would have otherwise taken years to derive a definite solution for. An amalgamation of the information provided in the multiple chapters will elucidate the revolutionary and riveting research being carried out in the brand-new domain encompassing quantum computation, quantum information and quantum mechanics. Each chapter gives a concise introduction to the topic.
The book comprises 18 chapters and describes the pioneering work on the interaction between artificial intelligence, machine learning, and quantum computing along with their applications and potential role in the world of big data. Subjects include:
* Combinational circuits called the quantum multiplexer with secured quantum gate (CSWAP);
* Detecting malicious emails and URLs by using quantum text mining algorithms to distinguish between phishing and benign sites;
* Quantum data traffic analysis for intrusion detection systems;
* Applications of quantum computation in banking, netnomy and vehicular ad-hoc networks, virtual reality in the education of autistic children, identifying bacterial diseases and accelerating drug discovery;
* The critical domain of traditional classical cryptography and quantum cryptography.
Audience
The book will be very useful for researchers in computer science, artificial intelligence and quantum physics as well as students who want to understand the history of quantum computing along with its applications and have a technical state-of-the-art overview.
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Persons
Sachi Nandan Mohanty received his PhD from IIT Kharagpur, India in 2015, with MHRD scholarship from the Govt of India. He is now an associate professor, VIT-AP University, Andhra Pradesh. He has published more than 100 research articles in international journals as well as edited 24 books including many with the Wiley-Scrivener imprint. His research areas include data mining, big data analysis, cognitive science, fuzzy decision-making, brain-computer interface, cognition, and computational intelligence.
Rajanikanth Aluvalu, PhD, is a professor in the Department of IT, Chaitanya Bharathi Institute of Technology, Hyderabad. He is a senior member of IEEE and his specialization is in high-performance computing. He has published 90 + research articles in peer-reviewed journals and conferences.
Sarita Mohanty is an assistant professor in the Department of Master in Computer Application, Centre for Post Graduate Study, OUAT, Govt of Odisha, India. Her research areas include digital forensics and cybersecurity. She has more than 10 years of teaching experience.
Content
Preface xvii
1 Introduction to Quantum Computing 1
V. Padmavathi, C. N. Sujatha, V. Sitharamulu, K. Sudheer Reddy and A. Mallikarjuna Reddy
1.1 Quantum Computation 2
1.2 Importance of Quantum Mechanics 2
1.3 Security Options in Quantum Mechanics 2
1.4 Quantum States and Qubits 3
1.5 Quantum Mechanics Interpretation 4
1.6 Quantum Mechanics Implementation 4
1.6.1 Photon Polarization Representation 4
1.7 Quantum Computation 6
1.7.1 Quantum Gates 7
1.8 Comparison of Quantum and Classical Computation 11
1.9 Quantum Cryptography 12
1.10 Qkd 12
1.11 Conclusion 12
References 13
2 Fundamentals of Quantum Computing and Significance of Innovation 15
Swapna Mudrakola, Uma Maheswari V., Krishna Keerthi Chennam and MVV Prasad Kantidpudi
2.1 Quantum Reckoning Mechanism 16
2.2 Significance of Quantum Computing 16
2.3 Security Opportunities in Quantum Computing 16
