Details

<p>1 Qubits, Gates, and Circuits 1</p> <p>1.1 Bits and Qubits . . . . . . . . . . . . . . . . . . . . . . . . 1</p> <p>1.1.1 Circuits in Space vs. Circuits in Time . . . . . . . 1</p> <p>1.1.2 Superposition . . . . . . . . . . . . . . . . . . . . . 2</p> <p>1.1.3 No Cloning . . . . . . . . . . . . . . . . . . . . . . 3</p> <p>1.1.4 Reversibility . . . . . . . . . . . . . . . . . . . . . 4</p> <p>1.1.5 Entanglement . . . . . . . . . . . . . . . . . . . . . 4</p> <p>1.2 Single-Qubit States . . . . . . . . . . . . . . . . . . . . . . 5</p> <p>1.3 Measurement and the Born Rule . . . . . . . . . . . . . . 6</p> <p>1.4 Unitary Operations and Single-Qubit Gates . . . . . . . . 7</p> <p>1.5 Two-Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . 9</p> <p>1.5.1 Two-Qubit States . . . . . . . . . . . . . . . . . . . 9</p> <p>1.5.2 Two-Qubit Gates . . . . . . . . . . . . . . . . . . . 11</p> <p>1.5.3 Controlled-NOT . . . . . . . . . . . . . . . . . . . 13</p> <p>1.6 Bell State . . . . . . . . . . . . . . . . . . . . . . . . . . . 14</p> <p>1.7 No Cloning, Revisited . . . . . . . . . . . . . . . . . . . . 15</p> <p>1.8 Example: Deutsch’s Problem . . . . . . . . . . . . . . . . 17</p> <p>1.9 Key Characteristics of Quantum Computing . . . . . . . . 20</p> <p>1.10 Quantum Computing Systems . . . . . . . . . . . . . . . . 22</p> <p>1.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 26</p> <p>2 Physics of Single Qubit Gates 29</p> <p>2.1 Requirements for a Quantum Computer . . . . . . . . . . 29</p> <p>2.2 Single Qubit Gates . . . . . . . . . . . . . . . . . . . . . . 30</p> <p>2.2.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . 30</p> <p>2.2.2 Two State Systems . . . . . . . . . . . . . . . . . . 38</p> <p>2.2.3 Creating Rotations: Rabi Oscillations . . . . . . . 44</p> <p>2.3 Quantum State Tomography . . . . . . . . . . . . . . . . 49</p> <p>2.4 Expectation Values and the Pauli Operators . . . . . . . . 51</p> <p>2.5 Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . 52</p> <p>2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 56</p> <p>iii</p> <p>iv CONTENTS</p> <p>3 Physics of Two Qubit Gates 59</p> <p>3.1 √</p> <p>iSWAP Gate . . . . . . . . . . . . . . . . . . . . . . . . 59</p> <p>3.2 Coupled Tunable Qubits . . . . . . . . . . . . . . . . . . . 61</p> <p>3.3 Fixed-frequency Qubits . . . . . . . . . . . . . . . . . . . 64</p> <p>3.4 Other Controlled Gates . . . . . . . . . . . . . . . . . . . 66</p> <p>3.5 Two-qubit States and the Density Matrix . . . . . . . . . 68</p> <p>3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 71</p> <p>4 Superconducting Quantum Computer Systems 73</p> <p>4.1 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . 73</p> <p>4.1.1 General Transmission Line Equations . . . . . . . 73</p> <p>4.1.2 Lossless Transmission Lines . . . . . . . . . . . . . 75</p> <p>4.1.3 Transmission Lines with Loss . . . . . . . . . . . . 77</p> <p>4.2 Terminated Lossless Line . . . . . . . . . . . . . . . . . . 82</p> <p>4.2.1 Reflection Coefficient . . . . . . . . . . . . . . . . . 82</p> <p>4.2.2 Power (Flow of Energy) and Return Loss . . . . . 84</p> <p>4.2.3 Standing Wave Ratio (SWR) . . . . . . . . . . . . 85</p> <p>4.2.4 Impedance as a Function of Position . . . . . . . . 86</p> <p>4.2.5 Quarter Wave Transformer . . . . . . . . . . . . . 88</p> <p>4.2.6 Coaxial, Microstrip, and Co-planar Lines . . . . . 