This chapter studies linear circuits built from capacitors, inductors, and waveguides. It shows how the excitations of these circuits are quantized and can be described as collections of quantum harmonic oscillators. It discusses the quantum states and quantum operations that are accessible by means of these circuits and external microwave drives. We show how to create coherent states, how microwave resonators decay and decohere, how to amplify and measure the quantum state of a resonator, and what states (e.g., Fock states, individual photons) require other, non-Gaussian means to be produced and detected.

This appendix provide a self-contained presentation of the open systems and quantum optics methods used in other parts of the book (e.g., studying the relaxation of a microwave cavity or a qubit, the driving of a quantum amplifier, etc.). Half of the appendix is devoted to the derivation of master equations for small systems that are in contact with Markovian environments. The other half of the appendix is devoted to the development of an alternative input–output description of how those systems absorb information from the environment and reflect it back.

In this chapter, we discuss the notion of a quantum simulator as a device that emulates a complex quantum many-body Hamiltonian, and a quantum annealer as an extension of such a paradigm that focuses on the preparation of the ground state in those Hamiltonians. Starting from the Landau–Zener processes and the adiabatic theorem, we illustrate how such ground states can be prepared by slow (adiabatic) deformations of a Hamiltonian. We discuss how this results in an adiabatic quantum computing algorithm and how, depending on the Hamiltonian we apply it to, we can solve problems of different classical and quantum complexity. The chapter closes with a thorough discussion of the D-Wave quantum annealer as a real-world superconducting quantum simulator of Ising-type models. This discussion centers both on the design of the superconducting annealers as well as on the conclusion from the literature on how this device works in practice, including how quantum operations are still possible even in the pressence of decoherence. We close the chapter with an outlook on the challenges that need to be overcome for making coherent quantum annealers and universal adiabatic quantum computers.

We discuss the building blocks of a universal quantum computer within the circuit model of computation and how this is implemented using superconducting quantum circuits. In particular, we discuss, one by one, the creation of quantum registers, resetting of quantum bits, qubit measurements, single-qubit operations, and universal two-qubit gates, and how these are all implemented using the tools from earlier chapters. We discuss how to calibrate the errors in the qubits and in the operations, assigning them complete descriptions via positive maps. We explain how these errors can be corrected and how to implement a fault-tolerant quantum computer, focusing on the paradigm of stabilizer codes and the surface code in particular. We close with a discussion on the outlook for quantum computers in the near term and the NISQ paradigm of computation.