LEARNING OUTCOMES

This course will provide the key conceptual tools needed to understand the most recent developments in the field of quantum computation and quantum information. Great attention will be devoted to explain quantum cryptography protocols, quantum algorithms (Deutsch, Grover, Shor) and to discuss the main physical implementation of qubits (trapped ions, superconducting qubits, quantum dots).

AIMS AND LEARNING OUTCOMES

Secure quantum key distributions for the data transfer among banks and the recent production of scientific papers sporting results obtained with quantum computer accessible in the Cloud (IBM, Rigetti) are only two of the various examples of how the relevance of quantum technologies is progressively growing in our everyday life. Starting from a critical revision of the basic concepts of quantum mechanics such as two level system (paradigm of a qubit, the fundamental building block of quantum logic) and harmonic oscillators, as well as their interaction, the students will acquire the fundamentals to understand and handle concepts like quantum superposition, entanglement and quantum correlations. These ideas are at the core of the development of quantum cryptography and quantum algorithms. The advantages and limitations of state-of-the-art technologies for the concrete development of finely-controllable two level systems (trapped ions, superconducting qubits, quantum dots) as well as possible future game-changers will be also discussed in details.

PREREQUISITES

None

TEACHING METHODS

Blackboard lectures.

SYLLABUS/CONTENT

0. Introduction to the course  

   0.1 What are quantum technologies? 

0.2 Quantum information and quantum communication 

1. Few words about classical logic  

   1.1 Abstract representation of bits 

   1.2 Classical logical operations 

   1.3 Single-bit reversible operations 

   1.4 Shannon entropy 

   1.5 von Neumann entropy 

   1.6 Two-bit reversible operations 

2. What is a quantum bit? 

   2.1 The polarization of light

   2.2 Photon polarization

   2.3 Two level system: a paradigm for a qubit

   2.4 Basic prerequisites: Pauli matrices, time evolution of discrete level systems

3. Manipulation of qubits  

   3.1 Dynamical evolution

   3.2 Rabi oscillations

   3.3 Quantum cryptography

   3.4 General solution of a two level system 

4. Quantum correlations   

   4.1 Two-qubit states 

   4.2 Entanglement of two-qubit states 

   4.2 Density operator: pure and mixed states 

   4.3 The Bell inequalities 

5.  Quantum algorithms   

   5.1 Quantum logic gates 

   5.2 Deutsch algorithm 

   5.3 Grover search algorithm 

   5.4 Quantum error correction protocols 

   5.5 Few words on the Shor algorithm  

   (Hands-on demo of the IBM Quantum Computer)

6. Physical realizations  

   6.1 Trapped ions 

   6.2 Josephson junction in the Feynman description 

   6.3 Quantum description of LC and superconducting circuits 

   6.4 Charge qubit 

   6.5 Simple model of decoherence 

7 Quantum harmonic oscillator reloaded 

   7.1 Number states 

   7.2 Coherent states

   7.3 Squeezed states 

   7.4 Wigner function description of quantum states of light 

   7.5 Physical realizations

RECOMMENDED READING/BIBLIOGRAPHY

M. Le Bellac “A short Introduction to Quantum Information and Quantum Computation”. Cambridge University Press (2006).