89</p> <p>4.3 S Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 92</p> <p>4.3.1 Lossless Condition . . . . . . . . . . . . . . . . . . 93</p> <p>4.3.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . 94</p> <p>4.4 Transmission (ABCD) Matrices . . . . . . . . . . . . . . . 94</p> <p>4.5 Attenuators . . . . . . . . . . . . . . . . . . . . . . . . . . 99</p> <p>4.6 Circulators and Isolators . . . . . . . . . . . . . . . . . . . 100</p> <p>4.7 Power Dividers/Combiners . . . . . . . . . . . . . . . . . 102</p> <p>4.8 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105</p> <p>4.9 Low-pass Filters . . . . . . . . . . . . . . . . . . . . . . . 111</p> <p>4.10 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112</p> <p>4.10.1 Thermal Noise . . . . . . . . . . . . . . . . . . . . 113</p> <p>4.10.2 Equivalent Noise Temperature . . . . . . . . . . . 116</p> <p>4.10.3 Noise Factor and Noise Figure . . . . . . . . . . . 117</p> <p>4.10.4 Attenuators and Noise . . . . . . . . . . . . . . . . 118</p> <p>4.10.5 Noise in Cascaded Systems . . . . . . . . . . . . . 120</p> <p>4.11 Low Noise Amplifiers . . . . . . . . . . . . . . . . . . . . . 121</p> <p>4.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 123</p> <p>5 Resonators: Classical Treatment 125</p> <p>5.1 Parallel Lumped Element Resonator . . . . . . . . . . . . 125</p> <p>5.2 Capacitive Coupling to a Parallel Lumped-Element Res[1]onator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128</p> <p>5.3 Transmission Line Resonator . . . . . . . . . . . . . . . . 130</p> <p>5.4 Capacitive Coupling to a Transmission Line Resonator . . 133</p> <p>5.5 Capacitively-Coupled Lossless Resonators . . . . . . . . . 136</p> <p>CONTENTS v</p> <p>5.6 Classical Model of Qubit Readout . . . . . . . . . . . . . 142</p> <p>5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 146</p> <p>6 Resonators: Quantum Treatment 149</p> <p>6.1 Lagrangian Mechanics . . . . . . . . . . . . . . . . . . . . 149</p> <p>6.1.1 Hamilton’s Principle . . . . . . . . . . . . . . . . . 149</p> <p>6.1.2 Calculus of Variations . . . . . . . . . . . . . . . . 150</p> <p>6.1.3 Lagrangian Equation of Motion . . . . . . . . . . . 151</p> <p>6.2 Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . 153</p> <p>6.3 Harmonic Oscillators . . . . . . . . . . . . . . . . . . . . . 153</p> <p>6.3.1 Classical Harmonic Oscillator . . . . . . . . . . . . 154</p> <p>6.3.2 Quantum Mechanical Harmonic Oscillator . . . . . 156</p> <p>6.3.3 Raising and Lowering Operators . . . . . . . . . . 158</p> <p>6.3.4 Can a Harmonic Oscillator be used as a Qubit? . . 160</p> <p>6.4 Circuit Quantum Electrodynamics . . . . . . . . . . . . . 162</p> <p>6.4.1 Classical LC Resonant Circuit . . . . . . . . . . . 162</p> <p>6.4.2 Quantization of the LC Circuit . . . . . . . . . . . 163</p> <p>6.4.3 Circuit Electrodynamic Approach for General Cir[1]cuits . . . . . . . . . . . . . . . . . . . . . . . . . . 164</p> <p>6.4.4 Circuit Model for Transmission Line Resonator . . 165</p> <p>6.4.5 Quantizing a Transmission Line Resonator . . . . 168</p> <p>6.4.6 Quantized Coupled LC Resonant Circuits . . . . . 169</p> <p>6.4.7 Schrödinger, Heisenberg, and Interaction Pictures 172</p> <p>6.4.8 Resonant Circuits and Qubits . . . . . . . . . . . . 175</p> <p>6.4.9 The Dispersive Regime . . . . . . . . . . . . . . . . 