Principles of Superconducting Quantum Computers

Explore the intersection of computer science, physics, and electrical and computer engineering with this discussion of the engineering of quantum computers

In Principles of Superconducting Quantum Computers, a pair of distinguished researchers delivers a comprehensive and insightful discussion of the building of quantum computing hardware and systems. Bridging the gaps between computer science, physics, and electrical and computer engineering, the book focuses on the engineering topics of devices, circuits, control, and error correction.

Using data from actual quantum computers, the authors illustrate critical concepts from quantum computing. Questions and problems at the end of each chapter assist students with learning and retention, while the text offers descriptions of fundamentals concepts ranging from the physics of gates to quantum error correction techniques.

The authors provide efficient implementations of classical computations, and the book comes complete with a solutions manual and demonstrations of many of the concepts discussed within. It also includes:

• A thorough introduction to qubits, gates, and circuits, including unitary transformations, single qubit gates, and controlled (two qubit) gates
• Comprehensive explorations of the physics of single qubit gates, including the requirements for a quantum computer, rotations, two-state systems, and Rabi oscillations
• Practical discussions of the physics of two qubit gates, including tunable qubits, SWAP gates, controlled-NOT gates, and fixed frequency qubits
• In-depth examinations of superconducting quantum computer systems, including the need for cryogenic temperatures, transmission lines, S parameters, and more

Ideal for senior-level undergraduate and graduate students in electrical and computer engineering programs, Principles of Superconducting Quantum Computers also deserves a place in the libraries of practicing engineers seeking a better understanding of quantum computer systems.

Daniel D. Stancil, PhD, is the Alcoa Distinguished Professor and Head of Electrical and Computer Engineering at North Carolina State University. In addition to quantum computing, his research interests include spin waves, and microwave and optical devices and systems..

Gregory T. Byrd, PhD, is Professor and Associate Head of Electrical and Computer Engineering at North Carolina State University. His research focuses on both classical and quantum computer architecture and systems.

1 Qubits, Gates, and Circuits 1

1.1 Bits and Qubits . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Circuits in Space vs. Circuits in Time . . . . . . . 1

1.1.2 Superposition . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 No Cloning . . . . . . . . . . . . . . . . . . . . . . 3

1.1.4 Reversibility . . . . . . . . . . . . . . . . . . . . . 4

1.1.5 Entanglement . . . . . . . . . . . . . . . . . . . . . 4

1.2 Single-Qubit States . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Measurement and the Born Rule . . . . . . . . . . . . . . 6

1.4 Unitary Operations and Single-Qubit Gates . . . . . . . . 7

1.5 Two-Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.1 Two-Qubit States . . . . . . . . . . . . . . . . . . . 9

1.5.2 Two-Qubit Gates . . . . . . . . . . . . . . . . . . . 11

1.5.3 Controlled-NOT . . . . . . . . . . . . . . . . . . . 13

1.6 Bell State . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7 No Cloning, Revisited . . . . . . . . . . . . . . . . . . . . 15

1.8 Example: Deutsch’s Problem . . . . . . . . . . . . . . . . 17

1.9 Key Characteristics of Quantum Computing . . . . . . . . 20

1.10 Quantum Computing Systems . . . . . . . . . . . . . . . . 22

1.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Physics of Single Qubit Gates 29

2.1 Requirements for a Quantum Computer . . . . . . . . . . 29

2.2 Single Qubit Gates . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 Two State Systems . . . . . . . . . . . . . . . . . . 38

2.2.3 Creating Rotations: Rabi Oscillations . . . . . . . 44

2.3 Quantum State Tomography . . . . . . . . . . . . . . . . 49

2.4 Expectation Values and the Pauli Operators . . . . . . . . 51

2.5 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

iii

iv CONTENTS

3 Physics of Two Qubit Gates 59

3.1 √

iSWAP Gate . . . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Coupled Tunable Qubits . . . . . . . . . . . . . . . . . . . 61

3.3 Fixed-frequency Qubits . . . . . . . . . . . . . . . . . . . 64

3.4 Other Controlled Gates . . . . . . . . . . . . . . . . . . . 66

3.5 Two-qubit States and the Density Matrix . . . . . . . . . 68

3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4 Superconducting Quantum Computer Systems 73

