Quantum Computing

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.

• Preface xiiiAuthor Biography xv1 Introduction of Quantum Computing 11.1 Introduction 11.2 What Is the Exact Meaning of Quantum Computing? 21.2.1 What Is Quantum Computing in Simple Terms? 21.3 Origin of Quantum Computing 31.4 History of Quantum Computing 51.5 Quantum Communication 191.6 Build Quantum Computer Structure 191.7 Principle Working of Quantum Computers 211.7.1 Kinds of Quantum Computing 211.8 Quantum Computing Use in Industry 231.9 Investors Invest Money in Quantum Technology 241.10 Applications of Quantum Computing 261.11 Quantum Computing as a Solution Technology 291.11.1 Quantum Artificial Intelligence 291.11.2 How Close Are We to Quantum Supremacy? 301.12 Conclusion 30References 312 Pros and Cons of Quantum Computing 332.1 Introduction 332.2 Quantum as a Numerical Process 332.3 Quantum Complexity 342.4 The Pros and Cons of the Quantum Computational Framework 362.5 Further Benefits of Quantum Computing 372.6 Further Drawbacks to Quantum Computing 382.7 Integrating Quantum and Classical Techniques 382.8 Framework of QRAM 392.9 Computing Algorithms in the Quantum World 402.9.1 Programming Quantum Processes 422.10 Modification of Quantum Building Blocks 42References 433 Methods and Instrumentation for Quantum Computing 453.1 Basic Information of Quantum Computing 453.2 Signal Information in Quantum Computing 473.3 Quantum Data Entropy 473.4 Basics of Probability in Quantum Computing 503.5 Quantum Theorem of No-Cloning 523.6 Measuring Distance 533.7 Fidelity in Quantum Theory 583.8 Quantum Entanglement 623.9 Information Content and Entropy 66References 714 Foundations of Quantum Computing 734.1 Single-Qubit 734.1.1 Photon Polarization in Quantum Computing 734.2 Multi-qubit 764.2.1 Blocks of Quantum States 764.2.2 Submission of Vector Space in Quantum Computing 774.2.3 Vector Spacing in Quantum Blocks 774.2.4 States of n-Qubit Technology 794.2.5 States of Entangled 814.2.6 Classical Measuring of Multi-Qubit 844.3 Measuring of Multi-Qubit 874.3.1 Mathematical Functions in Quantum Operations 87Example 884.3.2 Operator Measuring Qubits Projection 894.3.3 The Measurement Postulate 944.3.4 EPR Paradox and Bell’s Theorem 994.3.5 Layout of Bell’s Theorem 1014.3.6 Statistical Predicates of Quantum Mechanics 1014.3.7 Predictions of Bell’s Theorem 1024.3.8 Bell’s Inequality 1034.4 States of Quantum Metamorphosis 1054.4.1 Solitary Steps Metamorphosis 1064.4.2 Irrational Metamorphosis: The No-Cloning Principle 1074.4.3 The Pauli Transformations 1094.4.4 The Hadamard Metamorphosis 1094.4.5 Multi-Qubit Metamorphosis from Single-Qubit 1094.4.6 The Controlled-NOT and Other Singly Controlled Gates 1104.4.7 Opaque Coding 1134.4.8 Basic Bits in Opaque Coding 1144.4.9 Quantum Message Teleportation 1144.4.10 Designing and Constructing Quantum Circuits 1164.4.11 Single Qubit Manipulating Quantum State 1164.4.12 Controlling Single-Qubit Metamorphosis 1174.4.13 Controlling Multi Single-Qubit Metamorphosis 1174.4.14 Simple Metamorphosis 1194.4.15 Unique Setup Gates 1214.4.16 The Standard Circuit Model 122References 1235 Computational Algorithm Design in Quantum Systems 1255.1 Introduction 1255.2 Quantum Algorithm 1255.3 Rule 1 Superposition 1265.4 Rule 2 Quantum Entanglement 1305.5 Rule 3 Quantum Metrology 1325.6 Rule 4 Quantum Gates 1335.7 Rule 5 Fault-Tolerant Quantum Gates 1345.8 Quantum Concurrency 1385.9 Rule 7 Quantum Interference 1395.10 Rule 8 Quantum Parallelism 1415.11 Summary 143References 1446 Optimization of an Amplification Algorithm 1456.1 Introduction 1456.2 The Effect of Availability Bias 1466.2.1 Optimization of an Amplification Algorithm 1476.2.2 Specifications of the Mathematical Amplification Algorithm 1496.3 Quantum Amplitude Estimation and Quantum Counting 1496.4 An Algorithm for Quantitatively Determining Amplitude 1506.4.1 Mathematical Description of Amplitude Estimation Algorithm 1516.5 Counting Quantum Particles: An Algorithm 1516.5.1 Mathematical Description of Quantum Counting Algorithm 1526.5.2 Related Algorithms and Techniques 152References 1537 Error-Correction Code in Quantum Noise 1557.1 Introduction 1557.2 Basic Forms of Error-Correcting Code in Quantum Technologies 1567.2.1 Single Bit-Flip Errors in Quantum Computing 1567.2.2 Single-Qubit Coding in Quantum Computing 1617.2.3 Error-Correcting Code in Quantum Technology 1627.3 Framework for Quantum Error-Correcting Codes 1637.3.1 Traditional Based on Error-Correcting Codes 1647.3.2 Quantum