Efficient Approaches to Universal and Non-Universal Topological Quantum Computation

In topological quantum computation, information is encoded non-locally in exotic quasiparticles called anyons, and quantum gates are carried out by braiding the anyons in (2+1)-dimensional space-time. Universal topological quantum computation can be carried out by a universal set of quantum gates composed of single-qubit gates and controlled-phase gate and more efficiently together with controlled-controlled-phase gate. In this poster we show a unified scheme of encoding and computing for both universal and non-universal topological quantum computation with SU(2)$_k$ anyon models. We further give explicit construction of quantum gates, especially the controlled-controlled-phase gate, for universal topological quantum computation. Consequences of these ideas for the detection and correction of errors in topological quantum computation will be explored.