The theory of kernels offers a rich mathematical framework for the archetypical tasks of classification and regression. Its core insight consists of the representer theorem that asserts that an unknown target function underlying a dataset can be represented by a finite sum of evaluations of a singular function, the so-called kernel function. Together with the infamous kernel trick that provides a practical way of incorporating such a kernel function into a machine learning method, a plethora of algorithms can be made more versatile. This chapter first introduces the mathematical foundations required for understanding the distinguished role of the kernel function and its consequence in terms of the representer theorem. Afterwards, we show how selected popular algorithms, including Gaussian processes, can be promoted to their kernel variant. In addition, several ideas on how to construct suitable kernel functions are provided, before demonstrating the power of kernel methods in the context of quantum (chemistry) problems.

In this chapter, we change our viewpoint and focus on how physics can influence machine learning research. In the first part, we review how tools of statistical physics can help to understand key concepts in machine learning such as capacity, generalization, and the dynamics of the learning process. In the second part, we explore yet another direction and try to understand how quantum mechanics and quantum technologies could be used to solve data-driven task. We provide an overview of the field going from quantum machine learning algorithms that can be run on ideal quantum computers to kernel-based and variational approaches that can be run on current noisy intermediate-scale quantum devices.

In this chapter, we introduce the field of reinforcement learning and some of its most prominent applications in quantum physics and computing. First, we provide an intuitive description of the main concepts, which we then formalize mathematically. We introduce some of the most widely used reinforcement learning algorithms. Starting with temporal-difference algorithms and Q-learning, followed by policy gradient methods and REINFORCE, and the interplay of both approaches in actor-critic algorithms. Furthermore, we introduce the projective simulation algorithm, which deviates from the aforementioned prototypical approaches and has multiple applications in the field of physics. Then, we showcase some prominent reinforcement learning applications, featuring some examples in games; quantum feedback control; quantum computing, error correction and information; and the design of quantum experiments. Finally, we discuss some potential applications and limitations of reinforcement learning in the field of quantum physics.

This chapter discusses more specialized examples on how machine learning can be used to solve problems in quantum sciences. We start by explaining the concept of differentiable programming and its use cases in quantum sciences. Next, we describe deep generative models, which have proven to be an extremely appealing tool for sampling from unknown target distributions in domains ranging from high-energy physics to quantum chemistry. Finally, we describe selected machine learning applications for experimental setups such as ultracold systems or quantum dots. In particular, we show how machine learning can help in tedious and repetitive experimental tasks in quantum devices or in validating quantum simulators with Hamiltonian learning.

In this chapter, we describe basic machine learning concepts connected to optimization and generalization. Moreover, we present a probabilistic view on machine learning that enables us to deal with uncertainty in the predictions we make. Finally, we discuss various basic machine learning models such as support vector machines, neural networks, autoencoders, and autoregressive neural networks. Together, these topics form the machine learning preliminaries needed for understanding the contents of the rest of the book.

In this chapter, we review the growing field of research aiming to represent quantum states with machine learning models, known as neural quantum states. We introduce the key ideas and methods and review results about the capacity of such representations. We discuss in details many applications of neural quantum states, including but not limited to finding the ground state of a quantum system, solving its time evolution equation, quantum tomography, open quantum system dynamics and steady-state solution, and quantum chemistry. Finally, we discuss the challenges to be solved to fully unleash the potential of neural quantum states.

In this chapter, we introduce the reader to basic concepts in machine learning. We start by defining the artificial intelligence, machine learning, and deep learning. We give a historical viewpoint on the field, also from the perspective of statistical physics. Then, we give a very basic introduction to different tasks that are amenable for machine learning such as regression or classification and explain various types of learning. We end the chapter by explaining how to read the book and how chapters depend on each other.

This chapter gives an overview of the different topics covered in the book. As an outlook, we also share our view on potential exciting directions.