2.4 Quantum States of Qubit 17
2.5 Quantum Computing Analysis 17
2.6 Quantum Computing Development Mechanism 18
2.7 Representation of Photon Polarization 18
2.8 Theory of Quantum Computing 20
2.9 Quantum Logical Gates 21
2.9.1 I-Qubit GATE 21
2.9.2 Hadamard-GATE 22
2.9.3 NOT_GATE_QUANTUM or Pauli_X-GATE 22
2.9.3.1 Pauli_Y-GATE 23
2.9.3.2 Pauli_Z-GATE 23
2.9.3.3 Pauli_S-Gate 23
2.9.4 Two-Qubit GATE 24
2.9.5 Controlled NOT(C-NOT) 24
2.9.6 The Two-Qubits are Swapped Using SWAP_GATE 24
2.9.7 C-Z-GATE (Controlled Z-GATE) 24
2.9.8 C-P-GATE (Controlled-Phase-GATE) 25
2.9.9 Three-Qubit Quantum GATE 25
2.9.9.1 GATE: Toffoli Gate 25
2.9.10 F-C-S GATE (Fredkin Controlled Swap-GATE) 26
2.10 Quantum Computation and Classical Computation Comparison 27
2.11 Quantum Cryptography 27
2.12 Quantum Key Distribution - QKD 27
2.13 Conclusion 28
References 28
3 Analysis of Design Quantum Multiplexer Using CSWAP and Controlled-R Gates 31
Virat Tara, Navneet Sharma, Pravindra Kumar and Kumar Gautam
3.1 Introduction 32
3.2 Mathematical Background of Quantum Circuits 34
3.2.1 Hadamard Gate 34
3.2.2 CSWAP Gates 35
3.2.3 Controlled-R Gates 36
3.3 Methodology of Designing Quantum Multiplexer (QMUX) 36
3.3.1 QMUX Using CSWAP Gates 36
3.3.1.1 Generalization 37
3.3.2 QMUX Using Controlled-R Gates 37
3.4 Analysis and Synthesis of Proposed Methodology 39
3.5 Complexity and Cost of Quantum Circuits 41
3.6 Conclusion 42
References 42
4 Artificial Intelligence and Machine Learning Algorithms in Quantum Computing Domain 45
Syed Abdul Moeed, P. Niranjan and G. Ashmitha
4.1 Introduction 46
4.1.1 Quantum Computing Convolutional Neural Network 51
4.2 Literature Survey 52
4.3 Quantum Algorithms Characteristics Used in Machine Learning Problems 58
4.3.1 Minimizing Quantum Algorithm 58
4.3.2 K-NN Algorithm 58
4.3.3 K-Means Algorithm 60
4.4 Tree Tensor Networking 61
4.5 TNN Implementation on IBM Quantum Processor 62
4.6 Neurotomography 62
4.7 Conclusion and Future Scope 63
References 64
5 Building a Virtual Reality-Based Framework for the Education of Autistic Kids 67
Kanak Pandit, Aditya Mogare, Achal Shah, Prachi Thete and Megharani Patil
5.1 Introduction 68
5.2 Literature Review 71
5.3 Proposed Work 74
5.3.1 Methodology 74
5.3.2 Work Flow of Neural Style Transfer 75
5.3.3 A-Frame 75
5.3.3.1 Setting Up the Virtual World and Adding Components 75
5.3.3.2 Adding Interactivity Through Raycasting 76
5.3.3.3 Animating the Components 77
5.3.4 Neural Style Transfer 78
5.3.4.1 Choosing the Content and Styling Image 79
5.3.4.2 Image Preprocessing and Generation of a Random Image 79
5.3.4.3 Model Design and Extraction of Content and Style 81
5.3.4.4 Loss Calculation 81
5.3.4.5 Model Optimization 84
5.4 Evaluation Metrics 86
5.5 Results 89
5.5.1 A-Frame 89
5.5.2 Neural Style Transfer 90
5.6 Conclusion 90
References 91
6 Detection of Phishing URLs Using Machine Learning and Deep Learning Models Implementing a URL Feature Extractor 93
Abishek Mahesh, Prithvi Seshadri, Shruti Mishra and Sandeep Kumar Satapathy
6.1 Introduction 94
6.2 Related Work 94
6.3 Proposed Model 95
6.3.1 URL Feature Extractor 95
6.3.2 Dataset 103
6.3.3 Methodologies 104
6.3.3.1 AdaBoost Classifier 105
6.3.3.2 Gradient Boosting Classifier 105
6.3.3.3 K-Nearest Neighbors 105
6.3.3.4 Logistic Regression 106
6.3.3.5 Artificial Neural Networks 106
6.3.3.6 Support Vector Machines (SVM) 107
6.3.3.7 Naïve Bayes Classifier 107
6.4 Results 109
6.5 Conclusions 109