178</p> <p>6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 182</p> <p>7 Theory of Superconductivity 183</p> <p>7.1 Bosons and Fermions . . . . . . . . . . . . . . . . . . . . . 184</p> <p>7.2 Bloch Theorem . . . . . . . . . . . . . . . . . . . . . . . . 186</p> <p>7.3 Free Electron Model for Metals . . . . . . . . . . . . . . . 188</p> <p>7.3.1 Discrete States in Finite Samples . . . . . . . . . . 189</p> <p>7.3.2 Phonons . . . . . . . . . . . . . . . . . . . . . . . . 191</p> <p>7.3.3 Debye Model . . . . . . . . . . . . . . . . . . . . . 193</p> <p>7.3.4 Electron-Phonon Scattering and Electrical Con[1]ductivity . . . . . . . . . . . . . . . . . . . . . . . 194</p> <p>7.3.5 Perfect Conductor vs. Superconductor . . . . . . . 196</p> <p>7.4 Bardeen, Cooper and Schrieffer Theory of Superconduc[1]tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199</p> <p>7.4.1 Cooper Pair Model . . . . . . . . . . . . . . . . . . 199</p> <p>7.4.2 Dielectric Function . . . . . . . . . . . . . . . . . . 203</p> <p>7.4.3 Jellium . . . . . . . . . . . . . . . . . . . . . . . . 204</p> <p>7.4.4 Scattering Amplitude and Attractive Electron-Electron</p> <p>Interaction . . . . . . . . . . . . . . . . . . . . . . 208</p> <p>7.4.5 Interpretation of Attractive Interaction . . . . . . 209</p> <p>vi CONTENTS</p> <p>7.4.6 Superconductor Hamiltonian . . . . . . . . . . . . 210</p> <p>7.4.7 Superconducting Ground State . . . . . . . . . . . 211</p> <p>7.5 Electrodynamics of Superconductors . . . . . . . . . . . . 215</p> <p>7.5.1 Cooper Pairs and the Macroscopic Wave Function 215</p> <p>7.5.2 Potential Functions . . . . . . . . . . . . . . . . . . 216</p> <p>7.5.3 London Equations . . . . . . . . . . . . . . . . . . 217</p> <p>7.5.4 London Gauge . . . . . . . . . . . . . . . . . . . . 219</p> <p>7.5.5 Penetration Depth . . . . . . . . . . . . . . . . . . 220</p> <p>7.5.6 Flux Quantization . . . . . . . . . . . . . . . . . . 221</p> <p>7.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . 223</p> <p>7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 224</p> <p>8 Josephson Junctions 225</p> <p>8.1 Tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . 225</p> <p>8.1.1 Reflection from a Barrier . . . . . . . . . . . . . . 226</p> <p>8.1.2 Finite Thickness Barrier . . . . . . . . . . . . . . . 229</p> <p>8.2 Josephson Junctions . . . . . . . . . . . . . . . . . . . . . 231</p> <p>8.2.1 Current and Voltage Relations . . . . . . . . . . . 231</p> <p>8.2.2 Josephson Junction Hamiltonian . . . . . . . . . . 235</p> <p>8.2.3 Quantized Josephson Junction Analysis . . . . . . 237</p> <p>8.3 Superconducting Quantum Interference Devices (SQUIDs) 239</p> <p>8.4 Josephson Junction Parametric Amplifiers . . . . . . . . . 241</p> <p>8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 242</p> <p>9 Errors and Error Mitigation 245</p> <p>9.1 NISQ Processors . . . . . . . . . . . . . . . . . . . . . . . 245</p> <p>9.2 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . 246</p> <p>9.3 State Preparation and Measurement Errors . . . . . . . . 248</p> <p>9.4 Characterizing Gate Errors . . . . . . . . . . . . . . . . . 250</p> <p>9.5 State Leakage and Suppression using Pulse Shaping . . . 254</p> <p>9.6 Zero-Noise Extrapolation . . . . . . . . . . . . . . . . . . 257</p> <p>9.7 Optimized Control using Deep Learning . . . . . . . . . . 