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In a comment on my answer to the question: What exactly are anyons and how are they relevant to topological quantum computing? I was asked to give specific examples of occurrence of anyons in nature. I've spent 3 days searching, but every article refers to either "proposed experiments" or "nearly definitive evidence".

Abelian anyons:

Fractional charges have been measured directly since 1995, but in my search, all articles pointing to evidence of fractional statistics or an exchange factor $e^{i\theta}\ne\pm1$, point to this nearly 7-year old pre-print, where they say in the abstract that they "confirm" detecting the theoretically predicted phase of $\theta =2\pi/3$ in the $\nu=7/3$ state of a quantum Hall system. However, the paper seems never passed a journal's peer review. There is no link to a journal DOI on arXiv. On Google Scholar I clicked "see all 5 versions", but all 5 were arXiv versions. I then suspected the article's name might have changed at the time of publication so went hunting for it on the authors' websites. The last author has Princeton University's Department of Electrical Engineering listed as affiliation, but does not show up on that department's list of people (after clicking on "People", I clicked on "Faculty", "Technical", "Graduate Students", "Administrative", and "Research Staff" but nothing showed up). The same happened for the second-last author! The third-last author does have a lab website with a publication list, but nothing like this paper appears in the "Selected Publications out of more than 800" page. The fourth-last author is at a different university, but his website's publication list is given as a link to his arXiv page (still no published version visible). The 5th last, 6th last, and 7th last authors have an affiliation of James Franck Institute and Department of Physics at the University of Chicago, but none of their three names shows up on either website's People pages. One of the authors also has affiliation at a university in Taiwan, and her website there lists publications co-authored with some of the people from the pre-print in question, but never anything with a similar title or with a similar enough author list. Interestingly, even her automatically generated but manually adjustable Google Scholar page does not have even the arXiv version but does have earlier papers (with completely different titles and no mention of anyons) with some of the co-authors. That covers all authors. No correspondence emails were made available.

1. Is this pre-print the only claim of confirming an exchange factor $\ne\pm1$ ?
2. If yes, what is wrong with their claimed confirmation of this? (It appears to have not passed any journal's peer review, and it also appears that an author has even taken down the arXiv version from her Google Scholar page).

Non-abelian anyons:

I found here this quote: "Experimental evidence of non-abelian anyons, although not yet conclusive and currently contested [12] was presented in October 2013 [13]." The abstract of [12] says that the experiment in [13] is inconsistent with a plausible model and that the authors of [13] may have measured "Coulomb effects" rather than non-Abelian braiding. Interestingly the author list of [13] overlaps with the pre-print mentioned in the Abelian section of this question, though that pre-print was from 2 years earlier and said in the abstract "Our results provide compelling support for the existence of non-Abelian anyons" which is a much weaker statement than what they say in the same abstract for the Abelian case: "We confirm the Abelian anyonic braiding statistics in the $\nu=7/3$ FQH state through detection of the predicted statistical phase angle of $2\pi/3$, consistent with a change of the anyonic particle number by one."

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  • $\begingroup$ By "confirming existence" I mean confirming fractional or non-Abelian statistics, which some might say are the defining properties of Abelian and non-Abelian anyons respectively. $\endgroup$ – user1271772 May 14 '18 at 23:22
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It depends what you mean by the 'existence' of anyons.

One way is to engineer a Hamiltonian which leads to quasiparticles (or other defects) that have anyonic statistics. This will require the Hamiltonian to be implemented, the system to be cooled to sufficiently near the ground state, the anyons to be manipulated, etc. So there's a lot to be done, and I don’t think that the development of the systems required has a lot of other applications. So it suffers from being both hard to do, and quite a niche.

Hopefully, someone else will give you the answers you want on this kind of approach. However, I thought it is important to note that there is another way to get anyons. This is to not bother with the Hamiltonian. Instead, the eigenstates can be prepared and manipulated directly.

In this case, you aren’t getting any topological protection from the Hamiltonian. Instead, measurements are constantly made of what eigenstate you are in, in order to detect and help you mitigate the unwanted effects of errors.

The most realistic examples of this approach will be ones for which these operations can be easily performed on a quantum computer. All the development and progress towards building qubits and their gates can then be directly used in the search for anyons.

Anyons are systems that can be easily implemented with qubits or qubits are typically a specific form of quantum error correcting code. Specifically, they are stabilizer codes for which the states of the stabilizer space are topologically ordered, and syndrome measurements correspond to measuring whether anyons are present at each point throughout the system.

Th simplest example is the surface code. The basic quasiparticles of this are Abelian anyon. There have been experiments that create and manipulate these anyons to demonstrate their braiding behaviour. The first example was done over a decade ago in photonics systems.

The surface code can also host defects which behave as Majorana modes, and therefore non-Abelian anyons. I implemented a very minimal example of their braiding in this paper.

As quantum processors get larger, cleaner and more sophisticated, there will be a lot more of this kind of study. I would think that the majority of the anyons that we will see and use will be realized in this manner, rather than with an implementation of the Hamiltonian.

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    $\begingroup$ I have added a comment clarifying what I mean by 'existence'. The first paper you link to is not an experiment that creates and manipulates anyons: the abstract says that they use polarized photons (bosons) to simulate the behavior of anyons by encoding a model of anyons in the photonic qubits (analog quantum simulation). Likewise is the case for your paper, except with superconducting qubits instead of photonic ones. The question remains, whether or not an exchange factor different from $\pm1$ has ever been confirmed experimentally in a peer reviewed journal! $\endgroup$ – user1271772 May 14 '18 at 23:26
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    $\begingroup$ I don’t see much of a difference between a ‘simulation’ and a realization with a Hamiltonian. Is the latter not also something like a simulation, since the anyons are only quasiparticles? As long as topologically ordered states are used, I think they are both equally valid. $\endgroup$ – James Wootton May 15 '18 at 5:35
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    $\begingroup$ +1 Thanks @JamesWotton. This at least partly answers what I wanted to know. If I interpreted this correctly, for performing topological quantum computing, all we need to do is simulate "anyonic" behaviour/statistics. The world lines of these "simulated anyons" can be used to create logic gates which make up the computer (although I'm not aware of the exact method and might ask that as a fresh question). That is, as far as I understand: it isn't necessary for anyonic statistics to exist "in nature" for performing topological quantum computing; a simulation of that kind of statistics suffices. $\endgroup$ – Sanchayan Dutta May 15 '18 at 6:11
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    $\begingroup$ @JamesWootton: If I simulate a 10 qubit quantum computer by diagonalizing a $2^{10} \times 2^{10}$ matrix on a classical computer, have I made a quantum computer, or have I simulated one? The latter is not scalable in this case. Imagine that quantum theory existed without any experimental evidence (i.e. entangled states or superpositions were never confirmed in experiment, and neither was anything else quantum, such as discrete levels in the H atom). Then we can still use a classical computer to show that a 3-qubit Deutsch-Josza algorithm works, but still we have no evidence that qubits [cont] $\endgroup$ – user1271772 May 15 '18 at 17:32
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    $\begingroup$ This isn’t the same kind of simulation though. We aren’t just describing the quantum states involved on a classical computer, we are creating them using actual quantum systems. The only difference with a ‘true’ implementation is the lack of the Hamiltonian. But since the only job of the Hamiltonian is to create and protect the states (which we are doing manually instead) and not to induce dynamics, I don’t see why it’s absence makes the anyons any less anyonic. $\endgroup$ – James Wootton May 15 '18 at 18:39

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