After an introduction to classical computation, we will present, in a systematic fashion, the main concepts of quantum computation and communication. The physical consequences of quantum information theory will be illustrated by means of simple examples. The course will end with a more advance topic: quantum noise.

Course program:

1) Introduction to classical computation:
the Turing machine, the Church–Turing thesis, the universal Turing machine, the probabilistic Turing machine, the halting problem, the circuit model of computation, binary arithmetics, elementary logic gates, universal classical computation, computational complexity,
complexity classes, computing dynamical systems, deterministic chaos and algorithmic complexity,
energy and information, Maxwell’s demon, Landauer’s principle, extracting work from information, reversible computation, energy dissipation in computation.

2) Some basic tools of quantum mechanics:
the Schmidt decomposition, purification, generalized measurements, POVM measurements.

3) Quantum computation:
the qubit, the Bloch ball, the circuit model of quantum computation, single-qubit gates, controlled gates and entanglement generation, Hamiltonian model of one- and two-qubit gates, universal quantum gates, function evaluation, the quantum adder, adiabatic quantum computation, maximum speed of quantum gates, holonomic quantum computation.

4) Quantum algorithms:
Deutsch’s algorithm, the Deutsch–Jozsa problem, quantum search, searching by adiabatic quantum evolution, the quantum Fourier transform, quantum phase estimation, period finding and Shor’s algorithm, quantum computation of dynamical systems,
quantum simulation of the Schrödinger equation, quantum algorithms for quantum maps, information extraction for dynamical quantum systems, universal quantum simulation.

5) Quantum communication:
classical cryptography, the Vernam cypher, the public-key cryptosystem, the RSA protocol, the no-cloning theorem, quantum cryptography, the BB84 protocol, the E91 protocol, dense coding, quantum teleportation, quantum cryptography with continuous variables.

6) Quantum noise:
the Kraus representation, decoherence models for a single qubit, quantum circuits simulating noise channels, de-entanglement, the Bloch-Fano representation, the master equation, the master equation and quantum operations, non-Markovian quantum dynamics, quantum to classical transition and Schrödinger’s cat, decoherence and quantum measurements, weak measurements, decoherence and quantum trajectories.