Principles of Superconducting Quantum Computers

ISBN-13: 9781119750727 / Angielski / Twarda / 2022 / 384 str.

Daniel D. Stancil; Gregory T. Byrd

Spis treści

1 Qubits, Gates, and Circuits 11.1 Bits and Qubits . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Circuits in Space vs. Circuits in Time . . . . . . . 11.1.2 Superposition . . . . . . . . . . . . . . . . . . . . . 21.1.3 No Cloning . . . . . . . . . . . . . . . . . . . . . . 31.1.4 Reversibility . . . . . . . . . . . . . . . . . . . . . 41.1.5 Entanglement . . . . . . . . . . . . . . . . . . . . . 41.2 Single-Qubit States . . . . . . . . . . . . . . . . . . . . . . 51.3 Measurement and the Born Rule . . . . . . . . . . . . . . 61.4 Unitary Operations and Single-Qubit Gates . . . . . . . . 71.5 Two-Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . 91.5.1 Two-Qubit States . . . . . . . . . . . . . . . . . . . 91.5.2 Two-Qubit Gates . . . . . . . . . . . . . . . . . . . 111.5.3 Controlled-NOT . . . . . . . . . . . . . . . . . . . 131.6 Bell State . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 No Cloning, Revisited . . . . . . . . . . . . . . . . . . . . 151.8 Example: Deutsch's Problem . . . . . . . . . . . . . . . . 171.9 Key Characteristics of Quantum Computing . . . . . . . . 201.10 Quantum Computing Systems . . . . . . . . . . . . . . . . 221.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 Physics of Single Qubit Gates 292.1 Requirements for a Quantum Computer . . . . . . . . . . 292.2 Single Qubit Gates . . . . . . . . . . . . . . . . . . . . . . 302.2.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Two State Systems . . . . . . . . . . . . . . . . . . 382.2.3 Creating Rotations: Rabi Oscillations . . . . . . . 442.3 Quantum State Tomography . . . . . . . . . . . . . . . . 492.4 Expectation Values and the Pauli Operators . . . . . . . . 512.5 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . 522.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 56iiiiv CONTENTS3 Physics of Two Qubit Gates 593.1 square rootiSWAP Gate . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Coupled Tunable Qubits . . . . . . . . . . . . . . . . . . . 613.3 Fixed-frequency Qubits . . . . . . . . . . . . . . . . . . . 643.4 Other Controlled Gates . . . . . . . . . . . . . . . . . . . 663.5 Two-qubit States and the Density Matrix . . . . . . . . . 683.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 Superconducting Quantum Computer Systems 734.1 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . 734.1.1 General Transmission Line Equations . . . . . . . 734.1.2 Lossless Transmission Lines . . . . . . . . . . . . . 754.1.3 Transmission Lines with Loss . . . . . . . . . . . . 774.2 Terminated Lossless Line . . . . . . . . . . . . . . . . . . 824.2.1 Reflection Coefficient . . . . . . . . . . . . . . . . . 824.2.2 Power (Flow of Energy) and Return Loss . . . . . 844.2.3 Standing Wave Ratio (SWR) . . . . . . . . . . . . 854.2.4 Impedance as a Function of Position . . . . . . . . 864.2.5 Quarter Wave Transformer . . . . . . . . . . . . . 884.2.6 Coaxial, Microstrip, and Co-planar Lines . . . . . 894.3 S Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 924.3.1 Lossless Condition . . . . . . . . . . . . . . . . . . 934.3.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . 944.4 Transmission (ABCD) Matrices . . . . . . . . . . . . . . . 944.5 Attenuators . . . . . . . . . . . . . . . . . . . . . . . . . . 994.6 Circulators and Isolators . . . . . . . . . . . . . . . . . . . 1004.7 Power Dividers/Combiners . . . . . . . . . . . . . . . . . 