References 109
7 Detection of Malicious Emails and URLs Using Text Mining 111
Heetakshi Fating, Aditya Narawade, Sandeep Kumar Satapathy and Shruti Mishra
7.1 Introduction 112
7.2 Related Works 112
7.3 Dataset Description 114
7.4 Proposed Architecture 115
7.5 Methodology 116
7.5.1 Methodology for the URL Dataset 116
7.5.2 Methodology for the Email Dataset 118
7.5.2.1 Overcoming the Overfitting Problem 118
7.5.2.2 Tokenization 119
7.5.2.3 Applying Machine Learning Algorithms 119
7.5.3 Detecting Presence of Malicious URLs in Otherwise Non-Malicious Emails 119
7.5.3.1 Preparation of Dataset 119
7.5.3.2 Creation of Features 120
7.5.3.3 Applying Machine Learning Algorithms 120
7.6 Results 120
7.6.1 URL Dataset 120
7.6.2 Email Dataset 121
7.6.3 Final Dataset 121
7.7 Conclusion 122
References 122
8 Quantum Data Traffic Analysis for Intrusion Detection System 125
Anshul Harish Khatri, Vaibhav Gadag, Simrat Singh, Sandeep Kumar Satapathy and Shruti Mishra
8.1 Introduction 126
8.2 Literature Overview 127
8.3 Methodology 129
8.3.1 Autoviz 129
8.3.2 Dataset 132
8.3.3 Proposed Models 132
8.3.3.1 Decision Tree 135
8.3.3.2 Random Forest Classifier Algorithm 136
8.3.3.3 AdaBoost Classifier 136
8.3.3.4 Ridge Classifier 137
8.3.3.5 Logistic Regression 137
8.3.3.6 SVM-Linear Kernel 138
8.3.3.7 Naive Bayes 138
8.3.3.8 Quadratic Discriminant Analysis 139
8.4 Results 140
8.5 Conclusion 141
References 142
9 Quantum Computing in Netnomy: A Networking Paradigm in e-Pharmaceutical Setting 145
Sarthak Dash, Sugyanta Priyadarshini, Sachi Nandan Mohanty, Sukanya Priyadarshini and Nisrutha Dulla
9.1 Introduction 146
9.2 Discussion 148
9.2.1 Exploring Market Functioning via Quantum Network Economy 148
9.2.1.1 Internal Networking Marketing 149
9.2.1.2 Layered Marketing 149
9.2.1.3 Role of Marketing in Pharma Network Organizations 150
9.2.1.4 Role of Marketing in Vertical Networking Organizations 152
9.2.1.5 Generic e-Commerce Entity Model in Pharmaceutical Industry 153
9.2.2 Analyzing the Usability of Quantum Netnomics in Attending Economic Development 154
9.2.2.1 Theory of 4Ps in Pharma Marketing mix 155
9.2.2.2 Buying Behavior of the e-Consumers 156
9.2.2.3 Maintaining of Privacy and Security via Quantum Technology in e-Structure 157
9.2.2.4 Interface Influencing Sales 157
9.3 Results 158
9.4 Conclusion 159
References 159
10 Machine Learning Approach in the Indian Service Industry: A Case Study on Indian Banks 163
Pragati Priyadarshinee
10.1 Introduction 163
10.2 Literature Survey 164
10.3 Experimental Results 170
10.4 Conclusion 172
References 172
11 Accelerating Drug Discovery with Quantum Computing 175
Mahesh V. and Shimil Shijo
11.1 Introduction 175
11.2 Working Nature of Quantum Computers 176
11.3 Use Cases of Quantum Computing in Drug Discovery 178
11.4 Target Drug Identification and Validation 179
11.5 Drug Discovery Using Quantum Computers is Expected to Start by 2030 179
11.6 Conclusion 180
References 181
12 Problems and Demanding Situations in Traditional Cryptography: An Insistence for Quantum Computing to Secure Private Information 183
D. DShivaprasad, Mohamed Sirajudeen Yoosuf, P. Selvaramalakshmi, Manoj A. Patil and Dasari Promod Kumar
12.1 Introduction to Cryptography 184
12.1.1 Confidentiality 184
12.1.2 Authentication 185
12.1.3 Integrity 185
12.1.4 Non-Repudiation 186
12.2 Different Types of Cryptography 186
12.2.1 One-Way Processing 186
12.2.1.1 Hash Function (One-Way Processing) 186
