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What are the various approaches to fault tolerant quantum computation ? Two examples are 1. topological quantum computation which uses topological phases in quantum states (2-Dimensional for non-abelian statistics of anyons) and 2. spin-half BEC form (refer http://jqi.umd.edu/news/new-state-fifth-state (Dr. Victor Galitski's research at UMD) ) . Can you guide me to the references mentioning various approaches and the challenges associated with them ?

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If you downvote, please comment why and how can I improve the question. –  cleanplay Dec 1 '13 at 22:54
I'm not the downvoter, but maybe the question is a bit "open ended". I personally think what you're getting at is a good question - it seems as though you're trying to get abreast of knowledge in a particular field, so maybe try rewriting it as a reference request –  WetSavannaAnimal aka Rod Vance Dec 1 '13 at 23:25
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2 Answers

There are many good introductory textbooks and surveys on quantum error correction and fault tolerance out there. If you have never taken a course on quantum information I would suggest

Nielsen and Chuang is a very complete textbook but it will not introduce you to the most recent protocols and discoveries. Also, its section about fault tolerant quantum computing is rather brief. If the book does not suffice for your purposes then I would recommend you to take a look at:

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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.

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