Quantum Computing Fundamentals

Description

Preface xvii

Part I Preparatory Material

Chapter 1: Introduction to Essential Linear Algebra 2

1.1 What Is Linear Algebra?.. . . . . . . . . . . . . . . . . . . . 3

1.2 Some Basic Algebra.. . . . . . . . . . . . . . . . . . . . . 4

1.3 Matrix Math.. . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Vectors and Vector Spaces.. . . . . . . . . . . . . . . . . . 23

1.5 Set Theory.. . . . . . . . . . . . . . . . . . . . . . . . . 25

1.6 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 29

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 29

Chapter 2: Complex Numbers 32

2.1 What Are Complex Numbers?.. . . . . . . . . . . . . . . . . 32

2.2 Algebra of Complex Numbers.. . . . . . . . . . . . . . . . . 34

2.3 Complex Numbers Graphically.. . . . . . . . . . . . . . . . 38

2.4 Vector Representations of Complex Numbers.. . . . . . . . . . 45

2.5 Pauli Matrices.. . . . . . . . . . . . . . . . . . . . . . . 48

2.6 Transcendental Numbers.. . . . . . . . . . . . . . . . . . . 56

2.7 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter 3: Basic Physics for Quantum Computing 60

3.1 The Journey to Quantum.. . . . . . . . . . . . . . . . . . . 61

3.2 Quantum Physics Essentials.. . . . . . . . . . . . . . . . . 65

3.3 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 77

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 77

Chapter 4: Fundamental Computer Science for Quantum Computing 80

4.1 Data Structures.. . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Algorithms.. . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3 Computational Complexity.. . . . . . . . . . . . . . . . . . 93

4.4 Coding Theory.. . . . . . . . . . . . . . . . . . . . . . . 95

4.5 Logic Gates.. . . . . . . . . . . . . . . . . . . . . . . . 96

4.6 Computer Architecture.. . . . . . . . . . . . . . . . . . . 100

4.7 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 103

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 103

Chapter 5: Basic Information Theory 106

5.1 Basic Probability.. . . . . . . . . . . . . . . . . . . . . . 107

5.2 Set Theory.. . . . . . . . . . . . . . . . . . . . . . . . 108

5.3 Information Theory.. . . . . . . . . . . . . . . . . . . . . 112

5.4 Quantum Information.. . . . . . . . . . . . . . . . . . . . 118

5.5 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 120

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 120

Part II Basic Quantum Computing

Chapter 6: Basic Quantum Theory 122

6.1 Further with Quantum Mechanics.. . . . . . . . . . . . . . . 123

6.2 Quantum Decoherence.. . . . . . . . . . . . . . . . . . . 129

6.3 Quantum Electrodynamics.. . . . . . . . . . . . . . . . . . 131

6.4 Quantum Chromodynamics.. . . . . . . . . . . . . . . . . 133

6.5 Feynman Diagram.. . . . . . . . . . . . . . . . . . . . . 134

6.6 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 136

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 136

Chapter 7: Quantum Entanglement and QKD 138

7.1 Quantum Entanglement.. . . . . . . . . . . . . . . . . . . 138

7.2 Interpretation.. . . . . . . . . . . . . . . . . . . . . . . 143

7.3 QKE.. . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.4 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 151

