Principles of Superconducting Quantum Computers
Beschreibung
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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Andere Ausgaben
Personen
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.
Inhalt
List of Figures xiii
List of Tables xxv
Preface xxvii
Acknowledgments xxix
About the Companion Website xxxi
1 Qubits, Gates, and Circuits 1
1.1 Bits and Qubits 1
1.1.1 Circuits in Space vs. Circuits in Time 1
1.1.2 Superposition 1
1.1.3 No Cloning 3
1.1.4 Reversibility 3
1.1.5 Entanglement 3
1.2 Single-Qubit States 4
1.3 Measurement and the Born Rule 5
1.4 Unitary Operations and Single-Qubit Gates 6
1.5 Two-Qubit Gates 8
1.5.1 Two-Qubit States 8
1.5.2 Matrix Representation of Two-Qubit Gates 9
1.5.3 Controlled-NOT 11
1.6 Bell State 12
1.7 No Cloning, Revisited 13
1.8 Example: Deutsch's Problem 15
1.9 Key Characteristics of Quantum Computing 18
1.10 Quantum Computing Systems 18
1.11 Exercises 22
2 Physics of Single Qubit Gates 25
2.1 Requirements for a Quantum Computer 25
2.2 Single Qubit Gates 25
2.2.1 Rotations 25
2.2.2 Two State Systems 33
2.2.3 Creating Rotations: Rabi Oscillations 38
2.3 Quantum State Tomography 42
2.4 Expectation Values and the Pauli Operators 44
2.5 Density Matrix 45
2.6 Exercises 48
3 Physics of Two Qubit Gates 51
3.1 viSWAP Gate 51
3.2 Coupled Tunable Qubits 53
3.3 Cross Resonance Scheme 55
3.4 Other Controlled Gates 57
3.5 Two-Qubit States and the Density Matrix 59
3.6 Exercises 62
4 Superconducting Quantum Computer Systems 63
4.1 Transmission Lines 63
4.1.1 General Transmission Line Equations 63
4.1.2 Lossless Transmission Lines 65
4.1.3 Transmission Lines with Loss 67
4.2 Terminated Lossless Line 71
4.2.1 Reflection Coefficient 71
4.2.2 Power (Flow of Energy) and Return Loss 72
4.2.3 Standing Wave Ratio (SWR) 73
4.2.4 Impedance as a Function of Position 74
4.2.5 Quarter Wave Transformer 76
4.2.6 Coaxial, Microstrip, and Coplanar Lines 77
4.3 S Parameters 80
4.3.1 Lossless Condition 81
4.3.2 Reciprocity 81
4.4 Transmission (ABCD) Matrices 81
4.5 Attenuators 85
4.6 Circulators and Isolators 87
4.7 Power Dividers/Combiners 89
4.8 Mixers 92
4.9 Low-Pass Filters 95
4.10 Noise 97
4.10.1 Thermal Noise 97
4.10.2 Equivalent Noise Temperature 99
4.10.3 Noise Factor and Noise Figure 100
4.10.4 Attenuators and Noise 101
4.10.5 Noise in Cascaded Systems 103
4.11 Low Noise Amplifiers 104
4.12 Exercises 105
5 Resonators: Classical Treatment 107
5.1 Parallel Lumped Element Resonator 107
5.2 Capacitive Coupling to a Parallel Lumped-Element Resonator 109
5.3 Transmission Line Resonator 111
5.4 Capacitive Coupling to a Transmission Line Resonator 113
5.5 Capacitively-Coupled Lossless Resonators 117
5.6 Classical Model of Qubit Readout 120
5.7 Exercises 124
6 Resonators: Quantum Treatment 127
6.1 Lagrangian Mechanics 127
6.1.1 Hamilton's Principle 127
6.1.2 Calculus of Variations 128
6.1.3 Lagrangian Equation of Motion 129
6.2 Hamiltonian Mechanics 130
6.3 Harmonic Oscillators 131
6.3.1 Classical Harmonic Oscillator 131
6.3.2 Quantum Mechanical Harmonic Oscillator 133
6.3.3 Raising and Lowering Operators 135
6.3.4 Can a Harmonic Oscillator Be Used as a Qubit? 137
6.4 Circuit Quantum Electrodynamics 138
6.4.1 Classical LC Resonant Circuit 138
6.4.2 Quantization of the LC Circuit 139
6.4.3 Circuit Electrodynamic Approach for General Circuits 140
6.4.4 Circuit Model for Transmission Line Resonator 141
6.4.5 Quantizing a Transmission Line Resonator 144
6.4.6 Quantized Coupled LC Resonant Circuits 144
6.4.7 Schrödinger, Heisenberg, and Interaction Pictures 147
6.4.8 Resonant Circuits and Qubits 150
6.4.9 The Dispersive Regime 153
6.5 Exercises 156
7 Theory of Superconductivity 159
7.1 Bosons and Fermions 159
7.2 Bloch Theorem 161
7.3 Free Electron Model for Metals 163
7.3.1 Discrete States in Finite Samples 163
7.3.2 Phonons 166
7.3.3 Debye Model 167
7.3.4 Electron-Phonon Scattering and Electrical Conductivity 168