4.1 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . 73

4.1.1 General Transmission Line Equations . . . . . . . 73

4.1.2 Lossless Transmission Lines . . . . . . . . . . . . . 75

4.1.3 Transmission Lines with Loss . . . . . . . . . . . . 77

4.2 Terminated Lossless Line . . . . . . . . . . . . . . . . . . 82

4.2.1 Reflection Coefficient . . . . . . . . . . . . . . . . . 82

4.2.2 Power (Flow of Energy) and Return Loss . . . . . 84

4.2.3 Standing Wave Ratio (SWR) . . . . . . . . . . . . 85

4.2.4 Impedance as a Function of Position . . . . . . . . 86

4.2.5 Quarter Wave Transformer . . . . . . . . . . . . . 88

4.2.6 Coaxial, Microstrip, and Co-planar Lines . . . . . 89

4.3 S Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.1 Lossless Condition . . . . . . . . . . . . . . . . . . 93

4.3.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Transmission (ABCD) Matrices . . . . . . . . . . . . . . . 94

4.5 Attenuators . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.6 Circulators and Isolators . . . . . . . . . . . . . . . . . . . 100

4.7 Power Dividers/Combiners . . . . . . . . . . . . . . . . . 102

4.8 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.9 Low-pass Filters . . . . . . . . . . . . . . . . . . . . . . . 111

4.10 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.10.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . 113

4.10.2 Equivalent Noise Temperature . . . . . . . . . . . 116

4.10.3 Noise Factor and Noise Figure . . . . . . . . . . . 117

4.10.4 Attenuators and Noise . . . . . . . . . . . . . . . . 118

4.10.5 Noise in Cascaded Systems . . . . . . . . . . . . . 120

4.11 Low Noise Amplifiers . . . . . . . . . . . . . . . . . . . . . 121

4.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5 Resonators: Classical Treatment 125

5.1 Parallel Lumped Element Resonator . . . . . . . . . . . . 125

5.2 Capacitive Coupling to a Parallel Lumped-Element Res[1]onator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3 Transmission Line Resonator . . . . . . . . . . . . . . . . 130

5.4 Capacitive Coupling to a Transmission Line Resonator . . 133

5.5 Capacitively-Coupled Lossless Resonators . . . . . . . . . 136

CONTENTS v

5.6 Classical Model of Qubit Readout . . . . . . . . . . . . . 142

5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6 Resonators: Quantum Treatment 149

6.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . 149

6.1.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . 149

6.1.2 Calculus of Variations . . . . . . . . . . . . . . . . 150

6.1.3 Lagrangian Equation of Motion . . . . . . . . . . . 151

6.2 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . 153

6.3 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . 153

6.3.1 Classical Harmonic Oscillator . . . . . . . . . . . . 154

6.3.2 Quantum Mechanical Harmonic Oscillator . . . . . 156

6.3.3 Raising and Lowering Operators . . . . . . . . . . 158

6.3.4 Can a Harmonic Oscillator be used as a Qubit? . . 160

6.4 Circuit Quantum Electrodynamics . . . . . . . . . . . . . 162

6.4.1 Classical LC Resonant Circuit . . . . . . . . . . . 162

6.4.2 Quantization of the LC Circuit . . . . . . . . . . . 163

6.4.3 Circuit Electrodynamic Approach for General Cir[1]cuits . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.4.4 Circuit Model for Transmission Line Resonator . . 165

6.4.5 Quantizing a Transmission Line Resonator . . . . 168

6.4.6 Quantized Coupled LC Resonant Circuits . . . . . 169

6.4.7 Schrödinger, Heisenberg, and Interaction Pictures 172

6.4.8 Resonant Circuits and Qubits . . . . . . . . . . . . 175

6.4.9 The Dispersive Regime . . . . . . . . . . . . . . . . 178

6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

7 Theory of Superconductivity 183

7.1 Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . 184

7.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . 186

7.3 Free Electron Model for Metals . . . . . . . . . . . . . . . 188

7.3.1 Discrete States in Finite Samples . . . . . . . . . . 189

7.3.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . 191

7.3.3 Debye Model . . . . . . . . . . . . . . . . . . . . . 193

7.3.4 Electron-Phonon Scattering and Electrical Con[1]ductivity . . . . . . . . . . . . . . . . . . . . . . . 194

7.3.5 Perfect Conductor vs. Superconductor . . . . . . . 196

7.4 Bardeen, Cooper and Schrieffer Theory of Superconduc[1]tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.4.1 Cooper Pair Model . . . . . . . . . . . . . . . . . . 199

