Preface xiii
Author Biography xv
1 Introduction of Quantum Computing 1
1.1 Introduction 1
1.2 What Is the Exact Meaning of Quantum Computing? 2
1.2.1 What Is Quantum Computing in Simple Terms? 2
1.3 Origin of Quantum Computing 3
1.4 History of Quantum Computing 5
1.5 Quantum Communication 19
1.6 Build Quantum Computer Structure 19
1.7 Principle Working of Quantum Computers 21
1.7.1 Kinds of Quantum Computing 21
1.8 Quantum Computing Use in Industry 23
1.9 Investors Invest Money in Quantum Technology 24
1.10 Applications of Quantum Computing 26
1.11 Quantum Computing as a Solution Technology 29
1.11.1 Quantum Artificial Intelligence 29
1.11.2 How Close Are We to Quantum Supremacy? 30
1.12 Conclusion 30
References 31
2 Pros and Cons of Quantum Computing 33
2.1 Introduction 33
2.2 Quantum as a Numerical Process 33
2.3 Quantum Complexity 34
2.4 The Pros and Cons of the Quantum Computational Framework 36
2.5 Further Benefits of Quantum Computing 37
2.6 Further Drawbacks to Quantum Computing 38
2.7 Integrating Quantum and Classical Techniques 38
2.8 Framework of QRAM 39
2.9 Computing Algorithms in the Quantum World 40
2.9.1 Programming Quantum Processes 42
2.10 Modification of Quantum Building Blocks 42
References 43
3 Methods and Instrumentation for Quantum Computing 45
3.1 Basic Information of Quantum Computing 45
3.2 Signal Information in Quantum Computing 47
3.3 Quantum Data Entropy 47
3.4 Basics of Probability in Quantum Computing 50
3.5 Quantum Theorem of No-Cloning 52
3.6 Measuring Distance 53
3.7 Fidelity in Quantum Theory 58
3.8 Quantum Entanglement 62
3.9 Information Content and Entropy 66
References 71
4 Foundations of Quantum Computing 73
4.1 Single-Qubit 73
4.1.1 Photon Polarization in Quantum Computing 73
4.2 Multi-qubit 76
4.2.1 Blocks of Quantum States 76
4.2.2 Submission of Vector Space in Quantum Computing 77
4.2.3 Vector Spacing in Quantum Blocks 77
4.2.4 States of n-Qubit Technology 79
4.2.5 States of Entangled 81
4.2.6 Classical Measuring of Multi-Qubit 84
4.3 Measuring of Multi-Qubit 87
4.3.1 Mathematical Functions in Quantum Operations 87
Example 88
4.3.2 Operator Measuring Qubits Projection 89
4.3.3 The Measurement Postulate 94
4.3.4 EPR Paradox and Bell’s Theorem 99
4.3.5 Layout of Bell’s Theorem 101
4.3.6 Statistical Predicates of Quantum Mechanics 101
4.3.7 Predictions of Bell’s Theorem 102
4.3.8 Bell’s Inequality 103
4.4 States of Quantum Metamorphosis 105
4.4.1 Solitary Steps Metamorphosis 106
4.4.2 Irrational Metamorphosis: The No-Cloning Principle 107
4.4.3 The Pauli Transformations 109
4.4.4 The Hadamard Metamorphosis 109
4.4.5 Multi-Qubit Metamorphosis from Single-Qubit 109
4.4.6 The Controlled-NOT and Other Singly Controlled Gates 110
4.4.7 Opaque Coding 113
4.4.8 Basic Bits in Opaque Coding 114
4.4.9 Quantum Message Teleportation 114
4.4.10 Designing and Constructing Quantum Circuits 116
4.4.11 Single Qubit Manipulating Quantum State 116
4.4.12 Controlling Single-Qubit Metamorphosis 117
4.4.13 Controlling Multi Single-Qubit Metamorphosis 117
4.4.14 Simple Metamorphosis 119
4.4.15 Unique Setup Gates 121
4.4.16 The Standard Circuit Model 122
References 123
5 Computational Algorithm Design in Quantum Systems 125
5.1 Introduction 125
5.2 Quantum Algorithm 125
5.3 Rule 1 Superposition 126
5.4 Rule 2 Quantum Entanglement 130
5.5 Rule 3 Quantum Metrology 132
5.6 Rule 4 Quantum Gates 133
5.7 Rule 5 Fault-Tolerant Quantum Gates 134
5.8 Quantum Concurrency 138
5.9 Rule 7 Quantum Interference 139
5.10 Rule 8 Quantum Parallelism 141
5.11 Summary 143
References 144
6 Optimization of an Amplification Algorithm 145
6.1 Introduction 145
6.2 The Effect of Availability Bias 146
6.2.1 Optimization of an Amplification Algorithm 147
6.2.2 Specifications of the Mathematical Amplification Algorithm 149
6.3 Quantum Amplitude Estimation and Quantum Counting 149
6.4 An Algorithm for Quantitatively Determining Amplitude 150
6.4.1 Mathematical Description of Amplitude Estimation Algorithm 151
6.5 Counting Quantum Particles: An Algorithm 151
6.5.1 Mathematical Description of Quantum Counting Algorithm 152
6.5.2 Related Algorithms and Techniques 152
References 153
7 Error-Correction Code in Quantum Noise 155
7.1 Introduction 155
7.2 Basic Forms of Error-Correcting Code in Quantum Technologies 156
7.2.1 Single Bit-Flip Errors in Quantum Computing 156
7.2.2 Single-Qubit Coding in Quantum Computing 161
7.2.3 Error-Correcting Code in Quantum Technology 162
7.3 Framework for Quantum Error-Correcting Codes 163
7.3.1 Traditional Based on Error-Correcting Codes 164
7.3.2 Quantum Error Decode Mechanisms 166
