Extending the planar theory of anyons to quantum wire networks

1. Interesting & novel

2. Clearly written

1. The main weakness is that the authors did not prove 'braid coherence' for their anyon wire network models.

Report on the paper 'Extending the planar theory of anyons to quantum wire networks'

Due to the delay, I give a rather shorter report, than one could expect for a paper this length.
I will focus on the main text and appendix C.

The authors extend the theory of anyons (i.e., (unitary) (modular) tensor categories) to quantum wire networks. This is an important extension for several reasons. When restricting planar anyon models to quantum wire networks, it can happen that 'anyon' models that do not exhibit a consistent braiding on the plain, do have a consistent braiding on a wire network. Moreover, there are proposed models that do fall in this class.

In my opinion, this paper easily fullfils the requirements for publication in scipost. Below, I provide a list of questions and suggestions. The authors can use this list in order to improve the paper further.

Questions and suggestions.

Section 1.

The relation between the current paper and earlier work (such as [6,45]) is not entirely clear.
It could be beneficial if this is stated more clearly.

Section 2.

The authors mention that it could be interesting to consider non-commutative 'fusion' products, as the underlying graph provides a natural ordering. This is of course true. But, it would be good if the authors can mention any system (either experimental or theoretical), that hosts graph anyons that obey a non-commutative 'fusion' product.

A question concerning gauge factors and unitarity. In the multiplicity free case, the authors take u^ab_c in U(1), but the pentagon equations allow for arbitrary (non-zero) gauge coefficients. Are the authors restricting themselves to unitary models? If so, which result in the paper depend on this restriction?

A related question. Which results (if any) depend on the assumption that the fusion rules are multiplicity free?

When discussing Oceanu rigidity (or the number of independent solutions), one typically considers the number of solutions up to fusion automorphisms as well (as their number is finite, this does of course not change the rigidity property).

Section 3.

In section 3.1, the authors state that they did not prove that they actually considered all consistency relations in the case N=4. This is discussed in more detail in appendix C. It was, however, not entirely clear what additional work has to be done to prove braid coherence on wire networks.

In section 3.3, braiding on the H-graph is considered. The authors start with the anyons on one of the legs, and consider various consistency conditions. It was not immediately clear to me that if these consistency conditions are satisfied, it is implied that braiding two anyons on different legs (in the presence of other anyons as well), is als consistent. In other words, is it necessary to show 'coherence in the distribution of the anyons over the wire network'? Or is there a simple reason one does not have to worry about this?

Section 5.

I suggest the authors change 'lollipop' to 'tadpole' (throughout the paper), because tadpole is the established term for this graph/diagram.

Section 6.

It would be interesting to know if the authors think that some of the 'observations' in this section can be upgraded to more general results (and become conjectures), either in a reply or the in the paper.

Section 8.

The extend of the enhancement of 'computational power' on wire networks that occurs for certain models is not entirely clear to me. If a model is already universal, the only enhancement that can occur is that one needs fewer operations to approximate a gate for the same confidence level. However, if a model is not universal (i.e., provides a dense cover of some U(n)), it can happen that the different types of braids are inequivalent, and one can implement more gates. But, if one can still not obtain a sufficient set of gates (universal single qudit and an entangling two-qudit gate), has one really gained (much) in computational power?

A more naive question. In section 7, it is shown that the Theta graph yields effective planar anyon models. In section 8, the authors use the stadium graph, in which the Theta graph can be imbedded in two different ways. So, the braiding on the planar graph should also be like the planar case. So how can one get the inequivalent braidings the authors use in section 8 in the first place. Clearly, I am missing something here.

1- The study of anyons on wire networks is well-motivated from a physics point of view and the authors have done some important foundational work towards building the general theory.

2- There are many examples worked out to support and illustrate the new theory.

3-This effort is original and also very timely, with the development of the graph braid group being relatively recent and there being other recent physics papers on the subject of wire networks.

1- There are a few places where mathematical aspects of fusion categories versus anyon models get slightly confused, but it is nothing that can't be sorted out quickly.

2- A clear description of the graph braid group via generators and presentations for different types of graphs is central to the development of the theory but lacking.

3- A clearer delineation of the foundational assumptions of the theory (e.g. graph fusion, unitarity, compatibility of graph fusion and graph braiding) and what is not fully worked out (non-degeneracy of graph-braiding, topological twists) would improve the organization of the paper.

