Topological quantum computation

The third, last Loeb lecture by John Preskill was about topological quantum computation. It is an elegant, alternative approach to error correction which is more natural than the brute force error correction explained in the second talk. In the case of quantum computation, the method is inherently robust because the basic operations are local in space.



I still have not said what operations are involved in topological quantum computation. You must have a system in which you are able to pair-create and destroy particles living in a plane with generalized statistics - namely anyons. An anyon is a generalization of the concept of bosons and fermions in which you pick a general phase if two particles rotate around each other. This is only possible in 2+1 dimensions because a loop around a point does not exist in less than 2+1 dimensions, and it is contractible in more than 2+1 dimensions.

Because of this property, the mathematical concept of braiding is relevant. One has to learn things like the Moore-Seiberg constraints, the hexagon identity and the Rumsfeld identity. The latter two completely determine to what extent the elementary R,F moves are not independent from each other. The polynomial conditions written in terms of R,F visually resemble the anomaly polynomials, especially because the same letters R,F are used in both cases. ;-)

The ultimate purpose of having systems with anyons, especially non-Abelian anyons where the holonomy is represented by an element of U(N) rather than a simple phase, is to construct a robust quantum computer. One can show that a topological, anyon-based quantum computer can be well emulated by a qubit quantum computer. The converse statement also holds as long as you pick a sufficiently non-trivial set of anyons.

Because I missed the beginning of the talk, I can't tell you what the Fibonacci model really is, except that the golden ratio is important here, much like in Fibonacci trading. But what I can tell you is that you can try to construct a system of anyons based on the fusion rules of the SU(2) Chern-Simons theory at level "k". The fusion rules reproduce the rules for the tensor products of an SU(2) group truncated down to spins "j" smaller or equal to "k/2". It turns out that the level "k=2" gives you too simple moves that are not enough to emulate a qubit computer. However, the level "k=3" Chern-Simons theory is good enough.

Eventually one wants to realize this alternative architecture for quantum computing. One needs to find a physical system (a generalized "material") whose ground state looks like a condensate of string networks so that the elementary excitations look like the required anyons. Steps towards this goal have been done by Kitaev, Demler, and others.

If you're interested in topological quantum computation, I recommend you this PDF file.