7.3.5 Perfect Conductor vs. Superconductor 170
7.4 Bardeen, Cooper, and Schrieffer Theory of Superconductivity 172
7.4.1 Cooper Pair Model 172
7.4.2 Dielectric Function 175
7.4.3 Jellium 176
7.4.4 Scattering Amplitude and Attractive Electron-Electron Interaction 179
7.4.5 Interpretation of Attractive Interaction 180
7.4.6 Superconductor Hamiltonian 181
7.4.7 Superconducting Ground State 182
7.5 Electrodynamics of Superconductors 185
7.5.1 Cooper Pairs and the Macroscopic Wave Function 185
7.5.2 Potential Functions 186
7.5.3 London Equations 187
7.5.4 London Gauge 189
7.5.5 Penetration Depth 190
7.5.6 Flux Quantization 191
7.6 Chapter Summary 192
7.7 Exercises 193
8 Josephson Junctions 195
8.1 Tunneling 195
8.1.1 Reflection from a Barrier 196
8.1.2 Finite Thickness Barrier 198
8.2 Josephson Junctions 200
8.2.1 Current and Voltage Relations 200
8.2.2 Josephson Junction Hamiltonian 203
8.2.3 Quantized Josephson Junction Analysis 205
8.3 Superconducting Quantum Interference Devices (SQUIDs) 207
8.4 Josephson Junction Parametric Amplifiers 208
8.5 Exercises 209
9 Errors and Error Mitigation 211
9.1 NISQ Processors 211
9.2 Decoherence 212
9.3 State Preparation and Measurement Errors 214
9.4 Characterizing Gate Errors 215
9.5 State Leakage and Suppression Using Pulse Shaping 219
9.6 Zero-Noise Extrapolation 220
9.7 Optimized Control Using Deep Learning 223
9.8 Exercises 225
10 Quantum Error Correction 227
10.1 Review of Classical Error Correction 227
10.1.1 Error Detection 228
10.1.2 Error Correction: Repetition Code 228
10.1.3 Hamming Code 229
10.2 Quantum Errors 230
10.3 Detecting and Correcting Quantum Errors 232
10.3.1 Bit Flip 232
10.3.2 Phase Flip 234
10.3.3 Correcting Bit and Phase Flips: Shor's 9-Qubit Code 235
10.3.4 Arbitrary Rotations 236
10.4 Stabilizer Codes 238
10.4.1 Stabilizers 238
10.4.2 Stabilizers for Error Correction 239
10.5 Operating on Logical Qubits 242
10.6 Error Thresholds 243
10.6.1 Concatenation of Error Codes 243
10.6.2 Threshold Theorem 244
10.7 Surface Codes 245
10.7.1 Stabilizers 246
10.7.2 Error Detection and Correction 247
10.7.3 Logical X and Z Operators 250
10.7.4 Multiple Qubits: Lattice Surgery 253
10.7.5 CNOT 257
10.7.6 Single-Qubit Gates 258
10.8 Summary and Further Reading 259
10.9 Exercises 261
11 Quantum Logic: Efficient Implementation of Classical Computations 263
11.1 Reversible Logic 264
11.1.1 Reversible Logic Gates 264
11.1.2 Reversible Logic Circuits 266
11.2 Quantum Logic Circuits 268
11.2.1 Entanglement and Uncomputing 269
11.2.2 Multi-Qubit Gates 270
11.2.3 Qubit Topology 270
11.3 Efficient Arithmetic Circuits: Adder 272
11.3.1 Quantum Ripple-Carry Adder 273
11.3.2 In-Place Ripple-Carry Adder 275
11.3.3 Carry-Lookahead Adder 277
11.3.4 Adder Comparison 281
11.4 Phase Logic 283
11.4.1 Controlled-Z and Controlled-Phase Gates 283
11.4.2 Selective Phase Change 285
11.4.3 Phase Logic Gates 287
11.5 Summary and Further Reading 288
11.6 Exercises 289
12 Some Quantum Algorithms 291
12.1 Computational Complexity 291
12.1.1 Quantum Program Run-Time 292
12.1.2 Classical Complexity Classes 292
12.1.3 Quantum Complexity 293
12.2 Grover's Search Algorithm 294
12.2.1 Grover Iteration 294
12.2.2 Quantum Implementation 296
12.2.3 Generalizations 299
12.3 Quantum Fourier Transform 299
12.3.1 Discrete Fourier Transform 300
12.3.2 Inverse Discrete Fourier Transform 300
12.3.3 Quantum Implementation of the DFT 301
12.3.4 Encoding Quantum States 302
12.3.5 Quantum Implementation 304
12.3.6 Computational Complexity 306
12.4 Quantum Phase Estimation 307
12.4.1 Quantum Implementation 307
12.4.2 Computational Complexity and Other Issues 308
12.5 Shor's Algorithm 309
12.5.1 Hybrid Classical-Quantum Algorithm 309
12.5.2 Finding the Period 310
12.5.3 Computational Complexity 314
12.6 Variational Quantum Algorithms 314
12.6.1 Variational Quantum Eigensolver 316
12.6.2 Quantum Approximate Optimization Algorithm 320
