Daniel D. Stancil - Principles of Superconducting Quantum Computers

Daniel D. Stancil

North Carolina State University

Raleigh, North Carolina

Gregory T. Byrd

North Carolina State University

Raleigh, North Carolina

This edition first published 2022

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Dedicated to the pioneers of the first Quantum Revolution, who paved the way.

1 Cover

2 Title page Principles of Superconducting Quantum Computers Daniel D. Stancil North Carolina State University Raleigh, North Carolina Gregory T. Byrd North Carolina State University Raleigh, North Carolina

3 Copyright

4 Dedication

5 Preface

6 Acknowledgments

7 About the Companion Website

8 1 Qubits, Gates, and Circuits 1.1 Bits and Qubits1.1.1 Circuits in Space vs. Circuits in Time1.1.2 Superposition1.1.3 No Cloning1.1.4 Reversibility1.1.5 Entanglement1.2 Single-Qubit States1.3 Measurement and the Born Rule1.4 Unitary Operations and Single-Qubit Gates1.5 Two-Qubit Gates1.5.1 Two-Qubit States1.5.2 Matrix Representation of Two-Qubit Gates1.5.3 Controlled-NOT1.6 Bell State1.7 No Cloning, Revisited1.8 Example: Deutsch’s Problem1.9 Key Characteristics of Quantum Computing1.10 Quantum Computing Systems1.11 Exercises

9 2 Physics of Single Qubit Gates 2.1 Requirements for a Quantum Computer2.2 Single Qubit Gates2.2.1 Rotations2.2.2 Two State Systems2.2.3 Creating Rotations: Rabi Oscillations2.3 Quantum State Tomography2.4 Expectation Values and the Pauli Operators2.5 Density Matrix2.6 Exercises

10 3 Physics of Two Qubit Gates 3.1 √ iSWAP Gate3.2 Coupled Tunable Qubits3.3 Cross Resonance Scheme3.4 Other Controlled Gates3.5 Two-Qubit States and the Density Matrix3.6 Exercises

11 4 Superconducting Quantum Computer Systems 4.1 Transmission Lines4.1.1 General Transmission Line Equations4.1.2 Lossless Transmission Lines4.1.3 Transmission Lines with Loss4.2 Terminated Lossless Line4.2.1 Reflection Coefficient4.2.2 Power (Flow of Energy) and Return Loss4.2.3 Standing Wave Ratio (SWR)4.2.4 Impedance as a Function of Position4.2.5 Quarter Wave Transformer4.2.6 Coaxial, Microstrip, and Coplanar Lines4.3 SParameters4.3.1 Lossless Condition4.3.2 Reciprocity4.4 Transmission (ABCD) Matrices4.5 Attenuators4.6 Circulators and Isolators4.7 Power Dividers/Combiners4.8 Mixers4.9 Low-Pass Filters4.10 Noise4.10.1 Thermal Noise4.10.2 Equivalent Noise Temperature4.10.3 Noise Factor and Noise Figure4.10.4 Attenuators and Noise4.10.5 Noise in Cascaded Systems4.11 Low Noise Amplifiers4.12 Exercises

12 5 Resonators: Classical Treatment 5.1 Parallel Lumped Element Resonator5.2 Capacitive Coupling to a Parallel Lumped-Element Resonator5.3 Transmission Line Resonator5.4 Capacitive Coupling to a Transmission Line Resonator5.5 Capacitively-Coupled Lossless Resonators5.6 Classical Model of Qubit Readout5.7 Exercises

13 6 Resonators: Quantum Treatment 6.1 Lagrangian Mechanics6.1.1 Hamilton’s Principle6.1.2 Calculus of Variations6.1.3 Lagrangian Equation of Motion6.2 Hamiltonian Mechanics6.3 Harmonic Oscillators6.3.1 Classical Harmonic Oscillator6.3.2 Quantum Mechanical Harmonic Oscillator6.3.3 Raising and Lowering Operators6.3.4 Can a Harmonic Oscillator Be Used as a Qubit?6.4 Circuit Quantum Electrodynamics6.4.1 Classical LCResonant Circuit6.4.2 Quantization of the LCCircuit6.4.3 Circuit Electrodynamic Approach for General Circuits6.4.4 Circuit Model for Transmission Line Resonator6.4.5 Quantizing a Transmission Line Resonator6.4.6 Quantized Coupled LC Resonant Circuits6.4.7 Schrödinger, Heisenberg, and Interaction Pictures6.4.8 Resonant Circuits and Qubits6.4.9 The Dispersive Regime6.5 Exercises

14 7 Theory of Superconductivity 7.1 Bosons and Fermions7.2 Bloch Theorem7.3 Free Electron Model for Metals7.3.1 Discrete States in Finite Samples7.3.2 Phonons7.3.3 Debye Model7.3.4 Electron–Phonon Scattering and Electrical Conductivity7.3.5 Perfect Conductor vs. Superconductor7.4 Bardeen, Cooper, and Schrieffer Theory of Superconductivity7.4.1 Cooper Pair Model7.4.2 Dielectric Function7.4.3 Jellium7.4.4 Scattering Amplitude and Attractive Electron–Electron Interaction7.4.5 Interpretation of Attractive Interaction7.4.6 Superconductor Hamiltonian7.4.7 Superconducting Ground State7.5 Electrodynamics of Superconductors7.5.1 Cooper Pairs and the Macroscopic Wave Function7.5.2 Potential Functions7.5.3 London Equations7.5.4 London Gauge7.5.5 Penetration Depth7.5.6 Flux Quantization7.6 Chapter Summary7.7 Exercises