260</p> <p>9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 261</p> <p>10 Quantum Error Correction 265</p> <p>10.1 Review of Classical Error Correction . . . . . . . . . . . . 265</p> <p>10.1.1 Error Detection . . . . . . . . . . . . . . . . . . . . 266</p> <p>10.1.2 Error Correction: Repetition Code . . . . . . . . . 267</p> <p>10.1.3 Hamming Code . . . . . . . . . . . . . . . . . . . . 268</p> <p>10.2 Quantum Errors . . . . . . . . . . . . . . . . . . . . . . . 269</p> <p>10.3 Detecting and Correcting Quantum Errors . . . . . . . . . 272</p> <p>10.3.1 Bit Flip . . . . . . . . . . . . . . . . . . . . . . . . 272</p> <p>10.3.2 Phase Flip . . . . . . . . . . . . . . . . . . . . . . 274</p> <p>10.3.3 Correcting Bit and Phase Flips: Shor’s 9-qubit Code275</p> <p>10.3.4 Arbitrary Rotations . . . . . . . . . . . . . . . . . 277</p> <p>CONTENTS vii</p> <p>10.4 Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . 279</p> <p>10.4.1 Stabilizers . . . . . . . . . . . . . . . . . . . . . . . 279</p> <p>10.4.2 Stabilizers for Error Correction . . . . . . . . . . . 280</p> <p>10.5 Operating on Logical Qubits . . . . . . . . . . . . . . . . 283</p> <p>10.6 Error Thresholds . . . . . . . . . . . . . . . . . . . . . . . 285</p> <p>10.6.1 Concatenation of Error Codes . . . . . . . . . . . . 286</p> <p>10.6.2 Threshold Theorem . . . . . . . . . . . . . . . . . 286</p> <p>10.7 Surface Codes . . . . . . . . . . . . . . . . . . . . . . . . . 288</p> <p>10.7.1 Stabilizers . . . . . . . . . . . . . . . . . . . . . . . 289</p> <p>10.7.2 Error Detection and Correction . . . . . . . . . . . 291</p> <p>10.7.3 Logical X and Z Operators . . . . . . . . . . . . . 295</p> <p>10.7.4 Multiple Qubits: Lattice Surgery . . . . . . . . . . 297</p> <p>10.7.5 CNOT . . . . . . . . . . . . . . . . . . . . . . . . . 301</p> <p>10.7.6 Single-Qubit Gates . . . . . . . . . . . . . . . . . . 305</p> <p>10.8 Summary and Further Reading . . . . . . . . . . . . . . . 306</p> <p>10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 308</p> <p>11 Quantum Logic: Efficient Implementation of Classical</p> <p>Computations 309</p> <p>11.1 Reversible Logic . . . . . . . . . . . . . . . . . . . . . . . 310</p> <p>11.1.1 Reversible Logic Gates . . . . . . . . . . . . . . . . 311</p> <p>11.1.2 Reversible Logic Circuits . . . . . . . . . . . . . . 313</p> <p>11.2 Quantum Logic Circuits . . . . . . . . . . . . . . . . . . . 317</p> <p>11.2.1 Entanglement and Uncomputing . . . . . . . . . . 317</p> <p>11.2.2 Multi-qubit gates . . . . . . . . . . . . . . . . . . . 319</p> <p>11.2.3 Qubit topology . . . . . . . . . . . . . . . . . . . . 321</p> <p>11.3 Efficient Arithmetic Circuits: Adder . . . . . . . . . . . . 322</p> <p>11.3.1 Quantum Ripple Carry Adder . . . . . . . . . . . . 323</p> <p>11.3.2 In-place Ripple Carry Adder . . . . . . . . . . . . 326</p> <p>11.3.3 Carry-Lookahead Adder . . . . . . . . . . . . . . . 329</p> <p>11.3.4 Adder Comparison . . . . . . . . . . . . . . . . . . 334</p> <p>11.4 Phase Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 336</p> <p>11.4.1 Controlled-Z and Controlled-Phase Gates . . . . . 336</p> <p>11.4.2 Selective Phase Change . . . . . . . . . . . . . . . 339</p> <p>11.4.3 Phase Logic Gates . . . . . . . . . . . . . . . . . . 341</p> <p>11.5 Summary and Further Reading . . . . . . . . . . . . . . . 342</p> <p>11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 345</p> <p>12 Some Quantum Algorithms 347</p> <p>12.1 Computational Complexity . . . . . . . . . . . . . . . . . 