12.2.2 Two-Way Processing 187
12.2.2.1 Symmetric Cryptography 188
12.2.2.2 Asymmetric Cryptography 189
12.2.3 Algorithms Types 190
12.2.3.1 Stream Cipher 190
12.2.3.2 Block Cipher 191
12.2.4 Modes of Algorithm 192
12.2.4.1 Cipher Feedback Mode 192
12.2.4.2 Output Feedback Mode 192
12.2.4.3 Cipher Block Chaining Mode 192
12.2.4.4 Electronic Code Book 192
12.3 Common Attacks 193
12.3.1 Passive Attacks 193
12.3.1.1 Traffic Analysis 193
12.3.1.2 Eavesdropping 194
12.3.1.3 Foot Printing 195
12.3.1.4 War Driving 195
12.3.1.5 Spying 195
12.3.2 Active Attacks 196
12.3.2.1 Denial of Service 196
12.3.2.2 Distributed Denial of Service (DDOS) 197
12.3.2.3 Message Modification 197
12.3.2.4 Masquerade 197
12.3.2.5 Trojans 198
12.3.2.6 Replay Attacks 199
12.3.3 Programming Weapons for the Attackers 199
12.3.3.1 Dormant Phase 200
12.3.3.2 Propagation Phase 200
12.3.3.3 Triggering Phase 201
12.3.3.4 Execution Phase 201
12.4 Recent Cyber Attacks 201
12.5 Drawbacks of Traditional Cryptography 203
12.5.1 Cost and Time Delay 203
12.5.2 Disclosure of Mathematical Computation 203
12.5.3 Unsalted Hashing 204
12.5.4 Attacks 204
12.6 Need of Quantum Cryptography 204
12.6.1 Quantum Mechanics 204
12.7 Evolution of Quantum Cryptography 205
12.8 Conclusion and Future Work 205
References 205
13 Identification of Bacterial Diseases in Plants Using Re-Trained Transfer Learning in Quantum Computing Environment 207
Sri Silpa Padmanabhuni, B. Srikanth Reddy, A. Mallikarjuna Reddy and K. Sudheer Reddy
13.1 Introduction 208
13.2 Literature Review 218
13.3 Proposed Methodology 220
13.3.1 SVM Classifier 222
13.3.2 Random Forest to Classify the Rice Leaf 223
13.3.2.1 Image Pre-Processing 223
13.3.2.2 Feature Extraction 223
13.3.2.3 Classification 224
13.4 Experiment Results 226
Conclusion 230
References 230
14 Quantum Cryptography 233
Salma Fauzia
14.1 Fundamentals of Cryptography 234
14.2 Principle of Quantum Cryptography 237
14.2.1 Quantum vs. Conventional Cryptography 237
14.3 Quantum Key Distribution Protocols 238
14.3.1 Overview and BB84 Protocol 238
14.3.2 The B92 Protocol 240
14.3.3 E91 Protocol 241
14.3.4 SARG04 Protocol 243
14.4 Impact of the Sifting and Distillation Steps on the Key Size 243
14.5 Cryptanalysis 246
14.6 Quantum Key Distribution in the Real World 247
References 248
15 Security Issues in Vehicular Ad Hoc Networks and Quantum Computing 249
B. Veera Jyothi, L. Suresh Kumar and B. Surya Samantha
15.1 Introduction 250
15.2 Overview of VANET Security 250
15.2.1 Security of VANET 250
15.2.2 Attacks are Classified 251
15.3 Architectural and Systematic Security Methods 252
15.3.1 Solutions for Cryptography 252
15.3.2 Framework for Trust Groups 252
15.3.3 User Privacy Security System Based on ID 253
15.4 Suggestions on Particular Security Challenges 254
15.4.1 Content Delivery Integrity Metrics 254
15.4.2 Position Detection 254
15.4.3 Protective Techniques 255
15.5 Quantum Computing in Vehicular Networks 257
15.5.1 Securing Automotive Ecosystems: A Challenge 257
15.5.2 Generation of Quantum Random Numbers (QRNG) 258
15.6 Quantum Key Transmission (QKD) 258
15.7 Quantum Internet - A Future Vision 259
15.7.1 Quantum Internet Applications 259
15.7.2 Application Usage-Based Categorization 260
15.8 Conclusions 262
References 263
16 Quantum Cryptography with an Emphasis on the Security Analysis of QKD Protocols 265
Radhika Kavuri, Santhosh Voruganti, Sheena Mohammed, Sucharitha Inapanuri and B. Harish Goud
