Introduction to quantum circuit electrodynamics
Course description
Qubits are the basic units of systems for quantum information (quantum computer, quantum internet, ...). They are two  state quantum mechanical devices, the quantum version of the classical two  state devices. Classical two  state devices would have to be in one state or the other, instead qubits can be in a coherent superposition of both states, a fundamental property of quantum mechanics.
Examples of qubits include: the spin of the electron in which the two states can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. The most promising technology for the fabrication of qubits uses at present superconducting electrical circuits based on Josephson junctions (IBM, DWave Systems, Rigetti, Google, Quantum Circuits – Yale,…).
If decoherence due to uncontrolled degrees of freedom is sufficiently reduced electrical circuits can behave quantum mechanically. In a superconducting material, all super electrons can be in the same quantum coherent state. As a consequence, superconducting devices can be engineered in a way to behave as macroscopic artificial atoms. The research field of quantum state engineering with superconducting electrical circuits was born from fundamental questionings about the possibility of observing macroscopic quantum phenomena. Experiments have widely demonstrated that the quantum state of superconducting electrical circuits based on the Josephson junction can be effectively manipulated, controlled and read–out. Compared to real atoms, superconducting electrical circuits are macroscopic in size, leading to large electrical or magnetic dipoles, which facilitates their coupling to other circuits and systems. In particular, superconducting qubits can be strongly coupled to superconducting resonators, which offer architectures for quantum information processing. In fact, this enables single  qubit control and read–out, multi  qubit entanglement and gates. In addition, it is possible to couple superconducting circuits and resonators to other quantum systems such as spins or mechanical resonators, forming so called Hybrid Quantum Devices.
In this course we start from the Lagrangian and Hamiltonian formulations of classical electrical circuits, give the concept of quantum electrical circuit and, then, superconducting qubits are progressively introduced. We point out the links between electrical quantum gates, entanglement and quantum measurement, and illustrate these ideas using some examples.
Syllabus
 What are Quantum Circuits, Qubits and Quantum logic gates. The Bloch sphere.
 Classical Electrical Circuits: Lumped element circuits, distributed element circuits, Kirchhoff circuit laws, constitutive relations. Dissipative and nondissipative circuits. Lagrangian and Hamiltonian formulations for non dissipative circuits, conjugate electrical variable pairs, Poisson brackets. Circuits with dissipative elements, the Caldeira – Legget model. Fluctuation – dissipation theorem. Langevin equation.
 Non Dissipative Quantum Electrical Circuits: Basic notions of Quantum Mechanics. State of a non dissipative quantum electrical circuit: measurements, from electrical variables to operators, quantum state vector, commutators of charges and fluxes. Quantization of an electrical circuit. Schrödinger and Heisenberg Pictures. Quantum linear LC circuits. Entanglement. “Black Box” Quantization of Linear Circuits.
 Superconducting Qubits: Josephson junction. Nonlinear LC circuits. Charge  qubit circuits. Flux  qubit circuits. Phase  qubit circuits. Cooper  pair box. Transmon. Entangled qubits.
 Dissipative Quantum Electrical Circuits: Quantum dissipationfluctuation theorem. Quantum fluctuations in the damped LC oscillator. Nyquist model of resistance: semiinfinite transmission line. Heisenberg  Langevin equation. Environment and measurement operators. Noise and the environment. Decoherence, decay and dephasing. Noise induced Decoherence in Qubit Circuits. A glimpse to stocastic master equations.
 Qubit  cavity coupling: Resonant coupling and dispersive coupling. Amplification and feedback. Dispersive Readout of a Qubit in a Cavity. Quantum Control of Qubits in a Cavity. Multiqubit Dispersive Readout.
 Quantum state engineering and quantum gates: One qubit gates. Two qubit gates. The Grover search algorithm, error correction (ancilla qubit), steps toward quantum computers.
 A glimpse to Hybrid Quantum Devices: Spin superconducting circuits. Quantum dots. Hybrid quantum processor.
Prerequisites
Prior knowledge of quantum mechanics and solid state physics are helpful, but not required.
Tutorials
Textbooks
 U. Vool, M. Devoret, Introduction to quantum electromagnetic circuits, Special Issue on Quantum Technologies, International Journal of Circuit, Theory and Applications, 897934, 2017.

P.P. Civalleri, Finitedimensional open quantum systems, Lecture Notes, Politecnico di Torino, 2017.
 Zagoskin A. M. Quantum Engineering: Theory and Design of Quantum Coherent Structures, Cambridge University Press, 2011.

G. Mahler, V. A. Weberruß, Quantum Networks, Springer, 1998. I. Mayergoyz, Quantum Mechanics for Electrical Engineers, World Scientific, 2016.

P. Krantz , M. Kjaergaard , F. Yan, T. P. Orlando, S. Gustavsson, W. D. Oliver, A quantum engineer's guide to superconducting qubits , Appl. Phys. Rev. 6, 021318 (2019).