This chapter aims to illustrate how quantum theory provides useful technological solutions – applications that may be more integrated in our everyday lives than we tend to think. Some applications lend themselves to a particularly straightforward outline through examples already seen in the preceding chapters. These include scanning tunnelling microscopy and emission spectroscopy, which utilize tunnelling and energy quantization, respectively. Prior knowledge and readymade implementations allow these applications to be studied in a quantitative manner. Also, nuclear magnetic resonance is, albeit in a somewhat simplified model, studied quantitatively – within the framework of an oscillating spin-½ particle developed in Chapter 5. The remainder of the chapter is dedicated to quantum information technology. Also in this context, the notion of one or two spin-½ particles is applied frequently. A spin-½ particle is one possible realization of a quantum bit, and it serves well as a model even in cases when quantum bits are implemented differently. After having introduced some basic notions, two specific protocols for quantum communication are studied in some detail. The last part of the chapter addresses adiabatic quantum computing. This technology is studied in a manner that lies close to the last example of Chapter 5.

This chapter introduces the basic mathematical formalism for working with quantum information. We discover qubits, or quantum bits, how to combine them using the tensor product, and how to measure them by choosing a basis. We discuss unitary operations, which are elementary transformations on qubits. The chapter ends with a convenient representation of qubits as vectors on the 3-dimensional Bloch sphere, and a useful “cheat sheet,” which summarizes useful definitions and identities.

In this chapter, we develop the conditions to observe quantum fluctuations and quantum phenomena (entanglement, superpositions, etc.) in quantum circuits. Assuming the right conditions of temperature, we develop the quantum mechanical theory that models those fluctuations in a circuit built from nondissipative superconducting elements. We use this theory of circuit quantization to obtain the quantum Hamiltonians for microwave resonators and waveguides, for superconducting qubits of various types, and for other elements such as SQUIDs. The chapter closes with an illustration of how the same theory provides us with numerical methods to study the eigenstates, eigenenergies, and dynamics of said Hamiltonians.

In this chapter, we review basic concepts from quantum mechanics that will be required for the study of superconducting quantum circuits. We review the fundamental idea of energy quantization and how this can be formalized, using Dirac's ideas, to develop a quantum mechanical description that is consistent with the classical theory for a comparable object. We review the notions of quantum state, observable and projective and generalized measurements, particularizing some of these ideas to the simple case of a two-dimensional object or qubit.