Error Decode Mechanisms 1667.3.3 Correction Sets in Quantum Coding Error 1677.3.4 Quantum Errors Detection 1687.3.5 Basic Knowledge Representation of Error-Correcting Code 1707.3.6 Quantum Codes as a Tool for Error Detection and Correction 1737.3.7 Quantum Error Correction Across Multiple Blocks 1767.3.8 Computing on Encoded Quantum States 1777.3.9 Using Linear Transformation of Correctable Codes 1777.3.10 Model of Classical Independent Error 1787.3.11 Independent Quantum Inaccuracies Models 1797.4 Coding Standards for CSS 1827.4.1 Multiple Classical Identifiers 1827.4.2 Traditional CSS Codes Satisfying a Duality Consequence 1837.4.3 Code of Steane 1867.5 Codes for Stabilizers 1877.5.1 The Use of Binary Indicators in Quantum Correction of Errors 1887.5.2 Using Pauli Indicators to Fix Errors in Quantum Techniques 1887.5.3 Using Error-Correcting Stabilizer Algorithms 1897.5.4 Stabilizer State Encoding Computation 1917.6 A Stabilizer Role for CSS Codes 195References 1968 Tolerance for Inaccurate Information in Quantum Computing 1978.1 Introduction 1978.2 Initiating Stable Quantum Computing 1988.3 Computational Error Tolerance Using Steane’s Code 2008.3.1 The Complexity of Syndrome-Based Computation 2018.3.2 Error Removal and Correction in Fault-Tolerant Systems 2028.3.3 Steane’s Code Fault-Tolerant Gates 2048.3.4 Measurement with Fault Tolerance 2068.3.5 Readying the State for Fault Tolerance 2078.4 The Strength of Quantum Computation 2088.4.1 Combinatorial Coding 2088.4.2 A Threshold Theorem 210References 2119 Cryptography in Quantum Computing 2139.1 Introduction of RSA Encryption 2139.2 Concept of RSA Encryption 2149.3 Quantum Cipher Fundamentals 2169.4 The Controlled-Not Invasion as an Illustration 2199.5 Cryptography B92 Protocol 2209.6 The E91 Protocol (Ekert) 221References 22110 Constructing Clusters for Quantum Computing 22310.1 Introduction 22310.1.1 State of Clusters 22310.2 The Preparation of Cluster States 22410.3 Nearest Neighbor Matrix 22710.4 Stabilizer States 22810.4.1 Aside: Entanglement Witness 23010.5 Processing in Clusters 231References 23311 Advance Quantum Computing 23511.1 Introduction 23511.2 Computing with Superpositions 23611.2.1 The Walsh–Hadamard Transformation 23611.2.2 Quantum Parallelism 23711.3 Notions of Complexity 23911.3.1 Query Complexity 24011.3.2 Communication Complexity 24111.4 A Simple Quantum Algorithm 24211.4.1 Deutsch’s Problem 24211.5 Quantum Subroutines 24311.5.1 The Importance of Unentangling Temporary Qubits in Quantum Subroutines 24311.5.2 Phase Change for a Subset of Basis Vectors 24411.5.3 State-Dependent Phase Shifts 24611.5.4 State-Dependent Single-Qubit Amplitude Shifts 24711.6 A Few Simple Quantum Algorithms 24811.6.1 Deutsch–Jozsa Problem 24811.6.2 Bernstein–Vazirani Problem 24911.6.3 Simon’s Problem 25211.6.4 Distributed Computation 25311.7 Comments on Quantum Parallelism 25411.8 Machine Models and Complexity Classes 25511.8.1 Complexity Classes 25711.8.2 Complexity: Known Results 25811.9 Quantum Fourier Transformations 26011.9.1 The Classical Fourier Transform 26111.9.2 The Quantum Fourier Transform 26311.9.3 A Quantum Circuit for Fast Fourier Transform 26311.10 Shor’s Algorithm 26511.10.1 Core Quantum Phenomena 26611.10.2 Periodic Value Measurement and Classical Extraction 26711.10.3 Shor’s Algorithm and Its Effectiveness 26811.10.4 The Efficiency of Shor’s Algorithm 26911.11 Omitting the Internal Measurement 27011.12 Generalizations 27111.12.1 The Problem of Discrete Logarithms 27211.12.2 Hidden Subgroup Issues 27211.13 The Application of Grover’s Algorithm It’s Time to Solve Some Difficulties 27411.13.1 Explanation of the Superposition Technique 27511.13.2 The Black Box’s Initial Configuration 27511.13.3 The Iteration Step 27611.13.4 Various of Iterations 27711.14 Effective State Operations 27911.14.1 2D Geometry 28111.15 Grover’s Algorithm and Its Optimality 28311.15.1 Reduction to Three Inequalities 28411.16 Amplitude Amplification using Discrete Event Randomization of Grover’s Algorithm 28611.16.1 Altering Each Procedure 28611.16.2 Last Stage Variation 28711.16.3 Solutions: Possibly Infinite 28811.16.4 Varying the Number of Iterations 28911.16.5 Quantum Counting 29011.17 Implementing Grover’s Algorithm with Gain Boosting 291References 29212 Applications of Quantum Computing 29512.1 Introduction 29512.2 Teleportation 29512.3 The Peres Partial Transposition Condition 29812.4 Expansion of Transportation 30312.5 Entanglement Swapping 30412.6 Superdense Coding 305References 307Index 309