The superconducting circuit companion -- an introduction with worked examples

This article is a tutorial on the quantum treatment of superconducting electrical circuits. It is intended for new researchers with limited or no experience with the field, but should be accessible to anyone with a bachelors degree in physics or similar. The tutorial has three parts. The first part introduces the basic methods used in quantum circuit analysis, starting from a circuit diagram and ending with a quantized Hamiltonian truncated to the lowest levels. The second part introduces more advanced methods supplementing the methods presented in the first part. The third part is a collection of worked examples of superconducting circuits. Besides the examples in the third part, the two first parts also includes examples in parallel with the introduction of the methods.