Topological quantum computation. Lecture 1

This will be an introductory course on topological quantum computation. Combining physics and mathematics, topological quantum computation is a rapidly expanding research area focused on the physical realization of a quantum computer that is immune to errors. The makings of topological systems, their properties and their computational power will be presented in a pedagogical way. Special emphasis is given to the motivation and physical intuition behind mathematical concepts.
Physics should remain unchanged if we exchange two identical particles. This is a fundamental symmetry with far reaching consequences. In three dimensions it dictates the existence of bosons and fermions. In two dimensions a variety of statistical behaviors is possible: arbitrary phase factors or even non-trivial unitary evolutions are allowed when two particles are exchanged. Particles with such an exotic statistics are called anyons. Employing topological systems that support anyons provides a surprisingly effective and aesthetically appealing method for performing quantum computation.