Principles of Superconducting Quantum Computers
1. Edition April 2022
384 Pages, Hardcover
Wiley & Sons Ltd
ISBN:
978-1-119-75072-7
John Wiley & Sons
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.
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 square root
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
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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
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 square root
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
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.
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.
