Quantum computing. An applied approach

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Contents......Page 5 Preface......Page 11 Acknowledgements......Page 14 Navigating this Book......Page 16 Part I Foundations......Page 19 Superposition, Entanglement and Reversibility......Page 20 A Brief History of Quantum Computing......Page 27 Qubits, Operators and Measurement......Page 33 Unary Operators......Page 38 Binary Operators......Page 42 Ternary Operators......Page 44 3.2 Comparison with Classical Gates......Page 46 3.4 Gottesman-Knill and Solovay-Kitaev......Page 47 3.5 The Bloch Sphere......Page 48 3.6 The Measurement Postulate......Page 49 3.7 Computation in Place......Page 51 3.4 Representing Superposition of States......Page 36 Quantum Circuit Diagrams......Page 37 4.1 Problems vs. Algorithms......Page 53 4.2 Time Complexity......Page 54 4.3 Complexity Classes......Page 56 4.4 Quantum Computing and the Church-Turing Thesis......Page 59 Part II Hardware and Applications......Page 61 5.1 Assessing a Quantum Computer......Page 62 5.2 Neutral Atom......Page 64 5.3 NMR......Page 65 5.4 NV Center-in-Diamond......Page 66 5.5 Photonics......Page 67 Semiconductor quantum transistor......Page 68 5.6 Spin Qubits......Page 69 5.7 Superconducting Qubits......Page 71 5.8 Topological Quantum Computation......Page 72 5.9 Trapped Ion......Page 73 5.10 Summary......Page 74 Development Libraries for Quantum Computer Programming......Page 76 6.1 Quantum Computers and QC Simulators......Page 77 6.2 Cirq......Page 79 6.3 Qiskit......Page 81 6.4 Forest......Page 84 6.5 Quantum Development Kit......Page 86 6.6 Dev Libraries Summary......Page 89 Other Development Libraries......Page 90 Bell States......Page 91 Gates with Parameters......Page 92 7.1 Quantum Teleportation......Page 95 7.2 Superdense Coding......Page 98 7.3 Code for Quantum Teleportation and Superdense Communication......Page 99 7.4 Bell Inequality Test......Page 102 Summary......Page 106 The Canon: Code Walkthroughs......Page 108 8.1 The Deutsch-Jozsa Algorithm......Page 110 8.2 The Bernstein-Vazirani Algorithm......Page 117 8.3 Simon’s Problem......Page 120 8.4 Quantum Fourier Transform......Page 121 RSA Cryptography......Page 124 The Period of a Function......Page 126 Period of a Function as an Input to a Factorization Algorithm......Page 127 8.6 Grover’s Search Algorithm......Page 139 Summary......Page 142 9.1 Variational Quantum Eigensolver......Page 144 VQE with Noise......Page 149 More Sophisticated Ansatzes......Page 151 9.2 Quantum Chemistry......Page 152 9.3 Quantum Approximate Optimization Algorithm (QAOA)......Page 157 Example Implementation of QAOA......Page 160 9.4 Machine Learning on Quantum Processors......Page 167 9.5 Quantum Phase Estimation......Page 173 Implemention of QPE......Page 176 9.6 Solving Linear Systems......Page 179 Description of the HHL Algorithm......Page 181 Example Implementation of the HHL Algorithm......Page 183 9.7 Quantum Random Number Generator......Page 191 9.8 Quantum Walks......Page 193 Implementation of a Quantum Walk......Page 195 9.9 Summary......Page 200 Quantum Simulation and Chemistry......Page 202 10.2 Quantum Supremacy......Page 203 Random Circuit Sampling......Page 204 Other Problems for Demonstrating Quantum Supremacy......Page 208 Quantum Error Correction......Page 209 Conclusion......Page 210 Part III Toolkit......Page 212 11.1 Introduction and Self-Test......Page 213 Vectors and Notation......Page 215 Basic Vector Operations......Page 216 The Norm of a Vector......Page 220 The Dot Product......Page 223 Complex Numbers......Page 226 The Inner Product as a Refinement of the Dot Product......Page 228 The Polar Coordinate Representation of a Complex Number......Page 232 Basic Matrix Operations......Page 240 The Identity Matrix......Page 247 Transpose, Conjugate and Trace......Page 249 Matrix Exponentiation......Page 256 The Outer Product as a Way of Building Matrices......Page 257 The Tensor Product......Page 259 The Basics of Set Theory......Page 262 The Cartesian Product......Page 265 Relations and Functions......Page 266 Important Properties of Functions......Page 271 11.7 The Definition of a Linear Transformation......Page 276 11.8 How to Build a Vector Space From Scratch......Page 278 Groups......Page 279 Fields......Page 285 The Definition of a Vector Space......Page 292 Subspaces......Page 294 Span......Page 297 Linear Independence......Page 299 Bases and Dimension......Page 301 Orthonormal Bases......Page 304 12.1 Linear Transformations as Matrices......Page 306 An Introduction to the Determinant......Page 311 The Geometry of the Determinant......Page 316 Matrix Inversion......Page 317 12.3 Eigenvectors and Eigenvalues......Page 325 Change of Basis......Page 327 12.4 Further Investigation of Inner Products......Page 330 Why We Can’t Measure with Complex Numbers......Page 333 Hermitian Operators Have Real Eigenvalues......Page 335 12.6 Unitary operators......Page 337 The Direct Sum......Page 338 The Tensor Product......Page 340 Metrics, Cauchy Sequences and Completeness......Page 344 An Axiomatic Definition of the Inner Product......Page 348 The Definition of Hilbert Space......Page 349 12.9 The Qubit as a Hilbert Space......Page 350 13.1 Boolean Functions......Page 354 13.2 Logarithms and Exponentials......Page 355 13.3 Euler’s Formula......Page 357 Table of Quantum Operators and Core Circuits......Page 360 Works Cited......Page 363