347</p> <p>12.1.1 Quantum Program Run-Time . . . . . . . . . . . . 348</p> <p>12.1.2 Classical Complexity Classes . . . . . . . . . . . . 349</p> <p>12.1.3 Quantum Complexity . . . . . . . . . . . . . . . . 350</p> <p>12.2 Grover’s Search Algorithm . . . . . . . . . . . . . . . . . . 351</p> <p>12.2.1 Grover Iteration . . . . . . . . . . . . . . . . . . . 351</p> <p>viii CONTENTS</p> <p>12.2.2 Quantum Implementation . . . . . . . . . . . . . . 354</p> <p>12.2.3 Generalizations . . . . . . . . . . . . . . . . . . . . 357</p> <p>12.3 Quantum Fourier Transform . . . . . . . . . . . . . . . . . 358</p> <p>12.3.1 Frequencies and Quantum-encoded Signals . . . . 358</p> <p>12.3.2 Inverse QFT . . . . . . . . . . . . . . . . . . . . . 361</p> <p>12.3.3 Quantum Implementation . . . . . . . . . . . . . . 362</p> <p>12.3.4 Computational Complexity . . . . . . . . . . . . . 365</p> <p>12.4 Quantum Phase Estimation . . . . . . . . . . . . . . . . . 365</p> <p>12.4.1 Quantum Implementation . . . . . . . . . . . . . . 366</p> <p>12.4.2 Computational Complexity and Other Issues . . . 367</p> <p>12.5 Shor’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . 368</p> <p>12.5.1 Hybrid Classical-Quantum Algorithm . . . . . . . 368</p> <p>12.5.2 Finding the Period . . . . . . . . . . . . . . . . . . 370</p> <p>12.5.3 Computational Complexity . . . . . . . . . . . . . 373</p> <p>12.6 Variational Quantum Algorithms . . . . . . . . . . . . . . 375</p> <p>12.6.1 Variational Quantum Eigensolver . . . . . . . . . . 377</p> <p>12.6.2 Quantum Approximate Optimization Algorithm . 382</p> <p>12.6.3 Challenges and Opportunities . . . . . . . . . . . . 386</p> <p>12.7 Summary and Further Reading . . . . . . . . . . . . . . . 387</p> <p>12.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 388</p>

<p><b>Daniel D. Stancil, PhD,</b> 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.</p> <p><b> Gregory T. Byrd, PhD,</b> 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.

<p><b>Explore the intersection of computer science, physics, and electrical and computer engineering with this discussion of the engineering of quantum computers</b></p> <p>In <i>Principles of Superconducting Quantum Computers</i>, 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. <p>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. <p>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: <ul><li>A thorough introduction to qubits, gates, and circuits, including unitary transformations, single qubit gates, and controlled (two qubit) gates</li> <li>Comprehensive explorations of the physics of single qubit gates, including the requirements for a quantum computer, rotations, two-state systems, and Rabi oscillations</li> <li>Practical discussions of the physics of two qubit gates, including tunable qubits, SWAP gates, controlled-NOT gates, and fixed frequency qubits</li> <li>In-depth examinations of superconducting quantum computer systems, including the need for cryogenic temperatures, transmission lines, S parameters, and more</li></ul> <p>Ideal for senior-level undergraduate and graduate students in electrical and computer engineering programs, <i>Principles of Superconducting Quantum Computers</i> also deserves a place in the libraries of practicing engineers seeking a better understanding of quantum computer systems.

US