Principles of superconducting quantum computers. (2022)

• Record Type: Book
• Further Information: Note: Daniel D. Stancil, Gregory T. Byrd.
• Authors: Stancil, Daniel D; Byrd, Gregory T
• Contents: 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 . . . . . .
• Edition: 1st
• Publisher Details: Hoboken : John Wiley & Sons, Inc
• Publication Date: 2022
• Extent: 1 online resource
• Subjects: 006.3843; Quantum computers; Superconducting quantum interference devices
• Languages: English
• ISBNs: 9781119750741
• Related ISBNs: 9781119750727
• Notes: Note: Includes bibliographical references and index.; Note: Description based on CIP data; resource not viewed.
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