Quantum Computing, Seifert Surfaces, and Singular Fibers
Round 1
Reviewer 1 Report
Summary: This paper summarizes computations which supports the authors' previous work. It attempts to relate various topological constructions to known universal quantum computers.
General comments:
The indeterminacy of Seifert surfaces is never properly addressed. Conversely, the text frequently implies that the Alexander and Jones polynomials determine a braid. They are in fact not even complete invariants of links.
The claim that Thistlewaite's table of links is incomplete would be a result of great interest to knot theorists, and it should be treated with more care.
Unsupported claims that results are "good", "useful", or "remarkable" appear through the text. This enthusiasm is uplifting but may be out of place.
It is not apparent that the orientations of link components inherited from the knot coverings were preserved during the computations. In particular, Alexander's theorem can only be applied to oriented links, and the results of surgery are orientation-sensitive.
The motivation for this research should be provided independently from the paragraph describing the contents of the paper, while the conclusion does not do that, and is redundant and reads as a clumsy attempt to insert more self-citations.
The brief discussion of the nu invariant and the distinction between homeomorphic and diffeomorphic manifolds does not connect to original results in the paper. Similarly for the subsection about Reidemeister torsion.
Section 2.4. should be expended upon.
Specific comments:
line 7. operator-valued takes a dash.
line 8. What is Gamma?
line 9. Replace "trefoil knot 3-manifold" with "complement of the trefoil in S^3".
line 17. What is the exact Bravyi and Kitaev reference?
line 18. Is there text missing? This passage is not understandable.
line 21. Replace "irrespectively" with "irrespective".
line 26. What is the antecedent to "such"?
line 27. Use "obtained" instead of "follows"
line 28. Replace "seen as a permutation" with "realized as a"
line 30. Which calculation?
line 32. Please support the claim that the set of magic states is "rich". Is this a comment on its cardinality? On its density?
lines 35 and 36. Use L in lieu of K.
line 38. Precise $\pi_1(M^3)$.
line 39. The trefoil is denoted $3_1$. What is the meaning of "Sec [6]"?
line 46. Again, is there text missing?
line 48. What does "figure of Borromean rings as a basic geometry" mean?
line 49. Remove "turns out that".
line 50. What is the identification? What is the relation?
line 55. Say "algorithm" and not "methodology".
line 56. The links have yet to be described in this text.
line 57. What is the definition of "Seifert surface braid"?
line 58. What is the skein relation satisfied by anyons?
line 59. Use "yields" instead of "contains".
line 65. The claim of currently active scrutiny is not supported by the reference provided.
line 71. Use a semi-column and not a column.
line 76. What is topological about a skein relation?
line 86. There does not appear to be a relation between the two citations as is claimed.
line 87-88. Say "In practice, to" and not "Below, to practically". What is the reference for the SeifertView software? The rest of this paragraph would be more readable if it was itemized.
Table 1. Use consistent notation for sources. How is "$D_4$ Dynkin a link? What is "magic L6a5"? What is POVM? What are the d?
line 98. The variable M is overdetermined.
line 100. Please reword this.
line 101. What is L?
line 108. In what group are the elements a, A, b, B, etc?
line 109. $\pi_1$ is overdetermined.
line 110. What are types of UQC?
line 117. Say "found" and not "made explicit".
Figure 2. is missing (b) and (e).
line 120. What is Gamma?
line 122. What is $Z$?
line 123. $\pi_1$ is overdetermined.
line 127. What is $\omega_6$?
Figure 3. is missing (b) and (e).
line 143. "are described by" is mathematically unsound. Say "have".
line 151. Please provide a reference for this classification.
line 157 is unclear.
line 158. Which four links?
Figure 4. is missing (b).
line 183. What is an explicit interpretation of $\eta_d(BR_0)$ as counting sublattices?
line 187. How is $\tilde D_4$ related to $B_1$?
line 191. What degrees give each covering type?
line 192. Recall the definition of $\Sigma'$.
line 209. What is the importance of the 1-torus?
line 211. The authors do not appear to possess the full understanding of the facts needed to study knot coverings.
line 212. "in in" should be "is in".
line 214. "surface braids" are in fact called "braided surfaces".
Author Response
Thank you so much for the careful reading of this paper and the constructive
comments. This excellent review helped us a lot.
Most corrections and adds are highlighted (in yellow) in the attached pdf.
About the general comments
* It is explained (a bit better) that the braid associated to an oriented link
allows to define a unique Seifert surface from the braid word. This is the meaning
of calculations we performed. The reference [23] to the software SeifertView was
added to paper [21] and document [22].
* We tried to remove claims of being ’good’, ’useful’ or ’remarkable’.
* We reorganized the introduction and created the subsections ’Motivation’
and ’Contents’. And we removed unnecessary self-citations in the conclusion.
* We agree that the nu invariant is only weakly related to the present stage
of the research. We briefly commented this point. The (twisted) nu invariant
(that is the Alexander polynomial) is important in the sense that one introduces
the manifold ′ obtained by 0-surgery on the trefoil (in Sec. 3) and the manifold
Y obtained by 0-surgery on the figure-eight knot (in Sec. 2.4).
