Machine Learning with Quantum Computers

Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction
	1.1 Background
		1.1.1 Merging Two Disciplines
		1.1.2 The Rise of Quantum Machine Learning
		1.1.3 Four Intersections
		1.1.4 Fault-Tolerant Versus Near-Term Approaches
	1.2 A Toy Example of a Quantum Algorithm for Classification
		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
		1.2.5 Organisation of the Book
	References
2 Machine Learning
	2.1 Examples of Typical Machine Learning Problems
	2.2 The Three Ingredients of a Learning Problem
		2.2.1 Data
		2.2.2 Model
		2.2.3 Loss
	2.3 Risk Minimisation in Supervised Learning
		2.3.1 Minimising the Empirical Risk as a Proxy
		2.3.2 Quantifying Generalisation
		2.3.3 Optimisation
	2.4 Training in Unsupervised Learning
	2.5 Methods in Machine Learning
		2.5.1 Linear Models
		2.5.2 Neural Networks
		2.5.3 Graphical Models
		2.5.4 Kernel Methods
	References
3 Quantum Computing
	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 Qubits
		3.2.3 Quantum Gates
		3.2.4 Measuring Qubits in the Computational Basis
		3.2.5 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
	3.4 Strategies of Input Encoding
		3.4.1 Basis Encoding
		3.4.2 Amplitude Encoding
		3.4.3 Time-Evolution Encoding
		3.4.4 Hamiltonian Encoding
	3.5 Quantum Speedups
	3.6 Important Quantum Algorithms
		3.6.1 Measuring the Overlap of Quantum States
		3.6.2 Grover Search
		3.6.3 Quantum Phase Estimation
		3.6.4 Matrix Multiplication and Inversion
		3.6.5 Variational Quantum Algorithms
	3.7 Quantum Annealing and Other Computational Models
	References
4 Representing Data on a Quantum Computer
	4.1 Encoding Binary Inputs into Basis States
		4.1.1 Encoding a Single Input
		4.1.2 Encoding Data in Superposition
	4.2 Arbitrary State Preparation for Amplitude Encoding
		4.2.1 Amplitude-Efficient State Preparation
		4.2.2 Qubit-Efficient State Preparation
	4.3 Encoding Inputs as Time Evolutions
	4.4 Encoding a Dataset via the Hamiltonian
		4.4.1 Hamiltonian Simulation
		4.4.2 Qubit-Efficient Simulation of Hamiltonians
		4.4.3 Density Matrix Exponentiation
	4.5 Data Encoding as a Feature Map
		4.5.1 Why Data Encoding Is so Essential
		4.5.2 Examples of Data-Encoding Feature Maps
	References
5 Variational Circuits as Machine Learning Models
	5.1 How to Interpret a Quantum Circuit as a Model
		5.1.1 Deterministic Quantum Models
		5.1.2 Probabilistic Quantum Models
		5.1.3 An Example: Variational Quantum Classifier
		5.1.4 An Example: Variational Generator
	5.2 Which Functions Do Variational Quantum Models Express?
		5.2.1 Quantum Models as Linear Combinations of Periodic Functions
		5.2.2 An Example: The Pauli-Rotation Encoding
	5.3 Training Variational Quantum Models
		5.3.1 Gradients of Quantum Computations
		5.3.2 Parameter-Shift Rules
		5.3.3 Barren Plateaus
		5.3.4 Generative Training
	5.4 Quantum Circuits and Neural Networks
		5.4.1 Emulating Nonlinear Activations
		5.4.2 Variational Circuits as Deep Linear Neural Networks
		5.4.3 Time-Evolution Encoding as an Exponential Activation
	References
6 Quantum Models as Kernel Methods
	6.1 The Connection Between Quantum Models and Kernel Methods
	6.2 Quantum Computing, Feature Maps and Kernels
		6.2.1 Data Encoding as a Feature Map
		6.2.2 Quantum Kernels
		6.2.3 Making Sense of Matrix-Valued Feature Vectors
	6.3 Examples of Quantum Kernels
		6.3.1 Quantum Kernels Derived from Data Encoding
		6.3.2 Fourier Representation of Quantum Kernels
	6.4 The RKHS of Quantum Kernels
		6.4.1 Quantum Models as Linear Models
		6.4.2 Describing the RKHS
	6.5 Kernel-Based Training
		6.5.1 Training as Optimising Over the RKHS
		6.5.2 Optimal Measurements and the Representer Theorem
		6.5.3 The Impact of the Kernel on Regularisation
		6.5.4 Kernel-Based Learning Is Suprisingly Simple
	6.6 Comparing Kernel-Based and Variational Training
	References
7 Fault-Tolerant Quantum Machine Learning
	7.1 Linear Algebra Accelerators
		7.1.1 Basic Idea
		7.1.2 Matrix Inversion for Training
	7.2 Search and Amplitude Amplification
		7.2.1 Finding Closest Neighbours
		7.2.2 Adapting Grover's Search to Data Superpositions
		7.2.3 Amplitude Amplification for Perceptron Training
		7.2.4 Quantum Walks
	7.3 Sampling and Probabilistic Models
		7.3.1 Bayesian Networks
		7.3.2 Boltzmann Machines
		7.3.3 Other Proposals
	7.4 Superposition and Quantum Ensembles
	References
8 Approaches Based on the Ising Model
	8.1 Quantum Extensions of Ising Models
		8.1.1 The Quantum Ising Model
		8.1.2 Boltzmann Machines with a Transverse Field
		8.1.3 Quantum Hopfield Models
	8.2 Quantum Annealing
		8.2.1 Quadratic Unconstrained Optimisation
		8.2.2 Encoding Classifiers into an Annealer
		8.2.3 Annealing Devices as Samplers
	References
9 Potential Quantum Advantages
	9.1 Dissecting Quantum Advantage
		9.1.1 Do Quantum Models Generalise Well?
		9.1.2 Can Quantum Computers Speed up Machine Learning?
	9.2 Learning from Coherent Data
		9.2.1 Sample Complexity of Learning
		9.2.2 Exact Learning from Membership Queries
		9.2.3 PAC Learning from Examples
		9.2.4 Learning to Predict General Measurement Outcomes
	9.3 The Future of Quantum Machine Learning
	References
Index