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I am giving this bare list of references its own entry, so that it may be !include
-ed into related entries (such as topological quantum computation, anyon and Chern-Simons theory but maybe also elsewhere) for ease of updating and synchronizing
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I was looking for discussion whether pure braid gates are sufficient to model all topological quantum operations. These authors show that all “weaves” suffice (braids with a single “mobile” strand). All the example weaves in this article and the followup arXiv:2008.03542 are in fact pure, but the authors never comment on this.
My understanding is that their method shows that every braid is approximated by a pure weave up to, possibly, appending a single elementary braiding of a fixed pair of neighbouring strands. Which in practice should be good enough. But I am wondering if authors ever comment on this aspect.
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This shows many more examples of “weave” gates. Again, all of them are in fact pure braids (except for “injection weaves” and “F weaves”, but these are all meant to be intermediate).
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Jiannis K. Pachos, Quantum computation with abelian anyons on the honeycomb lattice, International Journal of Quantum Information 4 6 (2006) 947-954 (arXiv:quant-ph/0511273)
James Robin Wootton, Dissecting Topological Quantum Computation, 2010 (pdf)
“non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models.”
Seth Lloyd, Quantum computation with abelian anyons, Quantum Information Processing 1 1/2 (2002) (arXiv:quant-ph/0004010, doi:10.1023/A:1019649101654)
In
the authors write (first page):
all concrete known examples of non-abelian anyon theories with integer global dimension are constructed from a gauging of an abelian anyon theory.
I’d like to understand this statement in detail (preferably I’d like to understand if it can be expressed as a statement about something like “equivariant braid representations”?), but I have only a vague notion so far.
The authors don’t seem to come back to this statement in their main text, but from their preceding sentence it seems that it is meant to be taken from:
The discussion there is about taking a “unitary” modular tensor category equipped with a group action to something like the corresponding crossed product MTC. While I haven’t gone through it in much detail, that seems quite natural/plausible. But I don’t yet see an argument that “all concrete known examples” of MTCs (with integer global dimensions) arise this way.
I have a vague memory of statements like this from back when I was surrounded more by MTC theorists in Hamburg, but will need to remind myself.
On a different point:
I suspect that the relevant group of phases for physically realizable abelian anyons is not the full circle group $\mathbb{R}/\mathbb{Z}$, but just the “rational circle” $\mathbb{Q}/\mathbb{Z}$. Is there an authorative reference that would admit this?
Asking Google, I find a single decent hit:
Here the statement appears in the second paragraph, but just in passing and parenthetically, and then again on p. 2, above (11), in somewhat weaker form.
Is there a better reference?
Regarding #12:
I see from Kohno 2014 (pdf) that the flat line bundle on config space of interest in KZ-theory has phases given by a rational function of two real parameters (his (6.6)) and that the parameter values of interest form an open subset (his Thm. 7.1). So at least there is a large supply of “abelian anyons” of interest that have rational phases.
regarding #7:
Just to highlight a fun observation: What those physicist call “weave quantum gates” corresponds to the mondromy braid representations induced by the configuration space that is called “$Y_{\mathbf{z},1}$” in Etingof, Frenkel & Kirillov 1998, around Cor. 7.4.2.
It’s evident once one sees it. But I wonder if this has been made explicit anywhere before?
In fact it’s even better: it’s exactly the case of rational mondromy that corresponds to CFT-realizations: by the discussion in Etingof, Frenkel & Kirillov 1998, Sec. 13.4
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