16.1 Introduction 266
16.2 Basic Terminology and Concepts of Quantum Cryptography 267
16.2.1 Quantum Cryptography and Quantum Key Distribution 267
16.2.2 Quantum Computing and Quantum Mechanics 267
16.2.3 Post-Quantum Cryptography 267
16.2.4 Quantum Entanglement 267
16.2.5 Heisenberg's Uncertainty Principle 268
16.2.6 Qubits 268
16.2.7 Polarization 269
16.2.8 Traditional Cryptography vs. Quantum Cryptography 269
16.3 Trends in Quantum Cryptography 270
16.3.1 Global Quantum Key Distribution Links 271
16.3.2 Research Statistics on Quantum Cryptography 273
16.4 An Overview of QKD Protocols 274
16.4.1 Introduction to the Prepare-and-Measure Protocols 275
16.4.2 The BB84 Protocol 275
16.4.3 B92 Protocol 278
16.4.4 Six State Protocol (SSP) 278
16.4.5 SARG04 Protocol 279
16.4.6 Introduction to the Entanglement-Based Protocols 280
16.4.7 The E91 Protocol 280
16.4.8 The BBM92 Protocol 280
16.5 Security Concerns in QKD 282
16.6 Future Research Foresights 284
16.6.1 Increase in Bit Rate 284
16.6.2 Longer Distance Coverage 284
16.6.3 Long Distance Quantum Repeaters 285
16.6.4 Device Independent Quantum Cryptography 285
16.6.5 Development of Tools for Simulation and Measurements 285
16.6.6 Global Quantum Communication Network 285
16.6.7 Integrated Photonic Spaced QKD 285
16.6.8 Quantum Teleportation 286
References 286
17 Deep Learning-Based Quantum System for Human Activity Recognition 289
Shoba Rani Salvadi, Narsimhulu Pallati and Madhuri T.
17.1 Introduction 290
17.2 Related Works 292
17.3 Proposed Scheme 293
17.3.1 Datasets Description 294
17.3.2 Pre-Processing 294
17.3.3 Feature Extraction 295
17.3.4 Preliminaries 295
17.3.4.1 Quantum Computing 296
17.3.4.2 Convolutional Neural Networks 296
17.3.5 Proposed ORQC-CNN Model 296
17.3.5.1 Quantum Convolutional Layer 297
17.3.5.2 Convolutional Layer 299
17.3.5.3 Max-Pooling Layer 299
17.3.5.4 Fully Connected Layer 299
17.3.6 Parameter Selection Using Artificial Gorilla Troops Optimization Algorithm (AGTO) 300
17.3.6.1 Exploration Phase 301
17.3.6.2 Exploitation Phase 302
17.3.6.3 Follow the Silverback 303
17.3.6.4 Competition for Adult Females 303
17.3.7 Computational Difficulty 304
17.4 Results and Discussion 304
17.4.1 Performance Measure 305
17.4.2 Performance Analysis of Dataset 1 306
17.4.3 Performance Analysis of Dataset 2 307
17.4.4 Comparison 308
17.5 Conclusion 309
References 309
18 Quantum Intelligent Systems and Deep Learning 313
Bhagaban Swain and Debasis Gountia
18.1 Introduction 313
18.2 Quantum Support Vector Machine 315
18.3 Quantum Principal Component Analysis 318
18.4 Quantum Neural Network 319
18.5 Variational Quantum Classifier 321
18.6 Conclusion 323
References 323
Index 327
1
Introduction to Quantum Computing
V. Padmavathi1, C. N. Sujatha2, V. Sitharamulu3, K. Sudheer Reddy4* and A. Mallikarjuna Reddy5
1 Dept. of Computer Science and Engineering, Chaitanya Bharathi Institute of Technology, Hyderabad, India
2 Dept. of Computer Science and Engineering, Sreenidhi Institute of Science and Technology, Hyderabad, India
3 Dept. of Computer Science and Engineering, GITAM (Deemed to be University), Hyderabad, India
4 Dept. of Information Technology, Anurag University, Hyderabad, India
5 Dept. of Artificial Intelligence, Anurag University, Hyderabad, India
Abstract
