Principles of Superconducting Quantum Computers

List of Figures xiii List of Tables xxv Preface xxvii Acknowledgments xxix About the Companion Website xxxi 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 1 1.1.3 No Cloning 3 1.1.4 Reversibility 3 1.1.5 Entanglement 3 1.2 Single-Qubit States 4 1.3 Measurement and the Born Rule 5 1.4 Unitary Operations and Single-Qubit Gates 6 1.5 Two-Qubit Gates 8 1.5.1 Two-Qubit States 8 1.5.2 Matrix Representation of Two-Qubit Gates 9 1.5.3 Controlled-NOT 11 1.6 Bell State 12 1.7 No Cloning, Revisited 13 1.8 Example: Deutsch’s Problem 15 1.9 Key Characteristics of Quantum Computing 18 1.10 Quantum Computing Systems 18 1.11 Exercises 22 2 Physics of Single Qubit Gates 25 2.1 Requirements for a Quantum Computer 25 2.2 Single Qubit Gates 25 2.2.1 Rotations 25 2.2.2 Two State Systems 33 2.2.3 Creating Rotations: Rabi Oscillations 38 2.3 Quantum State Tomography 42 2.4 Expectation Values and the Pauli Operators 44 2.5 Density Matrix 45 2.6 Exercises 48 3 Physics of Two Qubit Gates 51 3.1 √iSWAP Gate 51 3.2 Coupled Tunable Qubits 53 3.3 Cross Resonance Scheme 55 3.4 Other Controlled Gates 57 3.5 Two-Qubit States and the Density Matrix 59 3.6 Exercises 62 4 Superconducting Quantum Computer Systems 63 4.1 Transmission Lines 63 4.1.1 General Transmission Line Equations 63 4.1.2 Lossless Transmission Lines 65 4.1.3 Transmission Lines with Loss 67 4.2 Terminated Lossless Line 71 4.2.1 Reflection Coefficient 71 4.2.2 Power (Flow of Energy) and Return Loss 72 4.2.3 Standing Wave Ratio (SWR) 73 4.2.4 Impedance as a Function of Position 74 4.2.5 Quarter Wave Transformer 76 4.2.6 Coaxial, Microstrip, and Coplanar Lines 77 4.3 S Parameters 80 4.3.1 Lossless Condition 81 4.3.2 Reciprocity 81 4.4 Transmission (ABCD) Matrices 81 4.5 Attenuators 85 4.6 Circulators and Isolators 87 4.7 Power Dividers/Combiners 89 4.8 Mixers 92 4.9 Low-Pass Filters 95 4.10 Noise 97 4.10.1 Thermal Noise 97 4.10.2 Equivalent Noise Temperature 99 4.10.3 Noise Factor and Noise Figure 100 4.10.4 Attenuators and Noise 101 4.10.5 Noise in Cascaded Systems 103 4.11 Low Noise Amplifiers 104 4.12 Exercises 105 5 Resonators: Classical Treatment 107 5.1 Parallel Lumped Element Resonator 107 5.2 Capacitive Coupling to a Parallel Lumped-Element Resonator 109 5.3 Transmission Line Resonator 111 5.4 Capacitive Coupling to a Transmission Line Resonator 113 5.5 Capacitively-Coupled Lossless Resonators 117 5.6 Classical Model of Qubit Readout 120 5.7 Exercises 124 6 Resonators: Quantum Treatment 127 6.1 Lagrangian Mechanics 127 6.1.1 Hamilton’s Principle 127 6.1.2 Calculus of Variations 128 6.1.3 Lagrangian Equation of Motion 129 6.2 Hamiltonian Mechanics 130 6.3 Harmonic Oscillators 131 6.3.1 Classical Harmonic Oscillator 131 6.3.2 Quantum Mechanical Harmonic Oscillator 133 6.3.3 Raising and Lowering Operators 135 6.3.4 Can a Harmonic Oscillator Be Used as a Qubit? 137 6.4 Circuit Quantum Electrodynamics 138 6.4.1 Classical LC Resonant Circuit 138 6.4.2 Quantization of the LC Circuit 139 6.4.3 Circuit Electrodynamic Approach for General Circuits 140 6.4.4 Circuit Model for Transmission Line Resonator 141 6.4.5 Quantizing a Transmission Line Resonator 144 6.4.6 Quantized Coupled LC Resonant Circuits 144 6.4.7 Schrödinger, Heisenberg, and Interaction Pictures 147 6.4.8 Resonant Circuits and Qubits 150 6.4.9 The Dispersive Regime 153 6.5 Exercises 156 7 Theory of Superconductivity 159 7.1 Bosons and Fermions 159 7.2 Bloch Theorem 161 7.3 Free Electron Model for Metals 163 7.3.1 Discrete States in Finite Samples 163 7.3.2 Phonons 166 7.3.3 Debye Model 167 7.3.4 Electron–Phonon Scattering and Electrical Conductivity 168 7.3.5 Perfect Conductor vs. Superconductor 170 7.4 Bardeen, Cooper, and Schrieffer Theory of Superconductivity 172 7.4.1 Cooper Pair Model 172 7.4.2 Dielectric Function 175 7.4.3 