I think this paper is a great fit for SciPost that will be ready to publish after the authors address some of the points raised in this report.

1- I think it is important to state that the results of the paper are for multiplicity-free anyon models somewhere in the abstract and again in the introduction, since otherwise it is not stated explicitly until Section 2.

2- I would have liked to see an explanation of why it is okay to assume that the pentagon equations for anyon on wire networks should be the same as the usual pentagon equations, either using a mathematical or physical argument. Especially since later in Section 3.1 the authors seem to argue that unlike in the planar case, not all choices of bases of the Hilbert space of states associated to a collection of anyons in a wire network are valid.

3- This is an annoying point, but typically when people call something Ocneanu rigidity they are only referring to the statement that there are only finitely many fusion categories with a given fusion rule. I realize that the authors are quoting directly from Kitaev's "exactly solvable models" paper though, so it is not necessarily to change it.

4- Since the definition of the graph braid group is so central to the theory developed by the authors I would have found it helpful to include a bit more background at the beginning of Section 3 and not just relegated to an Appendix, although I understand this is the prevailing style in paper in this journal. My bigger issue is that while the generators are clearly defined, the relations (or lack of!) are a bit obscured. For example, the fact that the three-strand braid group of a trijunction is free is not explicitly stated in the Appendix.

5- The topological twist is only defined ( and similarly Equation (6) only holds) if the category has a ribbon structure, so it could be good to mention much earlier that everything is unitary (as I'm sure is intended because the authors later mention so on page 8).

6- Several times the graph braid generators and their matrix representations are conflated, for example in the sentences before Equation 59. On this subject, my opinion is that it is not ideal that A,B,X, and Y were called A-symbols, B-symbols etc. This puts them on the same level semantically as the fundamental data of the graph-braided anyon model, like the R-symbols, Q-symbols, P-symbols etc., when the representations of the graph-braids are really just a consequence of R,Q, P, and so on.

7- In the last paragraph before Section 3 on page 8 the authors list Rep(G) as an anyon model, but this is a symmetric fusion category and not a modular fusion category. Probably you meant its Drinfeld center? But then later in Table 1 there is a UCC in the column for "Planar?" in the table of graph anyon models. But Rep(D_4) is not a planar anyon model, so I'm a bit confused what is meant.

8- In Section 3.1 there is an emphasis on picking good bases for the matrix representations of the graph braid generators. In order to characterize the image of the subgroup of the unitary group (set of quantum gates) generated by the graph braids, don't they all need to be in the same basis? Overall I found the discussion of good vs. bad choices of bases hard to follow.

9- This is only an opinion but the notation e.g. $f=b \times c$ and calling $f$ a composite anyon is a little misleading in the anyon graph braids in Figure 3, since in the Q-hexagon equations there is a sum over f. Rather $f$ is an anyon in the fusion channel of $b$ and $c$. This is again confusing in Equation 59. Later however I am worried this is important, for example at the top of page 14, where a composite anyon is used as an index for one of the graph braid generators. In the planar setting it is not valid to use composite anyons as labels of fusion trees, so it would be good to see some more explanation of what is meant by this notation.

10- Is the label in the superscript ofthe graph braid $\sigma_3^{1_d,2_c,1_b,2_a}$ supposed to match the one in Figure 5(b)?

11- An investigation of what is meant by non-degeneracy of graph-braiding seems to be missing. Depending on the authors' aims it is not so important but it could be good to mention.

12- It is not obvious to me that a given solution to the graph hexagon equations uniquely specifies a wire network anyon model. Does one need to find solutions to graph ribbon equations as well? In general I was a bit confused about the role of topological twists, independently of the underlying graph topology. This seems to be treated at the end of Section 4, where there is a definition given of topological twist and maybe partially in Appendix D but I was unclear about the big picture from the main body of the paper alone.

13- If Appendix A is meant to be a stand-alone section containing the mathematical background of the graph-braid group it may best best if it did not make any reference to anyons, since the graph braid group is a purely mathematical notion. A presentation of all generators and relations for each type of graph under consideration should be clearly stated.

14-In Appendix D it may be best to avoid the use of the word "morphism" in the description of half-twists in graph-braided anyon models, since (apart from the underlying fusion category that exists by fiat) the authors have not shown that the graph-braided anyon model described a category.

15- In Appendix E there are some sentences that allude to demanding that the F-symbols should be gauge invariant but this is not physically justified and it will rule out many interesting theories. I think this was stated unintentionally based on the content of later appendices.