Оглавление
Daniel D. Stancil. Principles of Superconducting Quantum Computers
Principles of Superconducting Quantum Computers
Contents
List of Figures
List of Tables
Guide
Pages
List of Figures
List of Tables
Preface
Acknowledgments
About the Companion Website
1 Qubits, Gates, and Circuits. 1.1 Bits and Qubits
1.1.1 Circuits in Space vs. Circuits in Time
1.1.2 Superposition
1.1.3 No Cloning
1.1.4 Reversibility
1.1.5 Entanglement
1.2 Single-Qubit States
1.3 Measurement and the Born Rule
1.4 Unitary Operations and Single-Qubit Gates
1.5 Two-Qubit Gates
1.5.1 Two-Qubit States
1.5.2 Matrix Representation of Two-Qubit Gates
1.5.3 Controlled-NOT
1.6 Bell State
1.7 No Cloning, Revisited
1.8 Example: Deutsch’s Problem
1.9 Key Characteristics of Quantum Computing
1.10 Quantum Computing Systems
Exercises
Notes
2 Physics of Single Qubit Gates. 2.1 Requirements for a Quantum Computer
2.2 Single Qubit Gates. 2.2.1 Rotations
2.2.1.1 Classical Rotations
2.2.1.2 Rotation of the Quantum Mechanical State Vector
2.2.1.3 Bloch Sphere
2.2.1.4 The Most General Unitary
2.2.2 Two State Systems
2.2.2.1 Eigenvalues of the Two State Spin System
2.2.2.2 Larmor Precession
2.2.2.3 Coupled Qubit States
2.2.3 Creating Rotations: Rabi Oscillations. 2.2.3.1 Rotation Operator Approach
2.2.3.2 Rotations about z
2.2.3.3 Coupled-Mode Theory Approach
2.3 Quantum State Tomography
2.4 Expectation Values and the Pauli Operators
2.5 Density Matrix
Exercises
Notes
3 Physics of Two Qubit Gates
3.1 iSWAP Gate
3.2 Coupled Tunable Qubits
3.3 Cross Resonance Scheme
3.4 Other Controlled Gates
3.5 Two-Qubit States and the Density Matrix
Exercises
Notes
4 Superconducting Quantum Computer Systems
4.1 Transmission Lines. 4.1.1 General Transmission Line Equations
4.1.2 Lossless Transmission Lines
4.1.3 Transmission Lines with Loss. 4.1.3.1 Sinusoidal Steady State
4.1.3.2 Low Loss Transmission Lines
4.2 Terminated Lossless Line
4.2.1 Reflection Coefficient
Power (Flow of Energy) and Return Loss
4.2.3 Standing Wave Ratio (SWR)
4.2.4 Impedance as a Function of Position
4.2.5 Quarter Wave Transformer
4.2.6 Coaxial, Microstrip, and Coplanar Lines
4.2.6.1 Coaxial Lines
4.2.6.2 Microstrip Lines
4.2.6.3 Coplanar Waveguide
4.3 S Parameters
4.3.1 Lossless Condition
4.3.2 Reciprocity
4.4 Transmission (ABCD) Matrices
4.5 Attenuators
4.6 Circulators and Isolators
4.7 Power Dividers/Combiners
4.8 Mixers
4.9 Low-Pass Filters
4.10 Noise
4.10.1 Thermal Noise
4.10.2 Equivalent Noise Temperature
4.10.3 Noise Factor and Noise Figure
4.10.4 Attenuators and Noise
4.10.5 Noise in Cascaded Systems
4.11 Low Noise Amplifiers
Exercises
Notes
5 Resonators: Classical Treatment
5.1 Parallel Lumped Element Resonator
5.2 Capacitive Coupling to a Parallel Lumped-Element Resonator
5.3 Transmission Line Resonator
5.4 Capacitive Coupling to a Transmission Line Resonator
5.5 Capacitively-Coupled Lossless Resonators
5.6 Classical Model of Qubit Readout
Exercises
Notes
6 Resonators: Quantum Treatment
6.1 Lagrangian Mechanics. 6.1.1 Hamilton’s Principle
6.1.2 Calculus of Variations
6.1.3 Lagrangian Equation of Motion
6.2 Hamiltonian Mechanics
6.3 Harmonic Oscillators
6.3.1 Classical Harmonic Oscillator
6.3.2 Quantum Mechanical Harmonic Oscillator
6.3.3 Raising and Lowering Operators
6.3.4 Can a Harmonic Oscillator Be Used as a Qubit?
6.4 Circuit Quantum Electrodynamics
6.4.1 Classical LC Resonant Circuit
6.4.2 Quantization of the LC Circuit
6.4.3 Circuit Electrodynamic Approach for General Circuits
6.4.4 Circuit Model for Transmission Line Resonator
6.4.5 Quantizing a Transmission Line Resonator
6.4.6 Quantized Coupled LC Resonant Circuits
6.4.7 Schrödinger, Heisenberg, and Interaction Pictures
6.4.8 Resonant Circuits and Qubits
6.4.9 The Dispersive Regime
Exercises
Notes
7 Theory of Superconductivity
7.1 Bosons and Fermions
7.2 Bloch Theorem
7.3 Free Electron Model for Metals
7.3.1 Discrete States in Finite Samples
7.3.2 Phonons
7.3.3 Debye Model
