Quantum Computing
Description
A helpful introduction to all aspects of quantum computing
Quantum computing is a field combining quantum mechanics--the physical science of nature at the scale of atoms and subatomic particles--and information science. Where ordinary computing uses bits, logical values whose position can either be 0 or 1, quantum computing is built around qubits, a fundamental unit of quantum information which can exist in a superposition of both states. As quantum computers are able to complete certain kinds of functions more accurately and efficiently than computers built on classical binary logic, quantum computing is an emerging frontier which promises to revolutionize information science and its applications.
This book provides a concise, accessible introduction to quantum computing. It begins by introducing the essentials of quantum mechanics that information and computer scientists require, before moving to detailed discussions of quantum computing in theory and practice. As quantum computing becomes an ever-greater part of the global information technology landscape, the knowledge in Quantum Computing will position readers to join a vital and highly marketable field of research and development.
The book's readers will also find:
* Detailed diagrams and illustrations throughout
* A broadly applicable quantum algorithm that improves on the best-known classical algorithms for a wide range of problems
* In-depth discussion of essential topics including key distribution, cluster state quantum computing, superconducting qubits, and more
Quantum Computing is perfect for advanced undergraduate and graduate students in computer science, engineering, mathematics, or the physical sciences, as well as for researchers and academics at the intersection of these fields who want a concise reference.
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Persons
Kuldeep Singh Kaswan, PhD, is Professor in the School of Computing Science and Engineering at Galgotias University, Greater Noida, India. He is co-editor of the Wiley-Scrivener title Swarm Intelligence: An Approach from Natural to Artificial.
Jagjit Singh Dhatterwal, PhD, is Associate Professor in the Department of Artificial Intelligence & Data Science at Koneru Lakshmaiah Education Foundation, Vaddeswaram, AP, India. He is co-editor of the Wiley-Scrivener title Swarm Intelligence: An Approach from Natural to Artificial.
Anupam Baliyan, PhD, is Additional Director with the University Institute of Engineering at Chandigarh University, Punjab, India.
Shalli Rani, PhD, is Professor at Chitkara University Institute of Engineering and Technology, Chitkara University, Punjab, India. She is co-editor of the Wiley title IoT-enabled Smart Healthcare Systems, Services and Applications.
Content
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
1
Introduction of Quantum Computing
1.1 Introduction
A significant advancement in computer science may take the form of a new algorithm that significantly outperforms the state of the art, or it may provide theoretical evidence that the state of the art cannot be significantly improved. The latter condition imposes a fundamental limit on the complexity of problems that any given computer can solve in a given amount of time. Increasing the computer's processing speed is the only way to increase the number of problems that can be solved. According to Moore's Law, the size of semiconductors (and, by extension, computing capability) has approximately doubled every two years since the 1960s. It is clear that, despite the fact that this development has been going on for decades, it cannot go on forever because of a number of basic physical constraints. As a result, quantum weirdness will dominate the behavior of the circuitry by 2020, and by 2050, the circuits will have achieved the lowest size at which knowledge can be permanently contained [1].
The results of this study have piqued the public's interest in how quantum theory may affect the future of computing over the next several decades. Is it possible, for instance, to make circuits immune to the influence of quantum effects? As an alternative, may quantum phenomena be exploited to do arithmetic? In order to do calculations, quantum computers take advantage of quantum phenomena. However, a quantum computer is not only a device with enhanced performance because of the faster speed of quantum-scale circuits. It is of more interest to the software programmer than to the theoretical physicist. After all, the computational complexity of algorithms executed on a certain CPU remains the same regardless of the CPU's clock speed. Different algorithms may provide better complexity in terms of the new variable P if the computer's architecture is altered to include some number P of processors. We may able to reduce the greatest feasible complexity for solving a specific problem from O(N) to O(N/P), if we have a good parallel extraction of processors. However, not all algorithms can be broken down into O(P)-independent portions that can be incorporated and enforced during the algorithm's operating time, therefore obtaining an O(P) complexity reduction is not always possible. To store and manipulate data, for instance, analog hardware and programmable real numbers may replace a discrete set of symbols, which would need a more radical redesign. It is possible that this design will prove to be far more powerful than the classic Turing machine. Because of the limitless precision with which a single physical value may be measured, it is possible to analyze massive amounts of data in parallel by treating them as a single unit cost. This is, of course, completely hypothetical since it assumes infinite precision can be maintained throughout those operations, and there is no reason to believe that such an infrastructure is physically conceivable. The potential of a quantum computer, which relies on the preservation of real, complex values, is underutilized [2].
