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"The two-eigenvalue problem and density of Jones representation of braid groups", Michael Freedman, Michael Larsen, Zhenghan Wang, 2002

Reviewed July 25, 2023

Citation: Freedman, Michael H., Michael J. Larsen, and Zhenghan Wang. "The Two-Eigenvalue Problem and Density of Jones Representation of Braid Groups." Commun. Math. Phys 228 (2002): 177-199.

Web: https://arxiv.org/abs/math/0103200

Tags: Foundational, TQFT, Universal-scheme, Quantum-groups

This paper has Freedman, Larsen, Wang doing what they do best: studying braid group representations. The just finished announcing topological quantum computation the year earlier in

>Freedman, Michael, et al. "Topological quantum computation." Bulletin of the American Mathematical Society 40.1 (2003): 31-38.

and now all that's left to do mathematically is find more universal models. Freedman sees this work as being really foundational, from the mathematical perspective. In his Simmons institute interview he says: "In the early 2000s, Zhenghan Wang and I with Michael Larsen wrote a paper where we proved that a large class of anyonic systems have dense braid group representations. So, at some level, you could say that's the end of the mathematical work now its the physicists and the engineers who have to build the computer. Because this density result, what it says is that if you have the correct kind of quantum material then its excitations [give universal quantum computation by braiding]."

Another nice quote from the interview: "One you know that braiding is dense it means you can approximate, and in fact efficiently approximate, any transformation of the Hilbert space by braiding."

One more nice quote: "So that would be a good place for mathematicians to retire from the fray. Well, its dense - you figure out how to do it. But then you don't feel fair to your physics colleagues. How do you actually make this quantum material? Then, you can start thinking with them about that."

Another reason to remember this paper: it is the origin of the (now classical) "Fibonacci anyon theory". They have a section where they prove that the Fibonacci TQFT gives universal braiding. In the introduction to this section, they say "These arise from Chern-Simons theory for r = 5 and G = SO(3), what G. Kuperberg calls the Fibonacci TQFT [KK]". The reference [KK] leads to this:

> A. Kitaev, and G. Kuperberg, work in progress.

Looking at newer versions of the paper and scouring the internet, I cannot find any reference to a work of Kitaev and Kuperberg about Fibonacci TQFTs. It seems that this "work in progress" was abandoned. The correct reference for the concept of Fibonacci anyons, I guess, would be to say that they originate from an unpublished work of Kitaev and Kuperberg, and this paper is the first recorded use of the term.

A good modern treatment of Fibonacci anyons is

> Trebst, Simon, et al. "A short introduction to Fibonacci anyon models." Progress of Theoretical Physics Supplement 176 (2008): 384-407.