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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 15th 2021
• (edited Feb 15th 2021)

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMay 19th 2021

• Ady Stern, Netanel H. Lindner, Topological Quantum Computation – From Basic Concepts to First Experiments, Science 08 Mar 2013: Vol. 339, Issue 6124, pp. 1179-1184 (doi:10.1126/science.1231473)
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 19th 2021

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• CommentAuthorUrs
• CommentTimeMay 20th 2021

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 23rd 2021

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeDec 23rd 2021

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeDec 23rd 2021

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.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeDec 23rd 2021

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeDec 23rd 2021

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).

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeDec 25th 2021
• (edited Dec 25th 2021)

• 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)

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJan 7th 2022
• (edited Jan 7th 2022)

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJan 8th 2022

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?

• Qiang Zhang, Bin Yan, Many-Anyons Wavefunction, State Capacity and Gentile Statistics (arXiv:1504.00290)

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?

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJan 8th 2022

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.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJan 9th 2022

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?

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeJan 9th 2022

regarding #12 and #13:

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

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeFeb 10th 2022

For when the edit functionality is back, to add pointer to this preprint from today:

• Nikita Kolganov, Sergey Mironov, Andrey Morozov, Large $k$ topological quantum computer (arXiv:2105.03980)
• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeMar 17th 2022

Here is a question on the (potential) experimental realization of anyon defects in crystalline materials:

For pure crystals, we may and usually do model them, mathematically, as the quotient space of Euclidean space by the given crystallographic group. For instance, this is done when classifying their topological phases by computing the twisted equivariant K-theory of these quotients.

This makes it appear natural to model punctures by puncturing this quotient space. While mathematically natural, in the real crystal this means, of course, to add an impurity not just in one single location – which might be most natural from an experimental perspective – but periodically, i.e. including for any one impurity also all its images under the crystallographic group.

So my question is: Do existing experimental realizations consider such periodic impurities?

It’s not easy to search for the answer to this question, but eventually I found this article:

• Ville Lahtinen, Andreas W. W. Ludwig, Simon Trebst,

Perturbed vortex lattices and the stability of nucleated topological phases

On p. 1 this has the following parenthetical remark:

… anyons, usually arranged in a regular array to enable systematic control…

This seems to mean that the answer to the above question is Yes. But is there a reference that would say this a little more explicitly?

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeMar 18th 2022
• (edited Mar 18th 2022)

I see that theorists, at least, are happy with anyons on tori (the special case of the above where the point group is trivial):

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeMar 18th 2022
• (edited Mar 18th 2022)

Ah, Guo Chuan Thiang kindly points out to me that punctures in the momentum-space torus are being considered and known to make good sense; these are the “Weyl points” in:

• Varghese Mathai, Guo Chuan Thiang, Differential topology of semimetals, Commun. Math. Phys. 355 561-602 (2017) (arXiv:1611.08961)
• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeMay 31st 2022

pointer to this textbook had been missing:

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeAug 23rd 2022
• (edited Aug 23rd 2022)

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeAug 26th 2022

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeSep 15th 2022

added a list of references (here) on compilation of quantum circuits to braid gate circuits

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeOct 20th 2022

• T. Andersen et al. Observation of non-Abelian exchange statistics on a superconducting processor $[$arXiv:2210.10255$]$
• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeDec 19th 2022
• (edited Dec 19th 2022)

added pointer to this early review:

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeDec 21st 2022

• Michael Freedman, P/NP, and the quantum field computer, Proc. Nat. Acad. Sci. 95 1 (1998) 98-101 $[$doi:10.1073/pnas.95.1.9$]$
• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeDec 26th 2022

• R. Walter Ogburn, John Preskill, Topological Quantum Computation, in: Quantum Computing and Quantum Communications, Lecture Notes in Computer Science 1509, Springer (1998) $[$doi:10.1007/3-540-49208-9_31$]$
• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeDec 26th 2022

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeFeb 19th 2023

• CommentRowNumber30.
• CommentAuthorUrs
• CommentTimeFeb 26th 2023

finally hyperlinked these author names, hoping that I identified the initials correctly:

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeMar 15th 2023