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.