7.3.4 Electron–Phonon Scattering and Electrical Conductivity
7.3.5 Perfect Conductor vs. Superconductor
7.4 Bardeen, Cooper, and Schrieffer Theory of Superconductivity
7.4.1 Cooper Pair Model
7.4.2 Dielectric Function
7.4.3 Jellium
7.4.4 Scattering Amplitude and Attractive Electron–Electron Interaction
7.4.5 Interpretation of Attractive Interaction
7.4.6 Superconductor Hamiltonian
7.4.7 Superconducting Ground State
7.5 Electrodynamics of Superconductors. 7.5.1 Cooper Pairs and the Macroscopic Wave Function
7.5.2 Potential Functions
7.5.3 London Equations
7.5.4 London Gauge
7.5.5 Penetration Depth
7.5.6 Flux Quantization
7.6 Chapter Summary
Exercises
Notes
8 Josephson Junctions
8.1 Tunneling
8.1.1 Reflection from a Barrier
8.1.2 Finite Thickness Barrier
8.2 Josephson Junctions
8.2.1 Current and Voltage Relations
8.2.2 Josephson Junction Hamiltonian
8.2.3 Quantized Josephson Junction Analysis
8.3 Superconducting Quantum Interference Devices (SQUIDs)
8.4 Josephson Junction Parametric Amplifiers
Exercises
Notes
9 Errors and Error Mitigation. 9.1 NISQ Processors
9.2 Decoherence
9.3 State Preparation and Measurement Errors
9.4 Characterizing Gate Errors
9.5 State Leakage and Suppression Using Pulse Shaping
9.6 Zero-Noise Extrapolation
9.7 Optimized Control Using Deep Learning
Exercises
Notes
10 Quantum Error Correction
10.1 Review of Classical Error Correction
10.1.1 Error Detection
10.1.2 Error Correction: Repetition Code
10.1.3 Hamming Code
10.2 Quantum Errors
10.3 Detecting and Correcting Quantum Errors. 10.3.1 Bit Flip
10.3.2 Phase Flip
10.3.3 Correcting Bit and Phase Flips: Shor’s 9-Qubit Code
10.3.4 Arbitrary Rotations
10.4 Stabilizer Codes
10.4.1 Stabilizers
10.4.2 Stabilizers for Error Correction
10.5 Operating on Logical Qubits
10.6 Error Thresholds
10.6.1 Concatenation of Error Codes
10.6.2 Threshold Theorem
10.7 Surface Codes
10.7.1 Stabilizers
10.7.2 Error Detection and Correction
10.7.3 Logical X and Z Operators
10.7.4 Multiple Qubits: Lattice Surgery
10.7.4.1 Lattice Merge
10.7.4.2 Lattice Split
10.7.5 CNOT
10.7.6 Single-Qubit Gates
10.8 Summary and Further Reading
Exercises
Notes
11 Quantum Logic: Efficient Implementation of Classical Computations
11.1 Reversible Logic
11.1.1 Reversible Logic Gates
11.1.2 Reversible Logic Circuits
11.2 Quantum Logic Circuits
11.2.1 Entanglement and Uncomputing
11.2.2 Multi-Qubit Gates
11.2.3 Qubit Topology
11.3 Efficient Arithmetic Circuits: Adder
11.3.1 Quantum Ripple-Carry Adder
11.3.2 In-Place Ripple-Carry Adder
11.3.3 Carry-Lookahead Adder
11.3.4 Adder Comparison
11.4 Phase Logic
11.4.1 Controlled-Z and Controlled-Phase Gates
11.4.2 Selective Phase Change
11.4.3 Phase Logic Gates
11.5 Summary and Further Reading
Exercises
Notes
12 Some Quantum Algorithms
12.1 Computational Complexity
12.1.1 Quantum Program Run-Time
12.1.2 Classical Complexity Classes
12.1.3 Quantum Complexity
12.2 Grover’s Search Algorithm
12.2.1 Grover Iteration
12.2.2 Quantum Implementation
12.2.3 Generalizations
12.3 Quantum Fourier Transform
12.3.1 Discrete Fourier Transform
12.3.2 Inverse Discrete Fourier Transform
12.3.3 Quantum Implementation of the DFT
12.3.4 Encoding Quantum States
12.3.5 Quantum Implementation
12.3.6 Computational Complexity
12.4 Quantum Phase Estimation
12.4.1 Quantum Implementation
12.4.2 Computational Complexity and Other Issues
12.5 Shor’s Algorithm
12.5.1 Hybrid Classical-Quantum Algorithm
12.5.2 Finding the Period
12.5.3 Computational Complexity
12.6 Variational Quantum Algorithms
12.6.1 Variational Quantum Eigensolver
12.6.1.1 Eigenvalues and Expectations
12.6.1.2 Ansatz
12.6.1.3 Measuring the Hamiltonian
12.6.1.4 Summary
12.6.2 Quantum Approximate Optimization Algorithm
12.6.2.1 Encoding the Objective Function
12.6.2.2 Ising and QUBO Formulations
12.6.2.3 QAOA Ansatz
12.6.2.4 Comparison with VQE
12.6.3 Challenges and Opportunities
12.7 Summary and Further Reading
Exercises
Notes
Bibliography
Index
WILEY END USER LICENSE AGREEMENT