7.4.2 Dielectric Function . . . . . . . . . . . . . . . . . . 203

7.4.3 Jellium . . . . . . . . . . . . . . . . . . . . . . . . 204

7.4.4 Scattering Amplitude and Attractive Electron-Electron

Interaction . . . . . . . . . . . . . . . . . . . . . . 208

7.4.5 Interpretation of Attractive Interaction . . . . . . 209

vi CONTENTS

7.4.6 Superconductor Hamiltonian . . . . . . . . . . . . 210

7.4.7 Superconducting Ground State . . . . . . . . . . . 211

7.5 Electrodynamics of Superconductors . . . . . . . . . . . . 215

7.5.1 Cooper Pairs and the Macroscopic Wave Function 215

7.5.2 Potential Functions . . . . . . . . . . . . . . . . . . 216

7.5.3 London Equations . . . . . . . . . . . . . . . . . . 217

7.5.4 London Gauge . . . . . . . . . . . . . . . . . . . . 219

7.5.5 Penetration Depth . . . . . . . . . . . . . . . . . . 220

7.5.6 Flux Quantization . . . . . . . . . . . . . . . . . . 221

7.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . 223

7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

8 Josephson Junctions 225

8.1 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.1.1 Reflection from a Barrier . . . . . . . . . . . . . . 226