1024.8 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.9 Low-pass Filters . . . . . . . . . . . . . . . . . . . . . . . 1114.10 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.10.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . 1134.10.2 Equivalent Noise Temperature . . . . . . . . . . . 1164.10.3 Noise Factor and Noise Figure . . . . . . . . . . . 1174.10.4 Attenuators and Noise . . . . . . . . . . . . . . . . 1184.10.5 Noise in Cascaded Systems . . . . . . . . . . . . . 1204.11 Low Noise Amplifiers . . . . . . . . . . . . . . . . . . . . . 1214.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235 Resonators: Classical Treatment 1255.1 Parallel Lumped Element Resonator . . . . . . . . . . . . 1255.2 Capacitive Coupling to a Parallel Lumped-Element Res[1]onator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.3 Transmission Line Resonator . . . . . . . . . . . . . . . . 1305.4 Capacitive Coupling to a Transmission Line Resonator . . 1335.5 Capacitively-Coupled Lossless Resonators . . . . . . . . . 136CONTENTS v5.6 Classical Model of Qubit Readout . . . . . . . . . . . . . 1425.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466 Resonators: Quantum Treatment 1496.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . 1496.1.1 Hamilton's Principle . . . . . . . . . . . . . . . . . 1496.1.2 Calculus of Variations . . . . . . . . . . . . . . . . 1506.1.3 Lagrangian Equation of Motion . . . . . . . . . . . 1516.2 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . 1536.3 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . 1536.3.1 Classical Harmonic Oscillator . . . . . . . . . . . . 1546.3.2 Quantum Mechanical Harmonic Oscillator . . . . . 1566.3.3 Raising and Lowering Operators . . . . . . . . . . 1586.3.4 Can a Harmonic Oscillator be used as a Qubit? . . 1606.4 Circuit Quantum Electrodynamics . . . . . . . . . . . . . 1626.4.1 Classical LC Resonant Circuit . . . . . . . . . . . 1626.4.2 Quantization of the LC Circuit . . . . . . . . . . . 1636.4.3 Circuit Electrodynamic Approach for General Cir[1]cuits . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.4.4 Circuit Model for Transmission Line Resonator . . 1656.4.5 Quantizing a Transmission Line Resonator . . . . 1686.4.6 Quantized Coupled LC Resonant Circuits . . . . . 1696.4.7 Schrödinger, Heisenberg, and Interaction Pictures 1726.4.8 Resonant Circuits and Qubits . . . . . . . . . . . . 1756.4.9 The Dispersive Regime . . . . . . . . . . . . . . . . 1786.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1827 Theory of Superconductivity 1837.1 Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . 1847.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . 1867.3 Free Electron Model for Metals . . . . . . . . . . . . . . . 1887.3.1 Discrete States in Finite Samples . . . . . . . . . . 1897.3.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . 1917.3.3 Debye Model . . . . . . . . . . . . . . . . . . . . . 1937.3.4 Electron-Phonon Scattering and Electrical Con[1]ductivity . . . . . . . . . . . . . . . . . . . . . . . 1947.3.5 Perfect Conductor vs. Superconductor . . . . . . . 1967.4 Bardeen, Cooper and Schrieffer Theory of Superconduc[1]tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.4.1 Cooper Pair Model . . . . . . . . . . . . . . . . . . 1997.4.2 Dielectric Function . . . . . . . . . . . . . . . . . . 2037.4.3 Jellium . . . . . . . . . . . . . . . . . . . . . . . . 2047.4.4 Scattering Amplitude and Attractive Electron-ElectronInteraction . . . . . . . . . . . . . . . . . . . . . . 