Students are expected to have a basic background in probability theory and linear algebra. Therefore, having passed the 1st year courses 'Statistics' and 'Lineal Algebra' is highly recommended.

- Understand the fundamental differences between classical and quantum probability theories. - Describe mathematically a quantum state of a single qubit and that of several qubits. - Know and use the axioms that govern the evolution of a quantum system. - Know and use the axioms that govern the measurement of a quantum state. - Model and analyze simple quantum communication channels and their cryptographic guarantees. - Interpret and implement a quantum computing algorithm.

This course introduces the fundamental concepts of quantum communication and computing. Starting from an experimental basis, we will motivate why the classical theory of probability is not able to model certain real physical systems. We will present a generalization of the concept of probability that allows us to model these experiments, as well as their (unexpected) consequences. The new quantum theory of probability will then be used to analyze several simple problems, including the transmission of information, the distribution of quantum entanglement and the teleportation protocol. Finally, the current state of the technology and its future perspectives will be discussed. · Contents Unit 1. Introduction: bits versus qubits 1.1. What is a qubit? 1.2. Probability theory 1.3. Bell's theorem Unit 2. Axioms of quantum mechanics 2.1. Principles of quantum mechanics 2.2. Combined systems: quantum entanglement 2.3. Temporal evolution of a quantum system 2.4. Simulating quantum systems Unit 3. Quantum communications 3.1. Classical versus quantum information 3.2. Transmission of information over quantum channels 3.3. Teleportation and other communication protocols 3.4. Secure links and quantum cryptography Unit 4. Quantum computing 4.1. Quantum computers and quantum gates 4.2. Quantum circuits and algorithms 4.3. Programming a quantum computer 4.4. Present and future of quantum computing

- 9 theoretical sessions presenting the generalization of the classical probability theory, basic concepts of quantum mechanics and illustrative examples. - 2 practical sessions to simulate simple quantum systems and quantum experiments. - 2 practical sessions to develop algorithms in real quantum computers. - 1 practical session to present and simulate a secure communications link. · Teaching material The material used in the course sessions will be uploaded to the platform 'Aula Global' in electronic format. Before each session, the students will have available all the information and recommended reading for the best understanding of the session. Exercises will also be given, some of which will be solved in practical sessions.