Over the past few decades, tremendous growth has been witnessed in cryptography in which different security techniques and concerns were projected and put into practice. The classical methods of cryptography are depended on binary bits, which are susceptible to predicting the key during transit. Hence, moving the classical cryptographic scheme to a new fast, and non-vulnerable scheme is time. The principles of quantum mechanics are applied in quantum computing to enhance security which uses qubits for communication. The advantage of using qubits is that it is impossible to make copies of qubits due to the no-cloning theorem. The computations are performed through photons or qubits produced using the photon's polarization. The qubits are disturbed when measured at an incorrect polarization angle due to the principle of uncertainty. The photons are quantized features used to encode the information. They can be applied in Quantum Key Distribution (QKD), in which distantly apart communicators share a standard secret key.
Keywords: Quantum computing, quantum mechanics, qubits, photon polarization, quantum gates, quantum cryptography, quantum key, Fredkin gate
1.1 Quantum Computation
The study of information processing based on quantum mechanics is known as quantum computing and quantum information. Quantum mechanics is a set of mathematical rules used to build physical hypotheses. Building tools to refine the notion of quantum mechanics is one of the objectives of quantum computation [7-11]. There was a debate on whether it was possible to make a duplicate of an unknown quantum state. Quantum mechanics might be used when replication is conceivable to send signals faster than light, which is highly impractical. Wooters and Zurek [3] proved that the theorem of no cloning, was the first result of quantum computation and information. Since then, there have been several improvements.
1.2 Importance of Quantum Mechanics
Quantum mechanics-based communications are safe since they stand on inalienable quantum mechanics concepts. The principles of Heisenberg Uncertainty and Photon Polarization are essential concepts.
The Heisenberg principle of Uncertainty refers to an entity's inability to discern two connected physical attributes [4]. In light of this assertion, two examples are worth considering. The first is the example generally stated: For every P, the momentum and the location cannot be calculated alongside. Second, the photon cannot be measured concurrently on either of it [1, 2, 18], then properties are impacted. According to the theorem of no-cloning [1, 3, 18], replicas of qubits are not feasible.
1.3 Security Options in Quantum Mechanics
Figure 1.1 demonstrates that a bit corresponds to a qubit by polarizing photons and with the help of a rectilinear and diagonal basis. In a rectilinear base, the binary 0 is represented by 0° photon polarization and 45° on a diagonal basis. In Figure 1.2 [4], binary one is represented by 90° rectilinear photon polarization and 135° diagonal photon polarization.
Figure 1.1 Rectilinear, diagonal basis.
Figure 1.2 Photon polarizations.
1.4 Quantum States and Qubits
Qubit exists in two states, symbolized as |0? and |1? in quantum mechanics. It is stated as a state, a ket, which Paul Dirac created [1, 6]. A bit can be 0 or 1, whereas a qubit can be |0? or |1? state. It also happens in the superposition state that is nothing more than a linear arrangement of the quantum states |0? and |1?. The symbol |?? indicates a form, and a superposition state is denoted by the symbol |f? = a|0? + ß|1? where a and ß are complex numbers [6].
