The first reference for Topological quantum computation is Kitaev's Fault Tolerant Quantum Computation with Anyons. This is also where the Toric code model is first laid out. Around the same time John Preskill wrote a chapter on Fault-Tolerant Quantum Computation which is more general but also includes a section based upon Kitaev's ideas.

An excellent review article for TQC is Nayak et. al's Non-Abelian anyons and topological quantum computation. I found that this really covers the lay of the land from a broad perspective ranging from the mathematical underpinnings to the possible experimental realizations. It certainly covers the challenges which exist in constructing a topological quantum computer. There is also FKLW, The Lecture notes of Jiannis Pachos, as well as those of John Preskill for a course on Quantum computing that includes notes on error correcting codes and TQC.

Book wise, there are two that I know of. Zhenghan Wang's Topological Quantum Computation is geared primarily towards mathematicians or mathematically minded physicists, but does an excellent job of providing an over view for the mathematics underlying TQC. I use this book a lot because it contains a treasure trove of information that's relatively well cited for getting further into other details. It is by no means complete, but it definitely serves as a map for the literature that was available at the time. The last chapter is also lists of open problems at the time the book was written, many if not most of which remain open.

The second is Pachos' Introduction to Topological Quantum Computation. I found this very readable but when I was going through it I'd already been through Wang's book as well as Nayak and several of the other references above. In hindsight I felt that it would have been a great introductory text to the subject.

This post imported from StackExchange Physics at 2014-06-11 15:02 (UCT), posted by SE-user Matthew Titsworth