1.2 What Is the Exact Meaning of Quantum Computing?
Large, complex datasets are no match for the speed with which quantum computers can process them. They use the foundations of quantum physics to speed up the process of doing complex computations. Quantum computers' ability to break cryptography and encrypted electronic communications is already changing portions of cybersecurity, and their usage in simulators with a practically endless quantity of variables has implications across fields, from biology to economics. The next large electronics race has already started [3], with some of the biggest names in industry, including Google, Microsoft, Intel, IBM, and Alibaba, exploring quantum computing to improve rates and other applications. Although Google has been studying quantum computing to speed up internet searches since at least 2009, the market for commercialized quantum entanglement is still in its infancy, and it is not yet obvious who will emerge as the market leader.
1.2.1 What Is Quantum Computing in Simple Terms?
Figure 1.1 depicts the interactions of matter in the universe at the level of fundamental particles, which provide the basis for special relativity, upon which quantum computing is founded. Bits can only be encrypted in classical computers if they have a value of 1 or 0.
Figure 1.1 David Deutsch father of quantum computing.
Source: Lulie Tanett (https://images.app.goo.gl/CQBoMf7JqWzXfr6r9).
1.3 Origin of Quantum Computing
Some types of computations now baffle today's computers and will continue to do so even if Moore's Law is extended indefinitely, although quantum computers may give a stronger correlation boost. Just imagine you have a phone book and need to find a certain number. A conventional computer would have to go through each listing in the phone book to find and provide the appropriate contact information. In theory, a computer system might scan an entire phone book in a fraction of a second, evaluating each line simultaneously and returning the result far faster than a modern computer [4]. The term "complex mathematical optimizing" is often used to describe the process of finding the best possible combination of elements and answers to a problem. Consider the costs of building the tallest building in the world, including machinery, food, labor, and permits. The challenge is in figuring out how to optimally allocate resources like money, time, and manpower. As a result, we may be able to plan for major projects with more efficiency with the aid of quantum computing if these factors are taken into account. Software development, supply chain management, finance, internet-based research, genomics, and other fields all face optimization challenges. The most challenging optimization problems in these fields are inherently well-suited for solution on a quantum machine [4] but stump conventional computers. In contrast to classical computers, which rely almost entirely on technological advances in transistors and microchips, quantum computers may evolve in ways that classical computers cannot. In quantum computers, transistors are not utilized (or classical bits). Substituting qubits for bits. In a quantum algorithm, qubits serve as the basic building blocks for pattern recognition. The example is shown in Figure 1.2.
Figure 1.2 Structure of bits and Qbits.
Source: Adapted from https://images.app.goo.gl/DeYCU9A7TeJvV5c16 Last accessed 25 Oct 2022.
Qubits may take on the characteristics of either a 0 or a 1, or they can have both at the same time. More choices exist to get accurate results quickly while doing computations. In addition, quantum entanglement and superposition are two important states of matter on which quantum computers depend. When applied to computing, these physical properties have the potential to greatly increase our ability to do very large computations [5].
Although Rigetti Computing's 19-qubit devices are the most powerful in the field of quantum computing, but after 2019, the business is moving on 128-qubit circuit. But as can be seen in Table 1.1, the race to build the most advanced quantum computer with the most qubits has been going on since at least the late 1990s.
Table 1.1 Quantum computing getting more powerful.
Year Labs Q-bits 1998 IBM, Oxford, Berkeley, Stanford, MIT 2 2000 Technical University of Munich 7 2006 Institute for Quantum Computing 12 2008 D-Wave System 28 2016 IBM 50 2018 Google 72 2020 Rigetti 1281.4 History of Quantum Computing
Conjugate coding was first developed in the 1960s by Stephen Wiesner. In the 1970s, James Park established the no-cloning theorem using his formulation. Alexander Holevo proved what is now known as Holevo's theorem, or Holevo's bound, in a paper that was published in 1973. This theorem states that even though n qubits may store more relevant data than n classical bits, only n conventional bits are obtainable. This is despite the fact that n qubits may store more information than n classical bits.
Research conducted by Charles H. Bennett demonstrates that it is feasible to carry out computing in a backward-compatible manner.
- In 1975, R. P. Poplavskii published (in Russian) thermodynamical models of information processing. This work highlights the computational difficulties of reproducing quantum systems on classical computers owing to the fact that the superposition principle is at play.
- In 1976, the Polish mathematician and physicist Roman Stanislaw Ingarden published Quantum Information Theory in the journal Reports on Mathematical Physics. Ingarden's paper "1976 Quantum Information Theory." This study, which was one of the early efforts to...