12.6.3 Challenges and Opportunities 323
12.7 Summary and Further Reading 324
12.8 Exercises 325
Bibliography 327
Index 339
List of Figures
- 1.1 Interpretation of classical versus quantum NOT gates. (a) Classical NOT Circuit diagram. The horizontal direction represents space, i.e., the input and output of the circuit are physically accessible from different points in the circuit, and they can be measured simultaneously. (b) Quantum X gate circuit (quantum version of the NOT gate). The horizontal direction represents time, i.e., the input and output of the circuit represent the state of the same qubit after performing the X gate operation. The lower part of the Figure shows an alternate symbol for the quantum NOT gate.
- 1.2 NAND circuit diagram.
- 1.3 Circuit representation of Eq. (1.31). In a quantum circuit diagram, the operation goes from left to right, while the matrix expression is shown going from right to left. The final box is a measurement in the standard basis, resulting in a classical bit.
- 1.4 Symbol for a CNOT gate, and its effect on basis states.
- 1.5 Circuit for creating an entangled state known as a Bell State. When the two qubits are measured, they will either both be 0, or they will both be 1.
- 1.6 Result of executing the circuit 1024 times on a quantum simulator, compared with executing the circuit 1024 times on a real IBM quantum computer.
- 1.7 Hypothetical cloning operator, that creates an exact and independent copy of unknown quantum state |a?. The text will show that such an operator cannot be implemented.
- 1.8 Conceptual illustration of the Deutsch Problem.
- 1.9 Reversible circuit for calculating f(x).
- 1.10 Implementations of black-box function Uf for Deutsch's problem. Top output is |y?f(x)?, and bottom output is |x?.
- 1.11 Implementation of Deutsch's algorithm. The dashed box is equivalent to U in Figure 1.8.
- 1.12 System diagram for a superconducting quantum computer.
- 2.1 Rotation of a vector of length r CCW around the z axis.
- 2.2 Illustration of how two consecutive rotations can be replaced with a single equivalent rotation.
- 2.3 Representation of a single qubit state on the Bloch Sphere (created in part using [2]).
- 2.4 Precession of spin vector for a particle with positive charge in a z-directed magnetic field.
- 2.5 Solutions to the coupled mode equations for ?=0 and ?/?=3. For the case of Rabi oscillations, |?t|=ORt/2.
- 2.6 Rotations enabling measurement of the projections of the state vector along the x and y axes. (a) The Hadamard gate corresponds to rotation of the state vector around an axis in the x-z plane making a 45o angle with the z axis. This rotates the x component to z axis. (b) z rotation of -90o followed by a Hadamard rotation to estimate the projection along y.
- 2.7 (a) Bloch sphere representation of a mixed state, and (b) the result of applying a Hadamard gate (H) to the mixed state. Plots generated using Qiskit [2].
- 3.1 Common symbols for the SWAP, iSWAP, and iSWAP two-qubit gates.
- 3.2 Operations needed to convert a iSWAP´ gate to a CNOT.
- 3.3 Controlled-U gate. If the control qubit is |1?, the U gate is applied to the target qubit. Otherwise, the gate is not applied.
- 3.4 Implementation of controlled-U gate using Eq. (3.39).
- 4.1 Ladder line used for radio frequency transmission.
- 4.2 Equivalent circuit for a transmission line. (a) Lumped elements can be used so long as the distance ?z is small compared to the distance traveled during a period of the signal. (b) Single section along the line for analysis.
- 4.3 A transmission line terminated with a load impedance. Note that two coordinate systems have their origins at the load location: the coordinate z increases to the right, and the coordinate l increases to the left.
- 4.4 Voltage standing wave pattern along a transmission line with a mismatched load.
- 4.5 Impedance looking into a terminated transmission line of length l.