8.1.2 Finite Thickness Barrier . . . . . . . . . . . . . . . 229

8.2 Josephson Junctions . . . . . . . . . . . . . . . . . . . . . 231

8.2.1 Current and Voltage Relations . . . . . . . . . . . 231

8.2.2 Josephson Junction Hamiltonian . . . . . . . . . . 235

8.2.3 Quantized Josephson Junction Analysis . . . . . . 237

8.3 Superconducting Quantum Interference Devices (SQUIDs) 239

8.4 Josephson Junction Parametric Amplifiers . . . . . . . . . 241

8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

9 Errors and Error Mitigation 245

9.1 NISQ Processors . . . . . . . . . . . . . . . . . . . . . . . 245

9.2 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . 246

9.3 State Preparation and Measurement Errors . . . . . . . . 248

9.4 Characterizing Gate Errors . . . . . . . . . . . . . . . . . 250

9.5 State Leakage and Suppression using Pulse Shaping . . . 254

9.6 Zero-Noise Extrapolation . . . . . . . . . . . . . . . . . . 257

9.7 Optimized Control using Deep Learning . . . . . . . . . . 260

9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

10 Quantum Error Correction 265

10.1 Review of Classical Error Correction . . . . . . . . . . . . 265

10.1.1 Error Detection . . . . . . . . . . . . . . . . . . . . 266

10.1.2 Error Correction: Repetition Code . . . . . . . . . 267

10.1.3 Hamming Code . . . . . . . . . . . . . . . . . . . . 268

10.2 Quantum Errors . . . . . . . . . . . . . . . . . . . . . . . 269

10.3 Detecting and Correcting Quantum Errors . . . . . . . . . 272

10.3.1 Bit Flip . . . . . . . . . . . . . . . . . . . . . . . . 272

10.3.2 Phase Flip . . . . . . . . . . . . . . . . . . . . . . 274

10.3.3 Correcting Bit and Phase Flips: Shor’s 9-qubit Code275

10.3.4 Arbitrary Rotations . . . . . . . . . . . . . . . . . 277

CONTENTS vii

10.4 Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . 279

10.4.1 Stabilizers . . . . . . . . . . . . . . . . . . . . . . . 279

10.4.2 Stabilizers for Error Correction . . . . . . . . . . . 280

10.5 Operating on Logical Qubits . . . . . . . . . . . . . . . . 283

10.6 Error Thresholds . . . . . . . . . . . . . . . . . . . . . . . 285

10.6.1 Concatenation of Error Codes . . . . . . . . . . . . 286

10.6.2 Threshold Theorem . . . . . . . . . . . . . . . . . 286

10.7 Surface Codes . . . . . . . . . . . . . . . . . . . . . . . . . 288

10.7.1 Stabilizers . . . . . . . . . . . . . . . . . . . . . . . 289

10.7.2 Error Detection and Correction . . . . . . . . . . . 291

10.7.3 Logical X and Z Operators . . . . . . . . . . . . . 295

10.7.4 Multiple Qubits: Lattice Surgery . . . . . . . . . . 297

10.7.5 CNOT . . . . . . . . . . . . . . . . . . . . . . . . . 301

10.7.6 Single-Qubit Gates . . . . . . . . . . . . . . . . . . 305

10.8 Summary and Further Reading . . . . . . . . . . . . . . . 306

10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

11 Quantum Logic: Efficient Implementation of Classical

Computations 309

11.1 Reversible Logic . . . . . . . . . . . . . . . . . . . . . . . 310

11.1.1 Reversible Logic Gates . . . . . . . . . . . . . . . . 311

11.1.2 Reversible Logic Circuits . . . . . . . . . . . . . . 313

11.2 Quantum Logic Circuits . . . . . . . . . . . . . . . . . . . 317

11.2.1 Entanglement and Uncomputing . . . . . . . . . . 317

11.2.2 Multi-qubit gates . . . . . . . . . . . . . . . . . . . 319

11.2.3 Qubit topology . . . . . . . . . . . . . . . . . . . . 321

11.3 Efficient Arithmetic Circuits: Adder . . . . . . . . . . . . 322

11.3.1 Quantum Ripple Carry Adder . . . . . . . . . . . . 323

11.3.2 In-place Ripple Carry Adder . . . . . . . . . . . . 326

11.3.3 Carry-Lookahead Adder . . . . . . . . . . . . . . . 329

11.3.4 Adder Comparison . . . . . . . . . . . . . . . . . . 334

11.4 Phase Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 336

11.4.1 Controlled-Z and Controlled-Phase Gates . . . . . 336

11.4.2 Selective Phase Change . . . . . . . . . . . . . . . 339

11.4.3 Phase Logic Gates . . . . . . . . . . . . . . . . . . 341

11.5 Summary and Further Reading . . . . . . . . . . . . . . . 342

11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

12 Some Quantum Algorithms 347

12.1 Computational Complexity . . . . . . . . . . . . . . . . . 347

12.1.1 Quantum Program Run-Time . . . . . . . . . . . . 348

12.1.2 Classical Complexity Classes . . . . . . . . . . . . 349

12.1.3 Quantum Complexity . . . . . . . . . . . . . . . . 350

12.2 Grover’s Search Algorithm . . . . . . . . . . . . . . . . . . 351

12.2.1 Grover Iteration . . . . . . . . . . . . . . . . . . . 351

viii CONTENTS

12.2.2 Quantum Implementation . . . . . . . . . . . . . . 354

12.2.3 Generalizations . . . . . . . . . . . . . . . . . . . . 357

12.3 Quantum Fourier Transform . . . . . . . . . . . . . . . . . 358

12.3.1 Frequencies and Quantum-encoded Signals . . . . 358

12.3.2 Inverse QFT . . . . . . . . . . . . . . . . . . . . . 361

12.3.3 Quantum Implementation . . . . . . . . . . . . . . 362

12.3.4 Computational Complexity . . . . . . . . . . . . . 365

12.4 Quantum Phase Estimation . . . . . . . . . . . . . . . . . 365

12.4.1 Quantum Implementation . . . . . . . . . . . . . . 366

12.4.2 Computational Complexity and Other Issues . . . 367

12.5 Shor’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . 368

12.5.1 Hybrid Classical-Quantum Algorithm . . . . . . . 368

12.5.2 Finding the Period . . . . . . . . . . . . . . . . . . 370

12.5.3 Computational Complexity . . . . . . . . . . . . . 373

12.6 Variational Quantum Algorithms . . . . . . . . . . . . . . 375

12.6.1 Variational Quantum Eigensolver . . . . . . . . . . 377

12.6.2 Quantum Approximate Optimization Algorithm . 382

12.6.3 Challenges and Opportunities . . . . . . . . . . . . 386

12.7 Summary and Further Reading . . . . . . . . . . . . . . . 387

12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.