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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Explore the intersection of computer science, physics, and electrical and computer engineering with this discussion of the engineering of quantum computers In Principles of Superconducting Quantum Computers, a pair of distinguished researchers delivers a comprehensive and insightful discussion of the building of quantum computing hardware and systems. Bridging the gaps between computer science, physics, and electrical and computer engineering, the book focuses on the engineering topics of devices, circuits, control, and error correction. Using data from actual quantum computers, the authors illustrate critical concepts from quantum computing. Questions and problems at the end of each chapter assist students with learning and retention, while the text offers descriptions of fundamentals concepts ranging from the physics of gates to quantum error correction techniques. The authors provide efficient implementations of classical computations, and the book comes complete with a solutions manual and demonstrations of many of the concepts discussed within. It also includes: * A thorough introduction to qubits, gates, and circuits, including unitary transformations, single qubit gates, and controlled (two qubit) gates * Comprehensive explorations of the physics of single qubit gates, including the requirements for a quantum computer, rotations, two-state systems, and Rabi oscillations * Practical discussions of the physics of two qubit gates, including tunable qubits, SWAP gates, controlled-NOT gates, and fixed frequency qubits * In-depth examinations of superconducting quantum computer systems, including the need for cryogenic temperatures, transmission lines, S parameters, and more Ideal for senior-level undergraduate and graduate students in electrical and computer engineering programs, Principles of Superconducting Quantum Computers also deserves a place in the libraries of practicing engineers seeking a better understanding of quantum computer systems.
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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
© 2022 by John Wiley & Sons, Inc.
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The right of Daniel D. Stancil and Gregory T. Byrd to be identified as the authors of this work has been asserted in accordance with law.
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A catalogue record for this book is available from the Library of Congress
Hardback ISBN: 9781119750727; ePub ISBN: 9781119750741; ePDF ISBN: 9781119750734; Obook ISBN: 9781119750758
Cover image: © Adaptrographics/Getty Images
Cover design by Wiley
Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India
Dedicated to the pioneers of the first Quantum Revolution, who paved the way.
Contents
Cover
Title page
Copyright
Dedication
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
1.11 Exercises
2 Physics of Single Qubit Gates
2.1 Requirements for a Quantum Computer
2.2 Single Qubit Gates
2.2.1 Rotations
2.2.2 Two State Systems
2.2.3 Creating Rotations: Rabi Oscillations
2.3 Quantum State Tomography
2.4 Expectation Values and the Pauli Operators
2.5 Density Matrix
2.6 Exercises
3 Physics of Two Qubit Gates
3.1
√
i
SWAP 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
3.6 Exercises
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
4.2 Terminated Lossless Line
4.2.1 Reflection Coefficient
4.2.2 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
4.3
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
4.12 Exercises
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
5.7 Exercises
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
6.5 Exercises
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
7.7 Exercises
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
8.5 Exercises
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
9.8 Exercises
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
10.7.5 CNOT
10.7.6 Single-Qubit Gates
10.8 Summary and Further Reading
10.9 Exercises
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-���� and Controlled-Phase Gates
11.4.2 Selective Phase Change
11.4.3 Phase Logic Gates
11.5 Summary and Further Reading
11.6 Exercises
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
12.6.2 Quantum Approximate Optimization Algorithm
12.6.3 Challenges and Opportunities
12.7 Summary and Further Reading
12.8 Exercises
Bibliography
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1 Useful ABCD matrices.
Chapter 6
Table 6.1 Classical parameters and the...
Chapter 7
Table 7.1 Energy and momentum conservation in electron-phonon...
Table 7.2 Comparison of...
Chapter 10
Table 10.1 Effects of bit-flip error channel...
Table 10.2 Effects of phase-flip error...
Table 10.3 Stabilizer generators for Steane [[7, 1, 3]] code.
Table 10.4 Syndrome measurements for all single-qubit errors...
Chapter 11
Table 11.1 Boolean algebra operators.
Table 11.2 Comparison of in-place...
Table 11.3 Comparison of adder circuits...
Table 11.4 Controlled-
Z
(CZ) gate...
Chapter 12
Table 12.1 Result of conditional phase rotations...
