Introduzione ai Circuiti Quantistici (Introduction to Circuit Quantum Electrodynamics)

CFU: 9

Prerequisites

None.

Preliminary Courses

None.

Learning Goals

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, D-Wave 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 two – level 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 readout.

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.

This course starts from the Lagrangian and Hamiltonian formulations of classical electrical circuits, give the concept of quantum electrical circuit and, then, superconducting qubits are progressively introduced. The links between electrical quantum gates, entanglement and quantum measurement are pointed out. The fundamental circuits for quantum computing are introduced.

Expected Learning Outcomes

Knowledge and understanding

This course, starting from the model of lumped and distributed circuits governed by quantum mechanics, introduces the new paradigms at the basis of quantum computers based on Josephson junction qubits.

Applying knowledge and understanding

Course Content - 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 non-dissipative 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. Non-linear LC circuits. Charge - qubit circuits. Flux - qubit circuits. Phase - qubit circuits. Cooper - pair box. Transmon. Entangled qubits.
Dissipative Quantum Electrical Circuits: Quantum dissipation-fluctuation theorem. Quantum fluctuations in the damped LC oscillator. Nyquist model of resistance: semi-infinite 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 Read-out of a Qubit in a Cavity. Quantum Control of Qubits in a Cavity. Multi-qubit 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.

Readings/Bibliography

- A. M. Zagoskin Quantum Engineering: Theory and Design of Quantum Coherent Structures, Cambridge University Press, 2011.

- U. Vool, M. Devoret, Introduction to quantum electromagnetic circuits, Special Issue on Quantum Technologies, International Journal of Circuit, Theory and Applications, 897-934, 2017.

- G. Wendin, Quantum information processing with superconducting circuits: a review, Rep. Prog. Phys. 80 106001, 2017.

- X. Gua, A. F. Kockum, A. Miranowicz, Y. Liu, F. Nori, Microwave photonics with superconducting quantum circuits, Physics Reports 718–719, 1–102, 2017.

- P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, A quantum engineer's guide to superconducting qubits, Appl. Phys. Rev. 6, 021318 (2019); https://doi.org/10.1063/1.5089550

- A. Blais, A. L. Grimsmo, S. M. Girvin, A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021).

- Lecture Notes.

Teaching Method

Lectures for approximate 90% of total hours and 10 % laboratories.

Examination/Evaluation criteria

Exam type

Only Oral.