# Quantum Computing Fundamentals, 1st edition

• William Chuck Easttom

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## Overview

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

9780136793816, TOC, 5/7/2021

ISBN-13: 9780137460328

Subject: Networking & Security

Category: Distributed Systems

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