1 Chapter 1 8
1.1. Overview 8
1.2 How Quantum Computing Differs From Classical Computing 12
1.3 Brief History 15
1.4 How To Compare Computing Performance? (Complexity theory) 17
1.5 Quantum Computing Architectures (Examples) 20
1.6 Quantum Computing Paradigms 28
Gate based QC, adiabatic QC, measurement-based QC 28
1.7 Measurement-Based Quantum Computing (MBQC) 29
1.8 Where Quantum Computers Might Shine 31
1.9 Conclusion 33
2 Chapter 2 Quantum Computing Basics 34
2.1 Overview Introduction 34
2.2 Bits and Qubits 34
2.3 Coherence and Decoherence 36
2.4 Bra-Ket Basics 38
2.5 Gates 39
2.6 Single Qubit Gates 40
2.7 Measurement Basics in Quantum Computing 47
2.8 Example Quantum Algorithms - Deutsch and Deutsch-Josza Algorithms 48
2.9 Groverâs Search Algorithm Example 52
2.10 Two-Qubit States 60
2.11 Tensor Products 61
2.12 Two-Qubit Gates 62
2.13 Unitarity and Itâs Importance for Quantum Computing 67
2.14 Entanglement 69
2.15 History of Bell Pairs 71
2.16 Two-Qubit Measurements Applied to Entangled Qubits 72
2.17 Entangled States and Separable States 74
2.18 The Power of Entanglement for Quantum Computing 74
2.19 Quantum Errors 75
2.20 Nature of Quantum Errors 75
2.20.1 Bit-Flip Errors 75
2.20.2 Phase-Flip Errors 76
2.20.3 Bit-and-Phase Flip Errors (Y Errors) 76
2.21 Amplitude Damping and Dephasing Errors 77
2.22 Depolarizing Errors 78
2.23 Quantum Error Correction Codes 78
2.24 Autonomous and Intrinsic Quantum Error Protection 82
2.25 Short Introduction to Qudits 83
2.26 Qudit Gates 84
2.27 Qutrit Based Grover Search Algorithm 85
2.28 Conclusions 87
3 Chapter 3: Superconducting Qubit 89
3.1 Overview Introduction 89
3.2 Superconducting Qubit Platform 90
3.3 Brief History 92
3.3.1 From Cooper Pair Box to Transmon 92
3.4 Digression: Quantum Mechanics Refresher 98
3.5 Hamiltonian and Conjugate Variables 98
3.6 The Momentum and Energy Operators 100
3.7 Commutators 101
3.8 Quantum Mechanics for Simple Harmonic Oscillator 102
3.9 The Operator Method 104
3.10 LC Circuit Quantum Harmonic Oscillator 108
3.11 Wave Functions for the SHO via Schrodinger Equation 111
3.12 Josephson junction 113
3.13 Cooper pair box 120
3.14 Wave Functions from Schr¶dinger Equation 124
3.15 Transmon Device 126
3.16 Two-Level Transmon Qubit 129
3.17 SQUIDS 131
3.18 Rabi Oscillations 133
3.19 Relaxation, Coherence and Dephasing Times 136
3.20 Implementing Gates with Transmons 138
3.20.1 Implementation of X and Y Rotations: 138
3.21 Direct Capacitive Coupling 140
3.22 Back Action Between Coupled Qubits 141
3.23 Tunable Coupling Using a Coupler 142
3.24 Implementation of a Hadamard Gate on a Transmon Qubit 143
3.25 Entangling Coupled Transmons 144
3.26 Coupling Transmons to Resonators 144
3.27 Jaynes-Cummings Hamiltonian 150
3.28 The Rotating Wave Approximation (RWA) 150
3.29 Strong Coupling Regime 151
3.29.1 Photon Blockade 153
3.30 Dispersive Regime 154
3.31 Entangling Transmons Coupled via Resonators 156
3.32 Engineering Qubit Coherence Through Resonator Coupling 160
3.33 Resonators as Quantum Memories in Circuit QED 161
3.34 Microwave Components 161
3.35 Purcell Filter 161
3.36 Amplifiers 163
3.37 Josephson Parametric Amplifier (JPA) 164
3.38 Attenuators 166
3.39 Filters 166
3.40 Isolator 166
3.41 Microwave Electronics Components 167
3.41.1 Oscillator 167
3.41.2 Arbitrary Waveform Generator 167
3.41.3 IQ mixer 167
3.41.4 Analog-to-digital converter 168
3.42 DC sources 168
3.43 Basic Experimental Setup 168
3.44 Room Temperature Wiring 168
3.45 Cryogenic Wiring 169
3.46 Dilution Refrigerator and Cryostat 171
3.47 Summary and Conclusion: Key Points 174
4 Chapter 4 177
4.1 Introduction 177
4.2 Benefits of Superconducting Cavity-Based Quantum Computing 177
4.3 Examples for 3D Cavities and Transmons 180
4.4 Creating Fock states 187
4.5 Bosonic Encoding 190
4.6 Coherent States 191
4.7 Qudits 193
4.8 Qudit Control via Gates 194
4.9 Errors, Detection and Correction 196
4.10 Qudit Error Correction 198
4.11 Dual Rail Encoding 198
4.12 Binomial codes 201
4.13 CAT States and CAT Codes 203
4.14 Cat State Generation 206
4.15 Two-Component Cat state 207
4.16 Four-component cat state 209
4.17 Gottesmanâ"Kitaevâ"Preskill (GKP) Coding 210
4.18 Architectures for Scaling Quantum Systems 218
4.18.1 Multiple Qubits 218
4.19 Scaling 3D Cavity-Based Quantum Processors 221
4.20 What is Multi-mode cavities 224
4.21 Quantum Memory and Quantum Processor 226
4.22 Cavity as a Computational Hilbert Space 228
4.23 Entangling Fock States 230
4.24 Preparing Qudits 231
4.25 Density Matrix and Trace 232
4.26 Wigner Tomography 238
4.27 Wigner Tomography Examples 240
4.28 Conclusion for Chapter 4 243
5 Chapter 5: Materials for High Coherence 246
5.1 Introduction 246
5.2 Materials for 2D 246
5.3 Quality Factors 246
5.4 Participation Ratios 248
5.5 Substrate Loss (Bulk Dielectric) 249
5.6 Metal Losses 251
5.7 Josephson Junction and Associated Losses 251
5.8 Example Transmon Loss Budget 252
5.9 TLS Losses 255
5.10 Lifetime Variability 259
5.11 Tantalum and Tantalum Capping 262
5.12 Materials for 3D QPU â" Niobium Cavities 263
5.13 Qubit Lifetimes and the Scaling Cost of Quantum Error Correction 267
5.14 Conclusion 269
6 Chapter 6 271
6.1 Applications of Quantum Computing 271
6.2 Introduction 271
6.3 Shorâs Algorithm for Factoring Large Numbers. 272
6.4 High Energy Physics 287
6.5 Chemistry 289
6.6 Materials Science 295
6.7 Finance and Economics 297
6.8 Conclusions 298
References 300
