Intro; Preface; Contents; Acronyms; 1 Introduction; 1.1 Background; 1.1.1 Merging Two Disciplines; 1.1.2 The Rise of Quantum Machine Learning; 1.1.3 Four Approaches; 1.1.4 Quantum Computing for Supervised Learning; 1.2 How Quantum Computers Can Classify Data; 1.2.1 The Squared-Distance Classifier; 1.2.2 Interference with the Hadamard Transformation; 1.2.3 Quantum Squared-Distance Classifier; 1.2.4 Insights from the Toy Example; 1.3 Organisation of the Book; References; 2 Machine Learning; 2.1 Prediction; 2.1.1 Four Examples for Prediction Tasks; 2.1.2 Supervised Learning; 2.2 Models

2.2.1 How Data Leads to a Predictive Model2.2.2 Estimating the Quality of a Model; 2.2.3 Bayesian Learning; 2.2.4 Kernels and Feature Maps; 2.3 Training; 2.3.1 Cost Functions; 2.3.2 Stochastic Gradient Descent; 2.4 Methods in Machine Learning; 2.4.1 Data Fitting; 2.4.2 Artificial Neural Networks; 2.4.3 Graphical Models; 2.4.4 Kernel Methods; References; 3 Quantum Information; 3.1 Introduction to Quantum Theory; 3.1.1 What Is Quantum Theory?; 3.1.2 A First Taste; 3.1.3 The Postulates of Quantum Mechanics; 3.2 Introduction to Quantum Computing; 3.2.1 What Is Quantum Computing?

3.2.2 Bits and Qubits3.2.3 Quantum Gates; 3.2.4 Quantum Parallelism and Function Evaluation; 3.3 An Example: The Deutsch-Josza Algorithm; 3.3.1 The Deutsch Algorithm; 3.3.2 The Deutsch-Josza Algorithm; 3.3.3 Quantum Annealing and Other Computational Models; 3.4 Strategies of Information Encoding; 3.4.1 Basis Encoding; 3.4.2 Amplitude Encoding; 3.4.3 Qsample Encoding; 3.4.4 Dynamic Encoding; 3.5 Important Quantum Routines; 3.5.1 Grover Search; 3.5.2 Quantum Phase Estimation; 3.5.3 Matrix Multiplication and Inversion; References; 4 Quantum Advantages; 4.1 Computational Complexity of Learning

4.2 Sample Complexity4.2.1 Exact Learning from Membership Queries; 4.2.2 PAC Learning from Examples; 4.2.3 Introducing Noise; 4.3 Model Complexity; References; 5 Information Encoding; 5.1 Basis Encoding; 5.1.1 Preparing Superpositions of Inputs; 5.1.2 Computing in Basis Encoding; 5.1.3 Sampling from a Qubit; 5.2 Amplitude Encoding; 5.2.1 State Preparation in Linear Time; 5.2.2 Qubit-Efficient State Preparation; 5.2.3 Computing with Amplitudes; 5.3 Qsample Encoding; 5.3.1 Joining Distributions; 5.3.2 Marginalisation; 5.3.3 Rejection Sampling; 5.4 Hamiltonian Encoding

5.4.1 Polynomial Time Hamiltonian Simulation5.4.2 Qubit-Efficient Simulation of Hamiltonians; 5.4.3 Density Matrix Exponentiation; References; 6 Quantum Computing for Inference; 6.1 Linear Models; 6.1.1 Inner Products with Interference Circuits; 6.1.2 A Quantum Circuit as a Linear Model; 6.1.3 Linear Models in Basis Encoding; 6.1.4 Nonlinear Activations; 6.2 Kernel Methods; 6.2.1 Kernels and Feature Maps; 6.2.2 The Representer Theorem; 6.2.3 Quantum Kernels; 6.2.4 Distance-Based Classifiers; 6.2.5 Density Gram Matrices; 6.3 Probabilistic Models; 6.3.1 Qsamples as Probabilistic Models