2087.4.5 Interpretation of Attractive Interaction . . . . . . 209vi CONTENTS7.4.6 Superconductor Hamiltonian . . . . . . . . . . . . 2107.4.7 Superconducting Ground State . . . . . . . . . . . 2117.5 Electrodynamics of Superconductors . . . . . . . . . . . . 2157.5.1 Cooper Pairs and the Macroscopic Wave Function 2157.5.2 Potential Functions . . . . . . . . . . . . . . . . . . 2167.5.3 London Equations . . . . . . . . . . . . . . . . . . 2177.5.4 London Gauge . . . . . . . . . . . . . . . . . . . . 2197.5.5 Penetration Depth . . . . . . . . . . . . . . . . . . 2207.5.6 Flux Quantization . . . . . . . . . . . . . . . . . . 2217.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . 2237.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 2248 Josephson Junctions 2258.1 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 2258.1.1 Reflection from a Barrier . . . . . . . . . . . . . . 2268.1.2 Finite Thickness Barrier . . . . . . . . . . . . . . . 2298.2 Josephson Junctions . . . . . . . . . . . . . . . . . . . . . 2318.2.1 Current and Voltage Relations . . . . . . . . . . . 2318.2.2 Josephson Junction Hamiltonian . . . . . . . . . . 2358.2.3 Quantized Josephson Junction Analysis . . . . . . 2378.3 Superconducting Quantum Interference Devices (SQUIDs) 2398.4 Josephson Junction Parametric Amplifiers . . . . . . . . . 2418.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 2429 Errors and Error Mitigation 2459.1 NISQ Processors . . . . . . . . . . . . . . . . . . . . . . . 2459.2 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . 2469.3 State Preparation and Measurement Errors . . . . . . . . 2489.4 Characterizing Gate Errors . . . . . . . . . . . . . . . . . 2509.5 State Leakage and Suppression using Pulse Shaping . . . 2549.6 Zero-Noise Extrapolation . . . . . . . . . . . . . . . . . . 2579.7 Optimized Control using Deep Learning . . . . . . . . . . 2609.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 26110 Quantum Error Correction 26510.1 Review of Classical Error Correction . . . . . . . . . . . . 26510.1.1 Error Detection . . . . . . . . . . . . . . . . . . . . 26610.1.2 Error Correction: Repetition Code . . . . . . . . . 26710.1.3 Hamming Code . . . . . . . . . . . . . . . . . . . . 26810.2 Quantum Errors . . . . . . . . . . . . . . . . . . . . . . . 26910.3 Detecting and Correcting Quantum Errors . . . . . . . . . 27210.3.1 Bit Flip . . . . . . . . . . . . . . . . . . . . . . . . 27210.3.2 Phase Flip . . . . . . . . . . . . . . . . . . . . . . 27410.3.3 Correcting Bit and Phase Flips: Shor's 9-qubit Code27510.3.4 Arbitrary Rotations . . . . . . . . . . . . . . . . . 277CONTENTS vii10.4 Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . 27910.4.1 Stabilizers . . . . . . . . . . . . . . . . . . . . . . . 27910.4.2 Stabilizers for Error Correction . . . . . . . . . . . 28010.5 Operating on Logical Qubits . . . . . . . . . . . . . . . . 28310.6 Error Thresholds . . . . . . . . . . . . . . . . . . . . . . . 28510.6.1 Concatenation of Error Codes . . . . . . . . . . . . 28610.6.2 Threshold Theorem . . . . . . . . . . . . . . . . . 28610.7 Surface Codes . . . . . . . . . . . . . . . . . . . . . . . . . 28810.7.1 Stabilizers . . . . . . . . . . . . . . . . . . . . . . . 28910.7.2 Error Detection and Correction . . . . . . . . . . . 29110.7.3 Logical X and Z Operators . . . . . . . . . . . . . 29510.7.4 Multiple Qubits: Lattice Surgery . . . . . . . . . . 29710.7.5 CNOT . . . . . . . . . . . . . . . . . . . . . . . . . 30110.7.6 Single-Qubit Gates . . . . . . . . . . . . . . . . . . 