Quantum Machine Learning with Python Using Cirq from Google Research and IBM Qiskit Santanu Pattanayak Quantum Machine Learning with Python Santanu Pattanayak Bangalore, Karnataka, India ISBN-13 (pbk): 978-1-4842-6521-5 ISBN-13 (electronic): 978-1-4842-6522-2 https://doi.org/10.1007/978-1-4842-6522-2 Copyright © 2021 by Santanu Pattanayak This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Trademarked names, logos, and images may appear in this book. 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Printed on acid-free paper To my wife, Sonia Table of Contents About the Author ..................................................................................................... xi About the Technical Reviewer ............................................................................... xiii Acknowledgments ...................................................................................................xv Introduction ...........................................................................................................xvii Chapter 1: Introduction to Quantum Computing ....................................................... 1 Quantum Bit .................................................................................................................................. 2 Realization of a Quantum Bit ................................................................................................... 4 Bloch Sphere Representation of a Qubit .................................................................................. 5 Stern–Gerlach Experiment .......................................................................................................... 10 Multiple Qubits ............................................................................................................................ 14 Bell State ..................................................................................................................................... 14 Multiple-Qubit State .............................................................................................................. 15 Dirac Notation ............................................................................................................................. 16 Ket Vector .............................................................................................................................. 17 Bra Vector .............................................................................................................................. 17 Inner Product ......................................................................................................................... 17 Magnitude of a Vector............................................................................................................ 18 Outer Product ........................................................................................................................ 19 Tensor Product....................................................................................................................... 20 Single-Qubit Gates ...................................................................................................................... 21 Quantum NOT Gate ................................................................................................................ 21 Hadamard Gate ...................................................................................................................... 24 Quantum Z Gate ..................................................................................................................... 25 v TABLE OF CONTENTS Multiple-Qubit Gates ................................................................................................................... 25 CNOT Gate.............................................................................................................................. 25 Controlled-U Gate .................................................................................................................. 28 Copying a Qubit: No Cloning Theorem ......................................................................................... 29 Measurements in Different Basis ................................................................................................ 31 Bell States with Quantum Gates ................................................................................................. 32 Quantum Teleportation ................................................................................................................ 35 Quantum Parallelism Algorithms ................................................................................................. 38 Quantum Interference ................................................................................................................. 42 Summary..................................................................................................................................... 43 Chapter 2: Mathematical Foundations and Postulates of Quantum Computing ..... 45 Topics from Linear Algebra ......................................................................................................... 45 Linear Independence of Vectors ............................................................................................ 46 Basis Vectors ............................................................................................................................... 47 Orthonormal Basis ...................................................................................................................... 48 Linear Operators ......................................................................................................................... 48 Interpretation of a Linear Operator as a Matrix ........................................................................... 49 Linear Operator in Terms of Outer Product ................................................................................. 51 Pauli Operators and Their Outer Product Representation ........................................................... 52 Eigenvectors and Eigenvalues of a Linear Operator ................................................................... 53 Diagonal Representation of an Operator ..................................................................................... 54 Adjoint of an Operator ................................................................................................................. 55 Self-Adjoint or Hermitian Operators ............................................................................................ 55 Normal Operators ........................................................................................................................ 56 Unitary Operators ........................................................................................................................ 56 Spectral Decomposition of Linear Operators .............................................................................