Jellium 176 7.4.4 Scattering Amplitude and Attractive Electron–Electron Interaction 179 7.4.5 Interpretation of Attractive Interaction 180 7.4.6 Superconductor Hamiltonian 181 7.4.7 Superconducting Ground State 182 7.5 Electrodynamics of Superconductors 185 7.5.1 Cooper Pairs and the Macroscopic Wave Function 185 7.5.2 Potential Functions 186 7.5.3 London Equations 187 7.5.4 London Gauge 189 7.5.5 Penetration Depth 190 7.5.6 Flux Quantization 191 7.6 Chapter Summary 192 7.7 Exercises 193 8 Josephson Junctions 195 8.1 Tunneling 195 8.1.1 Reflection from a Barrier 196 8.1.2 Finite Thickness Barrier 198 8.2 Josephson Junctions 200 8.2.1 Current and Voltage Relations 200 8.2.2 Josephson Junction Hamiltonian 203 8.2.3 Quantized Josephson Junction Analysis 205 8.3 Superconducting Quantum Interference Devices (SQUIDs) 207 8.4 Josephson Junction Parametric Amplifiers 208 8.5 Exercises 209 9 Errors and Error Mitigation 211 9.1 NISQ Processors 211 9.2 Decoherence 212 9.3 State Preparation and Measurement Errors 214 9.4 Characterizing Gate Errors 215 9.5 State Leakage and Suppression Using Pulse Shaping 219 9.6 Zero-Noise Extrapolation 220 9.7 Optimized Control Using Deep Learning 223 9.8 Exercises 225 10 Quantum Error Correction 227 10.1 Review of Classical Error Correction 227 10.1.1 Error Detection 228 10.1.2 Error Correction: Repetition Code 228 10.1.3 Hamming Code 229 10.2 Quantum Errors 230 10.3 Detecting and Correcting Quantum Errors 232 10.3.1 Bit Flip 232 10.3.2 Phase Flip 234 10.3.3 Correcting Bit and Phase Flips: Shor’s 9-Qubit Code 235 10.3.4 Arbitrary Rotations 236 10.4 Stabilizer Codes 238 10.4.1 Stabilizers 238 10.4.2 Stabilizers for Error Correction 239 10.5 Operating on Logical Qubits 242 10.6 Error Thresholds 243 10.6.1 Concatenation of Error Codes 243 10.6.2 Threshold Theorem 244 10.7 Surface Codes 245 10.7.1 Stabilizers 246 10.7.2 Error Detection and Correction 247 10.7.3 Logical X and Z Operators 250 10.7.4 Multiple Qubits: Lattice Surgery 253 10.7.5 CNOT 257 10.7.6 Single-Qubit Gates 258 10.8 Summary and Further Reading 259 10.9 Exercises 261 11 Quantum Logic: Efficient Implementation of Classical Computations 263 11.1 Reversible Logic 264 11.1.1 Reversible Logic Gates 264 11.1.2 Reversible Logic Circuits 266 11.2 Quantum Logic Circuits 268 11.2.1 Entanglement and Uncomputing 269 11.2.2 Multi-Qubit Gates 270 11.2.3 Qubit Topology 270 11.3 Efficient Arithmetic Circuits: Adder 272 11.3.1 Quantum Ripple-Carry Adder 273 11.3.2 In-Place Ripple-Carry Adder 275 11.3.3 Carry-Lookahead Adder 277 11.3.4 Adder Comparison 281 11.4 Phase Logic 283 11.4.1 Controlled-Z and Controlled-Phase Gates 283 11.4.2 Selective Phase Change 285 11.4.3 Phase Logic Gates 287 11.5 Summary and Further Reading 288 11.6 Exercises 289 12 Some Quantum Algorithms 291 12.1 Computational Complexity 291 12.1.1 Quantum Program Run-Time 292 12.1.2 Classical Complexity Classes 292 12.1.3 Quantum Complexity 293 12.2 Grover’s Search Algorithm 294 12.2.1 Grover Iteration 294 12.2.2 Quantum Implementation 296 12.2.3 Generalizations 299 12.3 Quantum Fourier Transform 299 12.3.1 Discrete Fourier Transform 300 12.3.2 Inverse Discrete Fourier Transform 300 12.3.3 Quantum Implementation of the DFT 301 12.3.4 Encoding Quantum States 302 12.3.5 Quantum Implementation 304 12.3.6 Computational Complexity 306 12.4 Quantum Phase Estimation 307 12.4.1 Quantum Implementation 307 12.4.2 Computational Complexity and Other Issues 308 12.5 Shor’s Algorithm 309 12.5.1 Hybrid Classical-Quantum Algorithm 309 12.5.2 Finding the Period 310 12.5.3 Computational Complexity 314 12.6 Variational Quantum Algorithms 314 12.6.1 Variational Quantum Eigensolver 316 12.6.2 Quantum Approximate Optimization Algorithm 320 12.6.3 Challenges and Opportunities 323 12.7 Summary and Further Reading 324 12.8 Exercises 325 Bibliography 327 Index 339