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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

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2021

    added this pointer:

    • 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)

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2021

    added pointer to:

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2021

    added this pointer:

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMay 23rd 2021

    added pointer to:

    diff, v8, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2021

    added pointer to:

    diff, v9, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2021

    have added pointer to:

    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.

    diff, v9, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2021

    added pointer to:

    diff, v9, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 23rd 2021

    added pointer to:

    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).

    diff, v9, current

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

    added pointers to:

    • 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)

    diff, v10, current

    • 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?

    Asking Google, I find a single decent hit:

    • 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 z,1Y_{\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 kk 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

      (’arXiv:1311.0794)

    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:

    diff, v12, current

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

    added pointer to today’s

    diff, v15, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeAug 26th 2022

    added pointer to today’s

    diff, v16, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeSep 15th 2022

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

    diff, v19, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2022

    added pointer to today’s

    • T. Andersen et al. Observation of non-Abelian exchange statistics on a superconducting processor [[arXiv:2210.10255]]

    diff, v21, current

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2022
    • (edited Dec 19th 2022)

    added pointer to this early review:

    diff, v23, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2022

    added pointer to

    diff, v24, current

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeDec 26th 2022

    added pointer to:

    • 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]]

    diff, v26, current

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeDec 26th 2022

    added pointer to:

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    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2023

    added pointer to:

    diff, v27, current

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2023

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

    diff, v28, current

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2023

    added pointer to:

    diff, v30, current

    • CommentRowNumber32.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 10th 2023
    • (edited May 10th 2023)
    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2023
    • (edited May 11th 2023)

    These and similar claims are curious, in that they start with a setup that is manifestly not a topological quantum system, in this case

    Quantinuum’s H2 trapped-ion quantum processor

    and then let that system do something which is then claimed to be a topological quantum system.

    I am not necessarily doubting that there is a way for this to be true, but since at face value it can’t be true just by definition of the terms, it can only by true in a more subtle sense, and therefore I’d like to see a careful explanation of what that more subtle sense really is.

    The evident guess that we are really looking at a non-topological quantum system just simulating aspects of a truly topological quantum system is biefly addressed by the authors in the paragraph starting with

    These experiments go beyond merely simulating non-Abelian order and statistics.

    The reason given in the following sentence

    The ions are entangled in precisely the same way as…

    maybe leaves room to be expanded on.

    We remember people tending to make an ontological leap of faith in these situations, for instance when they have their quantum computer exhibit a kind of entanglement expected of some kind of wormhole, and then claim it to be that wormhole. Here we have a quantum computer exhibit entanglement as expected of some kind of topologically ordered phase and next the claim for it to be in that phase.

    I guess there is some sense in which these claims are correct, but it seems subtle and I’d like to see that subtlety discussed in detail.

    • CommentRowNumber34.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 12th 2023

    Thanks for checking it out, since you’ve been immersing yourself in this literature, it’s a better take than mine. I agree that one has to be careful with what amounts to a press release from a QC company, hence the link to the paper for more details. But that needs to go through the usual publication scrutiny…

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2023

    the usual publication scrutiny…

    For a sobering reminder, recall the (de-)publication story of anyons in the form of Majorana zero modes.

    On this desaster, Das Sarma & Pan (2021) concluded (p. 1):

    serious problem of potential confirmation bias in the putative topological experimental discoveries often claimed in the literature since the theoretical prediction is precise, and condensed matter physics imposes no community standards on the definition of an experimental discovery as is common in high-energy physics.

    This is particularly problematic for topological discoveries since…

    But luckily Quantinuum is a commercial company and (or so is my understanding) will proceed with the proof of the pudding by building recognizably robust quantum gates in the forthfoming future using their latest results. Seeing these in action will change the face of the field.

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2023

    added pointer to today’s

    diff, v31, current

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2024

    added pointer to:

    diff, v38, current

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2024

    added a couple more references on computation with abelian anyons

    diff, v39, current