Principles of Superconducting Quantum Computers. Daniel D. Stancil
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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 Treatment6.1 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 LC Resonant Circuit6.4.2 Quantization of the LC Circuit6.4.3 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 Superconductivity7.1 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
15 8 Josephson Junctions8.1 Tunneling8.1.1 Reflection from a Barrier8.1.2 Finite Thickness Barrier8.2 Josephson Junctions8.2.1 Current and Voltage Relations8.2.2 Josephson Junction Hamiltonian8.2.3 Quantized Josephson Junction Analysis8.3 Superconducting Quantum Interference Devices (SQUIDs)8.4 Josephson Junction Parametric Amplifiers8.5 Exercises
16 9 Errors and Error Mitigation9.1 NISQ Processors9.2 Decoherence9.3 State Preparation and Measurement Errors9.4 Characterizing Gate Errors9.5 State Leakage and Suppression Using Pulse Shaping9.6 Zero-Noise Extrapolation9.7 Optimized Control Using Deep Learning9.8 Exercises
17 10 Quantum Error Correction10.1 Review of Classical Error Correction10.1.1 Error Detection10.1.2 Error Correction: Repetition Code10.1.3 Hamming Code10.2 Quantum Errors10.3 Detecting and Correcting Quantum Errors10.3.1 Bit Flip10.3.2 Phase Flip10.3.3 Correcting Bit and Phase Flips: Shor’s 9-Qubit Code10.3.4 Arbitrary Rotations10.4 Stabilizer Codes10.4.1 Stabilizers10.4.2 Stabilizers for Error Correction10.5 Operating on Logical Qubits10.6 Error Thresholds10.6.1 Concatenation of Error Codes10.6.2 Threshold Theorem10.7 Surface Codes10.7.1 Stabilizers10.7.2 Error Detection and Correction10.7.3 Logical X and Z Operators10.7.4 Multiple Qubits: Lattice Surgery10.7.5 CNOT10.7.6 Single-Qubit Gates10.8 Summary and Further Reading10.9 Exercises
18 11 Quantum Logic: Efficient Implementation of Classical Computations11.1 Reversible Logic11.1.1 Reversible Logic Gates11.1.2 Reversible Logic Circuits11.2 Quantum Logic Circuits11.2.1 Entanglement and Uncomputing11.2.2 Multi-Qubit Gates11.2.3 Qubit Topology11.3 Efficient Arithmetic Circuits: Adder11.3.1 Quantum Ripple-Carry Adder11.3.2 In-Place Ripple-Carry Adder11.3.3 Carry-Lookahead Adder11.3.4 Adder Comparison11.4 Phase Logic11.4.1 Controlled-���� and Controlled-Phase Gates11.4.2 Selective Phase Change11.4.3 Phase Logic Gates11.5 Summary and Further Reading11.6 Exercises
19 12 Some Quantum Algorithms12.1 Computational Complexity12.1.1 Quantum Program Run-Time12.1.2 Classical Complexity Classes12.1.3 Quantum Complexity12.2 Grover’s Search Algorithm12.2.1 Grover Iteration12.2.2 Quantum Implementation12.2.3 Generalizations12.3 Quantum Fourier Transform12.3.1 Discrete Fourier Transform12.3.2 Inverse Discrete Fourier Transform12.3.3 Quantum Implementation of the DFT12.3.4 Encoding Quantum States12.3.5 Quantum Implementation12.3.6 Computational Complexity12.4 Quantum Phase Estimation12.4.1 Quantum Implementation12.4.2 Computational Complexity and Other Issues12.5 Shor’s Algorithm12.5.1 Hybrid Classical-Quantum Algorithm12.5.2 Finding the Period12.5.3 Computational Complexity12.6 Variational Quantum Algorithms12.6.1 Variational Quantum Eigensolver12.6.2 Quantum Approximate Optimization Algorithm12.6.3 Challenges and Opportunities12.7 Summary and Further Reading12.8 Exercises
20 Bibliography
21 Index
List of Figures
1 Chapter 1Figure 1.1 Interpretation of classical versus quantum NOT gates.Figure 1.2 NAND circuit diagram.Figure 1.3 Circuit representation of Eq. (1.31)...Figure 1.4 Symbol for a CNOT gate, and its effect on basis states.Figure 1.5 Circuit for creating an entangled state...Figure 1.6 Result of executing the circuit 1024 times...Figure 1.7 Hypothetical cloning operator, that creates an exact...Figure 1.8 Conceptual illustration of the Deutsch Problem.Figure 1.9 Reversible circuit for calculating f(x).Figure 1.10 Implementations of black-box function...Figure 1.11 Implementation of Deutsch’s algorithm...Figure 1.12 System diagram for a superconducting quantum computer.
2 Chapter 2Figure 2.1 Rotation of a vector of length r CCW around...Figure 2.2 Illustration of how two consecutive rotations...Figure 2.3 Representation of a single qubit state...Figure 2.4 Precession of spin vector for a particle...Figure 2.5 Solutions to the coupled mode equations...Figure 2.6 Rotations enabling measurement of the projections...Figure 2.7 (a) Bloch sphere representation of a mixed state...
3 Chapter 3Figure 3.1 Common symbols for the SWAP...Figure 3.2 Operations needed to convert a...Figure 3.3 Controlled-U gate. If the...Figure 3.4 Implementation of controlled-U...
4 Chapter 4Figure 4.1 Ladder line used for radio frequency transmission...Figure 4.2 Equivalent circuit for a transmission line...Figure 4.3 A transmission line terminated with a load impedance...Figure 4.4 Voltage standing wave pattern along a transmission...Figure 4.5 Impedance looking into a terminated transmission line...Figure 4.6 A real impedance can be matched...Figure 4.7 Some commonly-used types of transmission lines...Figure 4.8 Incoming and outgoing wave amplitudes...Figure 4.9 Definition of voltages and currents for the ABCD...Figure 4.10 Definitions for constructing the...Figure 4.11 Circuit for an attenuator.Figure 4.12 Circulator Circuit diagrams. In actual devices...Figure 4.13 Wilkinson power divider.Figure 4.14 Quadrature hybrid 4-port network...Figure 4.15 Even and odd mode analysis...Figure 4.16 Commonly-used symbol for a quadrature hybrid...Figure 4.17 Mixer Circuit diagrams. In an ideal mixer...Figure 4.18 (a) Circuit to shape a microwave pulse...Figure 4.19 (a) Circuit to recover the cosine...Figure 4.20 Low-pass filter circuits.Figure 4.21 Frequency response of the T network...Figure 4.22 Circuit illustrating thermal...Figure 4.23 Quantum noise as a function of temperature normalized...Figure 4.24 Noise added by a circuit with power gain...Figure 4.25 Thermal noise from a passive element...Figure 4.26 Noise in a system of cascaded components.Figure 4.27 Layered structure of different...
5 Chapter 5Figure 5.1 Resonator circuits.Figure 5.2 Equivalent circuits for capacitively-coupled...Figure 5.3 Capacitively-coupled transmission line resonator.Figure 5.4 Near the...Figure 5.5 Equivalent circuits used to...Figure 5.6 Characteristics of capacitively-coupled transmission...Figure 5.7 Two LC resonant circuits coupled by a capacitor.Figure 5.8 Coupling between lossless LC resonators...Figure 5.9 Tire swings suspended from a common