- 4.6 A real impedance can be matched to a lossless transmission line using a quarter wavelength line whose impedance is the geometric mean of the load and line to be matched. This is referred to as a quarter wave matching transformer.
- 4.7 Some commonly-used types of transmission lines. Since there are multiple ways to drive the coplanar guide, the most common source connection is explicitly shown. All of the dielectric materials are assumed to be nonmagnetic for our purposes.
- 4.8 Incoming and outgoing wave amplitudes used in the definitions of S parameters.
- 4.9 Definition of voltages and currents for the ABCD transmission matrix.
- 4.10 Definitions for constructing the ABCD matrix of a section of transmission line.
- 4.11 Circuit for an attenuator.
- 4.12 Circulator Circuit diagrams. In actual devices, the three ports are symmetrically placed at 120º angles, as represented on the left. However, in drawing circuit diagrams, it is often convenient to show the ports at right angles, as shown in the center. As shown on the right, you can make an isolator by connecting one of the ports to a matched load.
- 4.13 Wilkinson power divider.
- 4.14 Quadrature hybrid 4-port network. For clarity each transmission line is represented by a single line, with the return conductors understood (e.g., an implied ground plane) [24].
- 4.15 Even and odd mode analysis of a quadrature hybrid coupler. All impedances are shown normalized to Zc [24].
- 4.16 Commonly-used symbol for a quadrature hybrid. A signal applied to port 1 is evenly split into quadrature signals at ports 2 and 3.
- 4.17 Mixer Circuit diagrams. In an ideal mixer, the signal at the output terminal is the product of the signals applied to the two input ports.
- 4.18 (a) Circuit to shape a microwave pulse. (b) Example of creating a pulse with a Gaussian pulse shape.
- 4.19 (a) Circuit to recover the cosine (in-phase, or I) and sine (quadrature, or Q) components of an RF signal. (b) The amplitude and phase plotted on the IQ plane. (c) Raw I-Q signals measured on ibm-q-armonk, a 1 qubit demonstration processor on the IBM Q Network. The means for each state are indicated by the large markers.
- 4.20 Low-pass filter circuits.
- 4.21 Frequency response of the T network low-pass filter.
- 4.22 Circuit illustrating thermal noise power from a resistor at temperature T. (a) Resistor at temperature T coupled to a load through a lossless bandpass filter with bandwidth B. (b) Equivalent circuit explicitly showing a noise voltage source that depends on temperature.
- 4.23 Quantum noise as a function of temperature normalized to the photon energy. Solid line is the exact expression (4.161), while the dashed line is the Rayleigh-Jeans approximation (4.160).
- 4.24 Noise added by a circuit with power gain G. For an amplifier, G>1, while for an attenuator G<1.
- 4.25 Thermal noise from a passive element such as an attenuator at temperature T. (a) Power applied to port 2 is partially reflected and partially transmitted to the output, with the balance dissipated as heat. The power dissipated is visualized as being conveyed to a fictitious port 3 connected to a thermal reservoir at temperature T. (b) The total power out has a contribution from the input as well as the thermal reservoir attached to the fictitious port 3.
- 4.26 Noise in a system of cascaded components.
- 4.27 Layered structure of different semiconductor materials used to separate the donor impurities from the donor electrons to achieve minimal scattering. Structures of different materials like this are referred to as heterostructures. Owing to the different band structure of AlGaAs and GaAs, donor electrons from the heavily-doped AlGaAs layer become trapped on the GaAs side of the interface between intrinsic GaAs and AlGaAs, forming a "2D electron gas." This concentration of electrons is depicted by the shaded region. A heterostructure similar to this is a key feature of HEMTs.
- 5.1 Resonator circuits.
- 5.2 Equivalent circuits for capacitively-coupled lumped-element resonator.
- 5.3 Capacitively-coupled transmission line resonator.
- 5.4 Near the nth resonant frequency of the transmission line, the capacitively-coupled transmission line resonator can be modeled as a capacitively-coupled lumped-element resonator.
- 5.5 Equivalent circuits used to calculate the insertion loss and return loss of a capacitively-coupled transmission line resonator.
- 5.6 Characteristics of capacitively-coupled transmission line resonator. Note the significantly different frequency scales on the horizontal axis as well as the different vertical scales for |S11|. Parameters used are Ln = 0.453µH/m, Zc=RL=50O, f0 = 5 GHz, Qint = 2.3, ×105, and l = 11.04 mm. These values are comparable to experimentally-measured parameters. Based on [26].
- 5.7 Two LC resonant circuits coupled by a capacitor.
- 5.8 Coupling between lossless LC resonators for Cg/CACB = 0.05.
- 5.9 Tire swings suspended...
