DanceQ: High-performance library for number conserving bases

I thank the authors for the detailed response.

I think the revised version easily surpasses the acceptance criteria and will present a valuable contribution to SciPost Codebases.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

I have read the Reports from all four Referees, the replies of the authors, and the revised manuscript. I am glad to observe that the authors have expanded their manuscript and improved it significantly. The original version of the manuscript was already good; with the improvements, this is now a very good contribution that in my opinion SciPost Physics Codebases should be proud to publish.

When rereading the manuscript, I nevertheless stumbled across a number of issues of a more typographic nature. Some can be considered as hints for proofreading. However, there are a few that in my opinion would merit to be addressed before the manuscript goes into production.

1- Line 73 on page 3: "maximization" $\to$ "magnetization". See also Report #1 by Anonymous (Referee 2) on 2024-11-7.
2- A number of equations lack full stops (".") at the end, notably Eqs. (5), (6), (12), (14), the unnumbered equation between lines 392 and 393, (26), (28), the unnumbered equation between lines 425 and 426, (30), the unnumbered equation between lines 450 and 451, the unnumbered equation between lines 455 and 456, (31), (32), (A.1), (A.3), (A.4), (A.7), (C.1), the unnumbered equation between lines 831 and 832, and (D.2).
3- I suspect that the "appendix B" on line 141 on page 5 is appendix B of Ref. [57]. If so, I recommend adding a "thereof" at the end of the sentence for clarity.
4- Line 170 on page 6: remove the "a" between "requires" and "three".
5- Figures 2 and 3: with the SciPost style file, the fonts are now on the small side. Maybe it would help if the figure could be typeset to page width.
6- Line 276 on page 9: I believe that the correct mathematical term is "monotonically" (not "monotonously").
7- Line 285 on page 10: the space before the footnote 1 appears unnecessarily large to me.
8- Line 369 on page 13: The summation upper bound $i-1$ contains the summation index $i$. Please check and correct, as appropriate.
9- There is a stray comma "," at the beginning of line 398 on page 14.
10- Is the a good reason to typeset the "${\rm index}(\vec{x})$" in the unnumbered equation between lines 425 and 426 one page 15 on a separate line?
11- Is there a good reason for typesetting the unnumbered equation between lines 455 and 456 on page 16 on two lines? And should there really be a full stop at the end of its first line?
12- The page break after line 647 on page 21 is strange.
13- When rearranging Fig. 5, the authors forgot to replace "left" and "right" by "top" and "bottom", respectively (second line of the caption of Fig. 5 and line 662 on page 23).
14- Line 722 on page 25: remove "the" between "where" and "more" (?).
15- Line 731 on page 25: I recommend to leave the judgment to the reader and thus remove the "elegant".
16- The index $i$ is used both for the product and the sum in Eq. (A.2). This looks ambiguous. Please check and correct, as appropriate.
17- Line 815 on page 29: "${\tt get\_local\_index}$" spills over. Can this be fixed?
18- If I am not mistaken, Algorithms 2-4 are not referenced in the text. I recommend adding at least one sentence that refers to these.
19- For clarity, I recommend to add parentheses ("$($" and "$)$") to the first sum in Eqs. (C.1) and (C.2).
20- Insert "to" between "have" and "execute" on line 829 on page 29.
21- Line 845 on page 32: "$Q^{N_{\rm tensor}} \times Q^{N_{\rm tensor}}$" spills over. Can this be fixed?
22- Eq. (D.2): There is risk of confusion of the multiplicative factor $i$ with the imaginary unit. So, change notation? In addition, parentheses ("$($" and "$)$") under the sum may help to improve clarity.
23- Refs. [16,20]: capitalize the "I" in the titles?
24- Some names in titles of references still need to be capitalized, in particular: "Bloch" [21], "Bose-Hubbard" [43,56], "Hamiltonian" [60], and "Gray" [60]. See also item 2 in Report #3 by Anonymous (Referee 3) on 2024-9-12.
25- Refs. [34,35]: Can you include the links to the corresponding repositories?
26- Ref [39]: Page number (238) and DOI (https://doi.org/10.1098/rspa.1963.0204) are missing.
27- Ref. [62]: "4" is the repository, not page number $\Rightarrow$ remove the "p.".

Ask for minor revision

The authors have improved their manuscript in a significant way, providing useful examples. I still do not understand their comment about the gray code since it can be readily applied to a number conservation case as well, e.g. enumerating in a lexographic order all binary numbers with a fixed number of "1", see
https://docs.python.org/3/library/itertools.html#itertools.combinations.

I think that this work can be quite useful so I recommend it for publication.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

The revised manuscript can be published after correcting the mistake in line 73: maximisation->magnetization.

Publish (easily meets expectations and criteria for this Journal; among top 50%)