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.

]]>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…

]]>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

areentangled 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.

]]>News for you, Urs:

With paper https://arxiv.org/abs/2305.03766

]]>added pointer to:

- Louis Kauffman,
*Quantum Topology and Quantum Computing*, in: Samuel J. Lomonaco (ed.),*Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium*, Proceedings of Symposia in Applied Mathematics**58**, AMS (2002) [pdf, doi:10.1090/psapm/058]

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

- Dmitry Melnikov, Andrei Mironov, Sergey Mironov, Alexei Morozov, Andrey Morozov,
*Towards topological quantum computer*, Nucl. Phys. B926 (2018) 491-508 (arXiv:1703.00431, doi:10.1016/j.nuclphysb.2017.11.016)

added pointer to:

- Ananda Roy, David P. DiVincenzo,
*Topological Quantum Computing*, Lecture notes of the 48th IFF Spring School (2017) $[$arXiv:1701.05052$]$

added pointer to:

Carlos Mochon,

*Anyons from non-solvable finite groups are sufficient for universal quantum computation*, Phys. Rev. A**67**022315 (2003) $[$arXiv:quant-ph/0206128, doi:10.1103/PhysRevA.67.022315$]$Carlos Mochon,

*Anyon computers with smaller groups*, Phys. Rev. A**69**032306 (2004) $[$arXiv:quant-ph/0306063, doi:10.1103/PhysRevA.69.032306$]$

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$]$

added pointer to

- 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$]$

added pointer to this early review:

- Gavin K. Brennen, Jiannis K. Pachos,
*Why should anyone care about computing with anyons?*, Proc. R. Soc. A**464**(2008) 1-24 $[$doi:10.1098/rspa.2007.0026$]$

added pointer to today’s

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

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

]]>added pointer to today’s

- Eric C. Rowell,
*Braids, Motions and Topological Quantum Computing*[arXiv:2208.11762]

added pointer to today’s

- Muhammad Ilyas,
*Quantum Field Theories, Topological Materials, and Topological Quantum Computing*[arXiv:2208.09707]

pointer to this textbook had been missing:

- Jiannis K. Pachos,
*Introduction to Topological Quantum Computation*, Cambridge University Press (2012) $[$doi:10.1017/CBO9780511792908$]$

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)

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

Roberto Iengo, Kurt Lechner,

*Quantum mechanics of anyons on a torus*, Nuclear Physics B**346**2–3 (1990) 551-575 (doi:10.1016/0550-3213(90)90292-L)Yutaka Hosotani, Choon-Lin Ho,

*Anyons on a Torus*, AIP Conference Proceedings**272**(1992) 1466 ; (arXiv:hep-th/9210112, doi:10.1063/1.43444)Songyang Pu, J. K. Jain,

*Composite anyons on a torus*, Phys. Rev. B**104**(2021) 115135 (arXiv:2106.15705, doi:10.1103/PhysRevB.104.115135)Songyang Pu,

*Study of Fractional Quantum Hall Effect in Periodic Geometries*(etda:21203sjp5650)

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?

]]>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)

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

]]>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?

]]>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.

]]>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?

]]>