A quantum computer is a (so far) hypothetical device that exploits the strange properties of quantum mechanics to perform qualitatively new kinds of computations. Thanks to remarkable experimental progress in realizing high quality quantum bits (qubits), as well as theoretical advances in our understanding of how to protect quantum information from errors and loss of coherence, the building of such a device no longer seems as far off in the future as once thought (i.e. the sun may not have expanded into a red giant before a fully functioning quantum computer is built!). One of the most promising ideas for carrying out fault-tolerant quantum computation using a quantum computer based on today's state-of-the-art qubits (for example, superconducting qubits) is to use what are known as "surface codes." Quantum computing with surface codes can be thought of, roughly, as using the computer to simulate a theory of anyons --- particle like excitations which live in 2 space dimensions whose world-lines form braids in 2+1 dimensional space-time, with different braids corresponding to different computations. In this talk I will provide the necessary background and describe the essential ideas behind this approach, focusing on one of the more speculative ideas based on simulating so-called "Fibonacci" anyons.