About the specific comments
Almost all of them have been accounted for and we agree that some statements
were not clear enough.
In particular, we expanded section 2.4 about braids of hyperbolic manifolds
and studied the link L10n46 that was missed before by us.
Some captions were lacking and this has been solved (the assistant editor,
while formatting the paper, did not use the new figures of version 2).
Reviewer 2 Report
I attach a PDF file with all comments.
in short: The subject is interesting and the work is original with a great impact. It should be published but not in the present form.
Comments for author File: 
Comments.pdf
Author Response
Referee 2 asked for a better presentation of the manuscript. We think that
some problems about unclear statements are solved by the responses to referee
1. They are highlighted in the attached pdf.
Comment 1:
The introduction has been reorganized to emphasize the motivation and the
contents of the paper. A brief sentence about the possible relation to quantum
gravity is in the conclusion.
Comment 2:
The surgery used in the paper is that offered by the software Snappy and
(we believe) corresponds to the explanations provided in Adam’s book [15]. In
this paper, branched coverings are given by Snappy (up to degree 8) and they
are made in correspondence with subgroups of the corresponding index using
the results of [26] by A. D. Meldnykh.
Comment 3:
There seems to be a misunderstanding here. We explain that ?(2), a con-
gruence subgroup of index 6 of the modular group ?, has to associated to the
link 633 (not to L6n1 or L8n3 as it incorrectly done in Snappy).
Comment 4:
Yes, we mean ’along the Dynkin diagram of D4’, this has been corrected.
Some singular fibers result from −2 surgeries upon the links in question. It
is why we looked at them from the point of view of UQC.
It would be nice to associate this to the shape of the universe at an upper
stage of the research, we agree.
Comment 5:
The Alexander polynomial is the natural invariant attached to a Seifert sur-
face, as explained. It helps to distinguish the type of UQC we develop compared
to the approach based on anyons and the Jones polynomial.
Comment 6:
Here, we think that the introduction to the research of the first author and
coauthors answers this comment.
Comment 7:
This is a difficult question to answer how we could built the links/braids
with a physical set-up. We do not know at this stage.
Comment 8:
It is related to comments 3 and 4.
Reviewer 3 Report
In this work, the authors defined braids whose closure is the L of such a quantum computer model and computes their Seifert surfaces and the corresponding Alexander polynomial. It seems that the work is interesting and novel. Based on this point, I am willing to recommend this manuscript to be published in QR.
The paper should be proofread carefully and thoroughly to remove some mistakes.
They include but are not limited to:
1.The authors should describe the novelty of their work more clearly.
2. Line 212. "Further work in in progress"--->"Further work is in progress"?
Author Response
The paper has been carefully improved based on detailed comments of referee
1. We think that it corresponds to your comments as well.
Round 2
Reviewer 1 Report
The authors have adequately addressed my previous comments.
Reviewer 2 Report
The authors improved the paper much. All nex text parts contribute much to increase the readiability of the work. There is only small thinks in the current version which can be corrected in the proofs.
An example is citation 37:
old: 37. M. Amaral, R. Aschheimand K. Irwin, Quantumgravity at the fifth root of unity, Preprint 1903.1085 [hrp-th].
new 37. M. Amaral, R. Aschheimand K. Irwin, Quantumgravity at the fifth root of unity, Preprint 1903.10851 [hep-th].
Good work.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
This paper gives an exposition of some three dimensional topology without explaining how it is supposed to be related to quantum computing. I do not recommend publication.
Reviewer 2 Report
The manuscript described concepts (knots, links, braids, Seifert Surfaces …) from topology and tried to show their applications in universal quantum computing. However, the manuscript is more of a mathematical paper introducing these concepts studied in earlier works, but not really about quantum computing or quantum physics. The connection to quantum computing is mainly through citing earlier works, but not clearly shown in this manuscript. I think the manuscript is not suitable for the journal Quantum Reports.
Besides, the authors should pay attention to the Author Contributions, Funding and Acknowledgement sections. The following is what I see:
Author Contributions: For research articles with several authors, a short paragraph specifying their individual contributions must be provided. The following statements should be used “conceptualization, X.X. and Y.Y.; methodology, X.X.; software, X.X.; validation, X.X., Y.Y. and Z.Z.; formal analysis, X.X.; investigation, X.X.; resources, X.X.; data curation, X.X.; writing–original draft preparation, X.X.; writing–review and editing, X.X.; visualization, X.X.; supervision, X.X.; project administration, X.X.; funding acquisition, Y.Y.”, please turn to the CRediT taxonomy for the term explanation. Authorship must be limited to those who have contributed substantially to the work reported.
Funding: Please add: “This research received no external funding” or “This research was funded by NAME OF FUNDER grant number XXX.” and and “The APC was funded by XXX”. Check carefully that the details given are accurate and use the standard spelling of funding agency names at https://search.crossref.org/funding, any errors may affect your future funding.
Acknowledgments: In this section you can acknowledge any support given which is not covered by the author contribution or funding sections. This may include administrative and technical support, or donations in kind (e.g., materials used for experiments).