Guide
Cover
Title page
Copyright
Dedication
Table of Contents
List of Figures
List of Tables
Preface
Acknowledgments
About the Companion Website
Begin Reading
Bibliography
Index
End User License Agreement
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List of Tables
4.1 Useful ABCD matrices.
6.1 Classical parameters and the corresponding quantum operators.
7.1 Energy and momentum conservation in electron-phonon single scattering events.
7.2 Comparison of 2Δ(0)/
k
B
T
c
with the BCS theory predition for some technologically important superconductors.
10.1 Effects of bit-flip error channel on a 3-qubit replication code.
10.2 Effects of phase-flip error channel on a 3-qubit phase replication code.
10.3 Stabilizer generators for Steane [[7, 1, 3]] code.
10.4 Syndrome measurements for all single-qubit errors in the Steane code. Each syndrome bit corresponds to the measurement of the corresponding operator from Table 10.3—e.g., bit 0 corresponds to measuring
M
0
.
11.1 Boolean algebra operators.
11.2 Comparison of in-place ripple-carry and carry-lookahead adder circuits. For this table, depth only counts Toffoli gates.
11.3 Comparison of adder circuits for 8-bit binary numbers. The bottom two rows show the results of compiling the circuit to a specific quantum computer topology.
Sparse
is the 53-qubit IBM Rochester topology, and
Full mesh
is Google Sycamore topology, both shown in Figure 11.18.
11.4 Controlled-
Z
(CZ) gate, where
q
0
is the control and
q
1
is the target.
12.1 Result of conditional phase rotations, the first four stages of Figure 12.13. The swap circuits at the end of the circuit reverse the order of the qubits, in order to match the desired rotations in Figure 12.14.
Preface
Over the past several years progress in quantum computing technology and algorithms has accelerated rapidly. We believe that Electrical and Computer Engineers have much to contribute to this work, and one of the goals of this book is to help introduce those with ECE backgrounds to this exciting area.
Our interest in quantum computing was initiated by the discussions between NC State and IBM about establishing the IBM Quantum Hub at NC State, and began in earnest in the spring of 2018. During this semester we both sat in on a Computer Science special topics seminar led by professors Frank Mueller and Patrick Dreher. This led to a joint CSC/ ECE special topics course on quantum computing in the fall of 2018 that was team-taught by professors Mueller and Dreher along with one of us (GTB).
As we have delved more deeply into quantum computing over the past several years, we have discovered that most of the rapidly-growing literature is addressed to those with backgrounds in physics, mathematics, and/or computer science, and often assumes a body of shared prerequisite knowledge and terminology that is not typical for Electrical and Computer Engineers. Electrical Engineers studying semiconductor physics do study quantum mechanics, but from a very different perspective. Band theory, effective mass, tunneling, and perhaps the hydrogen atom are covered, but quantum computing involves 2-level systems, state vectors and rotations, Hamiltonians, and unitary operators—topics not normally emphasized in device physics courses. Further, there is very little available in the current introductory literature explaining how the systems operate.
Just as semiconductor physics became a standard component of electrical engineering curricula beginning in the early 1960s, we believe that quantum computing will become integral parts of ECE curricula in the coming decades. The span of ECE covers the entire range of technologies underpinning quantum computing, including device physics and modeling, nanofabrication, RF and optical systems, signal processing, information theory, error correction and coding, transpilers and compilers, system architecture, and algorithms and applications.
At the time of this writing, there are competing technologies for the implementation of quantum computing, including trapped ions, quantum dots, topological structures, and artificial atoms made with superconducting devices. While we believe Electrical and Computer Engineers will be critical to the development of all of these systems, we have chosen in this volume to concentrate on superconducting technologies relying on Josephson junction transmons.
The intended audience is advanced undergraduate students and first-year graduate students in Electrical and Computer engineering. In presenting topics, we have tried to bridge the gaps that we have encountered between the prevailing literature and the backgrounds of Electrical and Computer engineers.
We offer this work not as an exposition by authorities, but rather as an introduction by technological “pilgrims” hoping to help other “pilgrims” along this exciting journey.
D. D. Stancil
G. T. Byrd
Raleigh, NC
Acknowledgments
We are grateful to our colleagues and students at NC State for many helpful conversations that have deepened and clarified our understanding. In addition, we would like to acknowledge helpful comments on portions of the manuscript by J. C. Bardin and N. Earnest-Noble. Of course, any remaining errors are ours.
We would also like to express appreciation to Brett Kurzman, Sarah Lemore, Becky Cowan, and the capable team at Wiley for their assistance and guidance.
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/stancil/principlesofsuperconductingquantumcomputers
The website includes:
Instructor manual