7.3.3 Correction Sets in Quantum Coding Error 167
7.3.4 Quantum Errors Detection 168
7.3.5 Basic Knowledge Representation of Error-Correcting Code 170
7.3.6 Quantum Codes as a Tool for Error Detection and Correction 173
7.3.7 Quantum Error Correction Across Multiple Blocks 176
7.3.8 Computing on Encoded Quantum States 177
7.3.9 Using Linear Transformation of Correctable Codes 177
7.3.10 Model of Classical Independent Error 178
7.3.11 Independent Quantum Inaccuracies Models 179
7.4 Coding Standards for CSS 182
7.4.1 Multiple Classical Identifiers 182
7.4.2 Traditional CSS Codes Satisfying a Duality Consequence 183
7.4.3 Code of Steane 186
7.5 Codes for Stabilizers 187
7.5.1 The Use of Binary Indicators in Quantum Correction of Errors 188
7.5.2 Using Pauli Indicators to Fix Errors in Quantum Techniques 188
7.5.3 Using Error-Correcting Stabilizer Algorithms 189
7.5.4 Stabilizer State Encoding Computation 191
7.6 A Stabilizer Role for CSS Codes 195
References 196
8 Tolerance for Inaccurate Information in Quantum Computing 197
8.1 Introduction 197
8.2 Initiating Stable Quantum Computing 198
8.3 Computational Error Tolerance Using Steane’s Code 200
8.3.1 The Complexity of Syndrome-Based Computation 201
8.3.2 Error Removal and Correction in Fault-Tolerant Systems 202
8.3.3 Steane’s Code Fault-Tolerant Gates 204
8.3.4 Measurement with Fault Tolerance 206
8.3.5 Readying the State for Fault Tolerance 207
8.4 The Strength of Quantum Computation 208
8.4.1 Combinatorial Coding 208
8.4.2 A Threshold Theorem 210
References 211
9 Cryptography in Quantum Computing 213
9.1 Introduction of RSA Encryption 213
9.2 Concept of RSA Encryption 214
9.3 Quantum Cipher Fundamentals 216
9.4 The Controlled-Not Invasion as an Illustration 219
9.5 Cryptography B92 Protocol 220
9.6 The E91 Protocol (Ekert) 221
References 221
10 Constructing Clusters for Quantum Computing 223
10.1 Introduction 223
10.1.1 State of Clusters 223
10.2 The Preparation of Cluster States 224
10.3 Nearest Neighbor Matrix 227
10.4 Stabilizer States 228
10.4.1 Aside: Entanglement Witness 230
10.5 Processing in Clusters 231
References 233
11 Advance Quantum Computing 235
11.1 Introduction 235
11.2 Computing with Superpositions 236
11.2.1 The Walsh–Hadamard Transformation 236
11.2.2 Quantum Parallelism 237
11.3 Notions of Complexity 239
11.3.1 Query Complexity 240
11.3.2 Communication Complexity 241
11.4 A Simple Quantum Algorithm 242
11.4.1 Deutsch’s Problem 242
11.5 Quantum Subroutines 243
11.5.1 The Importance of Unentangling Temporary Qubits in Quantum Subroutines 243
11.5.2 Phase Change for a Subset of Basis Vectors 244
11.5.3 State-Dependent Phase Shifts 246
11.5.4 State-Dependent Single-Qubit Amplitude Shifts 247
11.6 A Few Simple Quantum Algorithms 248
11.6.1 Deutsch–Jozsa Problem 248
11.6.2 Bernstein–Vazirani Problem 249
11.6.3 Simon’s Problem 252
11.6.4 Distributed Computation 253
11.7 Comments on Quantum Parallelism 254
11.8 Machine Models and Complexity Classes 255
11.8.1 Complexity Classes 257
11.8.2 Complexity: Known Results 258
11.9 Quantum Fourier Transformations 260
11.9.1 The Classical Fourier Transform 261
11.9.2 The Quantum Fourier Transform 263
11.9.3 A Quantum Circuit for Fast Fourier Transform 263
11.10 Shor’s Algorithm 265
11.10.1 Core Quantum Phenomena 266
11.10.2 Periodic Value Measurement and Classical Extraction 267
11.10.3 Shor’s Algorithm and Its Effectiveness 268
11.10.4 The Efficiency of Shor’s Algorithm 269
11.11 Omitting the Internal Measurement 270
11.12 Generalizations 271
11.12.1 The Problem of Discrete Logarithms 272
11.12.2 Hidden Subgroup Issues 272
11.13 The Application of Grover’s Algorithm It’s Time to Solve Some Difficulties 274
11.13.1 Explanation of the Superposition Technique 275
11.13.2 The Black Box’s Initial Configuration 275
11.13.3 The Iteration Step 276
11.13.4 Various of Iterations 277
11.14 Effective State Operations 279
11.14.1 2D Geometry 281
11.15 Grover’s Algorithm and Its Optimality 283
11.15.1 Reduction to Three Inequalities 284
11.16 Amplitude Amplification using Discrete Event Randomization of Grover’s Algorithm 286
11.16.1 Altering Each Procedure 286
11.16.2 Last Stage Variation 287
11.16.3 Solutions: Possibly Infinite 288
11.16.4 Varying the Number of Iterations 289
11.16.5 Quantum Counting 290
11.17 Implementing Grover’s Algorithm with Gain Boosting 291
References 292
12 Applications of Quantum Computing 295
12.1 Introduction 295
12.2 Teleportation 295
12.3 The Peres Partial Transposition Condition 298
12.4 Expansion of Transportation 303
12.5 Entanglement Swapping 304
12.6 Superdense Coding 305
References 307
Index 309