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 152

Chapter 8: Quantum Architecture 154

8.1 Further with Qubits.. . . . . . . . . . . . . . . . . . . . . 154

8.2 Quantum Gates.. . . . . . . . . . . . . . . . . . . . . . 158

8.3 More with Gates.. . . . . . . . . . . . . . . . . . . . . . 166

8.4 Quantum Circuits. . . . . . . . . . . . . . . . . . . . . . 167

8.5 The D-Wave Quantum Architecture.. . . . . . . . . . . . . . 169

8.6 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 172

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 172

Chapter 9: Quantum Hardware 174

9.1 Qubits.. . . . . . . . . . . . . . . . . . . . . . . . . . 174

9.2 How Many Qubits Are Needed?. . . . . . . . . . . . . . . . 181

9.3 Addressing Decoherence.. . . . . . . . . . . . . . . . . . 182

9.4 Topological Quantum Computing.. . . . . . . . . . . . . . . 186

9.5 Quantum Essentials.. . . . . . . . . . . . . . . . . . . . 187

9.6 Quantum Networking.. . . . . . . . . . . . . . . . . . . . 188

9.7 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 191

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 191

Chapter 10: Quantum Algorithms 194

10.1 What Is an Algorithm?. . . . . . . . . . . . . . . . . . . . 194

10.2 Deutsch’s Algorithm.. . . . . . . . . . . . . . . . . . . . 197

10.3 Deutsch-Jozsa Algorithm.. . . . . . . . . . . . . . . . . . 199

10.4 Bernstein-Vazirani Algorithm.. . . . . . . . . . . . . . . . . 201

10.5 Simon’s Algorithm.. . . . . . . . . . . . . . . . . . . . . 202

10.6 Shor’s Algorithm.. . . . . . . . . . . . . . . . . . . . . . 203

10.7 Grover’s Algorithm. . . . . . . . . . . . . . . . . . . . . 209

10.8 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 211

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 211

Part III Quantum Computing and Cryptography

Chapter 11: Current Asymmetric Algorithms 212

11.1 RSA. . . . . . . . . . . . . . . . . . . . . . . . . . . 213

11.2 Diffie-Hellman.. . . . . . . . . . . . . . . . . . . . . . . 216

11.3 Elliptic Curve.. . . . . . . . . . . . . . . . . . . . . . . 219

11.4 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 227

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 227

Chapter 12: The Impact of Quantum Computing on Cryptography 228

12.1 Asymmetric Cryptography.. . . . . . . . . . . . . . . . . . 229

12.2 Specific Algorithms.. . . . . . . . . . . . . . . . . . . . . 231

12.3 Specific Applications. . . . . . . . . . . . . . . . . . . . 233

12.3.1 Digital Certificates. . . . . . . . . . . . . . . . . . 233

12.3.2 SSL/TLS. . . . . . . . . . . . . . . . . . . . . . 234

12.4 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 241

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 241

Chapter 13: Lattice-based Cryptography 244

13.1 Lattice-Based Mathematical Problems.. . . . . . . . . . . . . 245

13.2 Cryptographic Algorithms. . . . . . . . . . . . . . . . . . 249

13.3 Solving Lattice Problems.. . . . . . . . . . . . . . . . . . 256

13.4 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 259

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 259

Chapter 14: Multivariate Cryptography 262

14.1 Mathematics.. . . . . . . . . . . . . . . . . . . . . . . 262

14.2 Matsumoto-Imai.. . . . . . . . . . . . . . . . . . . . . . 264

14.3 Hidden Field Equations. . . . . . . . . . . . . . . . . . . 266

14.4 Multivariate Quadratic Digital Signature Scheme (MQDSS).. . . . 268

14.5 SFLASH.. . . . . . . . . . . . . . . . . . . . . . . . . 269

14.6 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 271

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 271

Chapter 15: Other Approaches to Quantum Resistant Cryptography 274

15.1 Hash Functions.. . . . . . . . . . . . . . . . . . . . . . 274

15.2 Code-Based Cryptography.. . . . . . . . . . . . . . . . . 279

15.3 Supersingular Isogeny Key Exchange.. . . . . . . . . . . . . 281

15.4 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 289

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 289

Part IV Quantum Programming

Chapter 16: Working with Q# 292

16.1 Basic Programming Concepts.. . . . . . . . . . . . . . . . 292

16.2 Getting Started with Q#.. . . . . . . . . . . . . . . . . . . 298

16.3 Grover’s Algorithm. . . . . . . . . . . . . . . . . . . . . 303

16.4 Deutsch-Jozsa Algorithm.. . . . . . . . . . . . . . . . . . 307

16.5 Bit Flipping.. . . . . . . . . . . . . . . . . . . . . . . . 310

16.6 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 311

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 311

Chapter 17: Working with QASM 314

17.1 Basic Programming Concepts.. . . . . . . . . . . . . . . . 315

17.2 Getting Started with QASM.. . . . . . . . . . . . . . . . . 319

17.3 Quantum Error Correction. . . . . . . . . . . . . . . . . . 320

17.4 Grover’s Algorithm. . . . . . . . . . . . . . . . . . . . . 322

17.5 Deutsch-Jozsa Algorithm.. . . . . . . . . . . . . . . . . . 326

17.6 Summary.. . . . . . . . . . . . . . . . . . . . . . . . . 328

Test Your Skills. . . . . . . . . . . . . . . . . . . . . . . 328

Appendix: Answers to Test Your Skills Questions 330

9780136793816, TOC, 5/7/2021

ONE-VOLUME INTRODUCTION TO QUANTUM COMPUTING

• Clearly explains core concepts, terminology, and techniques
• Covers the foundational physics, math, and information theory you need
• Provides hands-on practice with quantum programming
• The perfect beginner’s guide for anyone interested in a quantum computing career

Dr. Chuck Easttom brings together complete coverage of basic quantum computing concepts, terminology, and issues, along with key skills to get you started. Drawing on 30+ years as a computer science instructor, consultant, and researcher, Easttom demystifies the field’s underlying technical concepts and math, shows how quantum computing systems are designed and built, explains their implications for cyber security, and previews advances in quantum-resistant cryptography. Writing clearly and simply, he introduces two of today’s leading quantum programming languages, Microsoft Q# and QASM, and guides you through sample projects. Throughout, tests, projects, and review questions help you deepen and apply your knowledge. Whether you’re a student, professional, or manager, this guide will prepare you for the quantum computing revolution–and expand your career options, too.

• Master the linear algebra and other mathematical skills you’ll need
• Explore key physics ideas such as quantum states and uncertainty
• Review data structures, algorithms, and computing complexity
• Work with probability and set theory in quantum computing
• Familiarize yourself with basic quantum theory and formulae
• Understand quantum entanglement and quantum key distribution
• Discover how quantum computers are architected and built
• Explore several leading quantum algorithms
• Compare quantum and conventional asymmetric algorithms
• See how quantum computing might break traditional cryptography
• Discover several approaches to quantum-resistant cryptography
• Start coding with Q#, Microsoft’s quantum programming language
• Simulate quantum gates and algorithms with QASM

Dr. Chuck Easttom is the author of 31 books, including several on computer security, forensics, and cryptography. His books are used at more than 60 universities. He has also authored scientific papers (more than 70 so far) on digital forensics, cyber warfare, cryptography, and applied mathematics. He is an inventor with 22 computer science patents. He holds a Doctor of Science in cyber security (dissertation topic: a study of lattice-based cryptographic algorithms for post-quantum computing). He also has a Ph.D. in Technology, focusing on nanotechnology (dissertation title: “The Effects of Complexity on Carbon Nanotube Failures”) and a Ph.D. in Computer Science (dissertation title: “On the Application of Graph Theory to Digital Forensics”). He also has three master’s degrees (one in applied computer science, one in education, and one in systems engineering). He is a senior member of the IEEE and a senior member of the ACM (Association of Computing Machinery) as well as a member of IACR (International Association of Cryptological Research) and INCOSE (International Council on Systems Engineering). He is also a distinguished speaker of the ACM and a distinguished visitor of the IEEE Computer Society. He currently is an adjunct lecturer for Georgetown University.

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