The ABCs of TQC: An introduction to the mathematics of Topological Quantum Computing

Computation is a mechanical process. Computers process information by manipulating physical systems encoding bits, and quantum computers manipulate encodings in quantum mechanical systems. This process is extremely delicate and error-prone, so we must develop fault-tolerant computation protocols to make quantum computers useful. Quantum error-correcting codes provide a means of developing fault-tolerance at the software level. This course will explore Topological Quantum Computing (TQC) as a means of achieving fault-tolerance at the hardware level instead, by encoding information in topological phases of matter that are intrinsically protected from local deformations and interactions. TQC promises scalable quantum computing and it lies at the crossroads of cutting-edge research in Physics, Engineering, and Mathematics. This course will introduce the mathematical machinery modeling TQC. The main players are Anyons, Braids, and Categories: braiding anyons, which are certain quasiparticles existing only in two-dimensional systems, results in unitary state transformations implementing logical gates on encoded qubits. The mathematical theory of anyons, which are neither bosons nor fermions, as simple objects in unitary modular tensor categories is quite interesting, and this course will develop it from the ground up.