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 ?
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
- Chapter 10 of Michael A. Nielsen, Isaac L. Chuang, Quantum Computation and Quantum Information. This is a standard textbook in quantum information and computation, and that chapter summarizes very neatly the key concepts on which quantum error correction is based and gives an introduction to the threshold theorem, a central result in the theory of fault-tolerance quantum computing.
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:
Daniel Gottesman, An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation, 2009. A very complete survey. I recommend it, in particular, for its fault tolerance part. It is open access.
Barbara Terhal, Quantum Error Correction for Quantum Memories, 2013. The most recent review available, and a very good one; I recommend it if you want to know about the latest results in the field. It is oriented more towards error correction (quantum memories) than towards fault tolerance (quantum computers). It is also open access.
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.