A qubit exists in superposition state |0? and |1? in most cases, but the state is not calculated. When it is calculated, though, it is either in |0? or |1?. The chance of realizing a qubit is either |0? or |1? equals the square of the modulus of a, ß. It is stated that in state |0?, the likelihood of obtaining |?? is |a|2, and in state |1?, the likelihood of obtaining |?? is |ß|2. Squaring coefficients yield likelihood of attaining a measurement's result. Thus, |a|2 + |ß|2 = 1 [6, 18] is the state.
1.5 Quantum Mechanics Interpretation
The state is shown by the expression a|0? + ß|1?. It can alternatively be represented as a vector or a column. It is represented as a two-dimensional complex unit vector by stacking two complex integers [1, 12].
And
1.6 Quantum Mechanics Implementation
The following notations are used to implement quantum mechanics:
- 1) |0? = | -?
- 2) |1?= | | ?
- 3) (|0? + |1?) = |/?
- 4) (|0? - |1?) = |\?
In rectilinear basis, 1 and 2 indicates 0° and 90°, respectively. In diagonal basis, the 3 and 4 indicates 45° and 135°, respectively.
1.6.1 Photon Polarization Representation
- i) A binary 0 is represented in rectilinear basis as a 0° state using polarization, as demonstrated below using a linear layout with x and y axes both set to 1. Figure 1.3 depicts the situation.
Figure 1.3 Representation of photon polarization.
- ii) Using the linear configurations of 0 on the x axis and 1 on the y axis, respectively. Polari-zation isused to represent the binary 1 in rectilinear basis as a 90° state. The illustration is shown in Figure 1.4.
Figure 1.4 Representation of photon polarization.
- iii) Using polarization and the linear combinations listed below, a 0 in binary is shown as 45° state on a diagonal basis. Figure 1.5 depicts the situation.
Figure 1.5 Photon polarization in diagonal basis (of binary 0).
- iv) Using polarization and the linear combinations listed below, 1 in binary corresponds to 135° in diagonal basis. The representation is shown in Figure 1.6.
Figure 1.6 Representation of photon polarization.
1.7 Quantum Computation
Quantum computation has led to a new way of thinking about computation. The notion of quantum computation explains changes in state stirring. A classical computer, like a method, is built with the help of an electrical circuit with logic wires and gates, whereas a quantum computer is built with the help of a quantum circuit with quantum wires and gates. The quantum gates are employed to control or manipulate and act on the quantum bit [5].
The matrix representation is the best way to understand linear combinations. Unitary matrix U is the proper state on the matrix to represent the gate. Specifically, U┼U = I, in which U┼ is the adjoint of U, that is the complex conjugate of the transpose of matrix U, and I is a two-by-two identity matrix. It is simple to validate X┼X = I for a gate, for example. Surprisingly, quantum gates are only limited by the unitary check. It denotes the presence of a logical quantum gate [5].
1.7.1 Quantum Gates
Gates are used to carry out computations in quantum computers. Quantum gates are unitary operators with n inputs and outputs that may be expressed using matrices. As a result, they have a 2n degree. A two-degree matrix is used to represent one qubit. As a result, a two-by-two matrix is required for a quantum gate acting on a qubit. A matrix of 22 degrees is used to represent a two-qubit gate. As a result, a four-by-four unitary matrix [5, 6] is used to represent it. As a result, the quantum gates have a temporal complexity of 2n. It has been proved that exponential complexities are secure. The tensor product is used to obtain the matrix depiction of quantum gates. Tensor product is represented as ?.
It creates a single vector space out of two. XY is a space with mxn dimension if X and Y are vector spaces with mxn n dimensions. The components of |x| of X and |y| of Y are the elements of XY, which are linear combinations of these tensor products. The matrix is represented as given if A is an m x n matrix and B is a p x q matrix.
A. One-Qubit Gate
A quantum gate that operates on a single qubit requires a single qubit as input and output. The computation requires a two-by-two unitary matrix. The several forms of one-qubit gates are explained in the following...
