This chapter focuses on three key problems that underlie the formulation of many machine learning methods for inference and learning, namely variational inference (VI), amortized VI, and variational expectation maximization (VEM). We have already encountered these problems in simplified forms in previous chapters, and they will be essential in developing the more advanced techniques to be covered in the rest of the book. Notably, VI and amortized VI underpin optimal Bayesian inference, which was used, e.g., in Chapter 6 to design optimal predictors for generative models; and VEM generalizes the EM algorithm that was introduced in Chapter 7 for training directed generative latent-variable models.

The previous chapters have adopted a limited range of probabilistic models, namely Bernoulli and categorical distributions for discrete rvs and Gaussian distributions for continuous rvs. While these are common modeling choices, they clearly do not represent many important situations of interest for machine learning applications. For instance, discrete data may a priori take arbitrarily large values, making categorical models unsuitable. Continuous data may need to satisfy certain constraints, such as non-negativity, rendering Gaussian models far from ideal.

So far, this book has focused on conventional centralized learning settings in which data are collected at a central server, which carries out training. When data originate at distributed agents, such as personal devices, organizations, or factories run by different companies, this approach has two clear drawbacks: • First, it requires transferring data from the agents to the server, which may incur a prohibitive communication load. • Second, in the process of transferring, storing, and processing the agents’ data, sensitive information may be exposed or exploited.

This appendix collects a series of analytical methods that are needed in various parts of the book. All the tools are oriented toward the diagonalization of Hamiltonian, including some cases that allow a complete analytical diagonalization (e.g., coupled resonators) as well as perturbation theory methods for cases where the fully interacting model is too complex for an exact solution.

We introduce the notion of qubit as unit of quantum information, illustrating how this notion can be implemented in nonlinear superconducting circuits via the charge and current degrees of freedom. Within these two types of qubits, we discuss the charge qubit, the transmon, and the flux qubit, illustrating the nature of the states that implement the qubit subspace and how they can be controlled and measured. We discuss how qubits can interact with each other directly or through mediators, illustrating different limits of interaction, introducing the notion of dipolar electric and magnetic moments, and demonstrating the tunability of interactions by different means. The chapter closes with a brief study of qubit coherence along the history of this field, with an outlook to potential near-term improvements.

Almost all superconducting quantum technologies are built using a combination of qubits and microwave resonators. In this chapter, we develop the theory to study coherent qubit–photon interaction in such devices. We start with the equivalent of an atom in free space, studying a qubit in an open waveguide. We develop the spin-boson Hamiltonian, with specific methods to solve its dynamics in the limits of few excitations. Using these tools, we can study how an excited qubit can relax to the ground state, producing a photon, and how a propagating photon can interact with a qubit. We then move to closed environments where the photons are confined in cavities or resonators, developing the theory of cavity-QED. Using this theory, we study the Purcell enhancement of interactions, the Jaynes–Cummings model, Rabi oscillations, and vacuum Rabi splitting. We close the chapter illustrating some limits in which cavities can be used to control and measure qubits.