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 returns to the zero-field limit of MHD replacing the isotropic pressure force density in ideal HD with force densities arising from the viscous stress tensor for viscid HD. As tensor analysis is not a prerequisite for this course, the stress tensor is developed purely from a vector analysis of all stresses applied at a single point in a viscid fluid. This leads to the introduction of bulk and kinetic viscosity in a Newtonian fluid and the identification of ordinary thermal pressure with the trace of the stress tensor. Various flavours of the Navier–Stokes equation are developed including compressible and incompressible forms. The Reynold’s number is introduced as a result of scaling the Navier–Stokes equation which leads to a qualitative discussion on turbulent and laminar flow. Numerous examples are given in which a simplified form of the Navier–Stokes equation can be solved analytically, including plane-parallel flow, open channel flow, Hagen–Poiseuille flow, and Couette flow.

This chapter begins with a formal definition of a fluid (what it means to be a continuum rather than an ensemble of particles) followed by a review of kinetic theory of gases where the connections between pressure and particle momentum and between specific energy (temperature) and average particle kinetic energy are made. A distinction is made between extensive and intensive variables, from which the Theorem of Hydrodynamics is postulated and proven. From this theorem, the basic equations of ideal hydrodynamics (zero-field limit of MHD) are derived including continuity, total energy equation, and the momentum equation. Alternate equations of HD such as the internal energy, pressure, and Euler’s equations are also introduced. The equations of HD are then assembled into two sets–conservative and primitive–with the distinction between the two explained.

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.

The content of this chapter may serve as, yet, another supplemental topic to meet the needs and interests beyond those of a usual course curriculum. Here we shall present an oversimplified, but hopefully totally transparent, description of some of the fundamental ideas and concepts of quantum mechanics, using a pure linear algebra formalism.

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 a steady-state, axisymmetric atmosphere surrounding a gravitating point mass, three constants of flow along lines of induction (equivalently, streamlines) are identified, collectively referred to as the Weber–Davis constants. The MHD Bernoulli function, the fourth constant along a line of induction, is derived from examining Euler’s equation in a rotating reference frame, and a link is made between the centrifugal terms and the magnetic terms found in an inertial reference frame. From the four constants, two types of magneto-rotational forces arise which, acting in tandem, can accelerate material from an accretion disc to escape velocities provided the line of induction emerges from the disc at an angle less than 60°. Two astrophysical examples are then described. The first is a quantitative account of Weber and Davis’ model for a stellar wind, including the derivation of specific fluid profiles along a poloidal line of induction. The second looks at how the four constants can arise naturally in an axisymmetric, non-steady-state simulation of an astrophysical jet.

In this chapter, we consider vector spaces over a field that is either the real or complex numbers. We shall start from the most general situation of scalar products. We then consider the situations when scalar products are nondegenerate and positive definite, respectively.

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 present an introduction to an important area of contemporary quantum physics: quantum information and quantum entanglement. After a brief introduction regarding why and how linear algebra is so useful in this area, we first consider the concepts of quantum bits and quantum gates in quantum information theory. We next explore some geometric features of quantum bits and quantum gates. Then we study the phenomenon of quantum entanglement. In particular, we shall clarify the notions of untangled and entangled quantum states and establish a necessary and sufficient condition to characterize or divide these two different categories of quantum states. Finally, we present Bell’s theorem which is of central importance for the mathematical foundation of quantum mechanics implicating that quantum mechanics is nonlocal.

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.