30510.8 Summary and Further Reading . . . . . . . . . . . . . . . 30610.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 30811 Quantum Logic: Efficient Implementation of ClassicalComputations 30911.1 Reversible Logic . . . . . . . . . . . . . . . . . . . . . . . 31011.1.1 Reversible Logic Gates . . . . . . . . . . . . . . . . 31111.1.2 Reversible Logic Circuits . . . . . . . . . . . . . . 31311.2 Quantum Logic Circuits . . . . . . . . . . . . . . . . . . . 31711.2.1 Entanglement and Uncomputing . . . . . . . . . . 31711.2.2 Multi-qubit gates . . . . . . . . . . . . . . . . . . . 31911.2.3 Qubit topology . . . . . . . . . . . . . . . . . . . . 32111.3 Efficient Arithmetic Circuits: Adder . . . . . . . . . . . . 32211.3.1 Quantum Ripple Carry Adder . . . . . . . . . . . . 32311.3.2 In-place Ripple Carry Adder . . . . . . . . . . . . 32611.3.3 Carry-Lookahead Adder . . . . . . . . . . . . . . . 32911.3.4 Adder Comparison . . . . . . . . . . . . . . . . . . 33411.4 Phase Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 33611.4.1 Controlled-Z and Controlled-Phase Gates . . . . . 33611.4.2 Selective Phase Change . . . . . . . . . . . . . . . 33911.4.3 Phase Logic Gates . . . . . . . . . . . . . . . . . . 34111.5 Summary and Further Reading . . . . . . . . . . . . . . . 34211.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 34512 Some Quantum Algorithms 34712.1 Computational Complexity . . . . . . . . . . . . . . . . . 34712.1.1 Quantum Program Run-Time . . . . . . . . . . . . 34812.1.2 Classical Complexity Classes . . . . . . . . . . . . 34912.1.3 Quantum Complexity . . . . . . . . . . . . . . . . 35012.2 Grover's Search Algorithm . . . . . . . . . . . . . . . . . . 35112.2.1 Grover Iteration . . . . . . . . . . . . . . . . . . . 351viii CONTENTS12.2.2 Quantum Implementation . . . . . . . . . . . . . . 35412.2.3 Generalizations . . . . . . . . . . . . . . . . . . . . 35712.3 Quantum Fourier Transform . . . . . . . . . . . . . . . . . 35812.3.1 Frequencies and Quantum-encoded Signals . . . . 35812.3.2 Inverse QFT . . . . . . . . . . . . . . . . . . . . . 36112.3.3 Quantum Implementation . . . . . . . . . . . . . . 36212.3.4 Computational Complexity . . . . . . . . . . . . . 36512.4 Quantum Phase Estimation . . . . . . . . . . . . . . . . . 36512.4.1 Quantum Implementation . . . . . . . . . . . . . . 36612.4.2 Computational Complexity and Other Issues . . . 36712.5 Shor's Algorithm . . . . . . . . . . . . . . . . . . . . . . . 36812.5.1 Hybrid Classical-Quantum Algorithm . . . . . . . 36812.5.2 Finding the Period . . . . . . . . . . . . . . . . . . 37012.5.3 Computational Complexity . . . . . . . . . . . . . 37312.6 Variational Quantum Algorithms . . . . . . . . . . . . . . 37512.6.1 Variational Quantum Eigensolver . . . . . . . . . . 37712.6.2 Quantum Approximate Optimization Algorithm . 38212.6.3 Challenges and Opportunities . . . . . . . . . . . . 38612.7 Summary and Further Reading . . . . . . . . . . . . . . . 38712.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

O autorze

Daniel D. Stancil, PhD, is the Alcoa Distinguished Professor and Head of Electrical and Computer Engineering at North Carolina State University. In addition to quantum computing, his research interests include spin waves, and microwave and optical devices and systems.Gregory T. Byrd, PhD, is Professor and Associate Head of Electrical and Computer Engineering at North Carolina State University. His research focuses on both classical and quantum computer architecture and systems.