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.

Distinguishing between different phases of matter and detecting phase transitions are some of the most central tasks in many-body physics. Traditionally, these tasks are accomplished by searching for a small set of low-dimensional quantities capturing the macroscopic properties of each phase of the system, so-called order parameters. Because of the large state space underlying many-body systems, success generally requires a great deal of human intuition and understanding. In particular, it can be challenging to define an appropriate order parameter if the symmetry breaking pattern is unknown or the phase is of topological nature and thus exhibits nonlocal order. In this chapter, we explore the use of machine learning to automate the task of classifying phases of matter and detecting phase transitions. We discuss the application of various machine learning techniques, ranging from clustering to supervised learning and anomaly detection, to different physical systems, including the prototypical Ising model that features a symmetry-breaking phase transition and the Ising gauge theory which hosts a topological phase of matter.

This chapter delves into the theory and application of reversible Markov Chain Monte Carlo (MCMC) algorithms, focusing on their role in Bayesian inference. It begins with the Metropolis–Hastings algorithm and explores variations such as component-wise updates, and the Metropolis-Adjusted Langevin Algorithm (MALA). The chapter also discusses Hamiltonian Monte Carlo (HMC) and the importance of scaling MCMC methods for high-dimensional models or large datasets. Key challenges in applying reversible MCMC to large-scale problems are addressed, with a focus on computational efficiency and algorithmic adjustments to improve scalability.

This chapter provides a comprehensive overview of the foundational concepts essential for scalable Bayesian learning and Monte Carlo methods. It introduces Monte Carlo integration and its relevance to Bayesian statistics, focusing on techniques such as importance sampling and control variates. The chapter outlines key applications, including logistic regression, Bayesian matrix factorization, and Bayesian neural networks, which serve as illustrative examples throughout the book. It also offers a primer on Markov chains and stochastic differential equations, which are critical for understanding the advanced methods discussed in later chapters. Additionally, the chapter introduces kernel methods in preparation for their application in scalable Markov Chain Monte Carlo (MCMC) diagnostics.