Bosonic entanglement renormalization circuits from wavelet theory

The authors
1- illustrate a new approach for discretizing bosonic fields using biorthogonal wavelets
2- provide an in-depth mathematical illustration for translation invariant chains of harmonic oscillators
3- introduce a thought-threw estimation for the approximation error introduced by performing the entanglement renormalisation
4- provide an extensive appendix supporting their approximation theorem

1- The manuscript is of theoretical nature and could be imporved by a more extensive numerical study benchmarking the approach e.g. against exactly solvable ground state preparations or alternative techniques
2- There is already a rich literature on the connection between wavelet theory and entanglement renormalisation. In this context, the results might not necessarily come unexpected.

The authors address all the points I raised in my previous report. Especially the restructured introduction improves the readability of the manuscript and offers the reader a better introduction into the topic including some of the relevant literature. I appreciate as well, the expanded sections 5.2 and 5.3 on the continuous wavelet functions adding more details to the discussion on the continuum limit. With all the changes made, I believe the manuscript is a round, solid and sound work.

The authors raise a valid point that the fermionic case is yet restricted to a critical model of hopping fermions while the here presented approach, in general, works for arbitrary free bosonic models. With this in mind, I came to appreciate the work more and can see that this work might be suited for a publication in SciPost Physics. However, compared to other works, I am still left with the feeling that the paper is indeed interesting, though extremely technical, and the main results somehow expected (as the referee report of Luca states as well).

After I went through the literature again, I have to admit that, while I am active in the field of Tensor Networks and numerical renormalisation, I might not have the profound overview on the field of wavelet theory to confidently judge whether the scientific importance is high enough for a publication in SciPost Physics or if the manuscript is better suited for SciPost Physics Core. Thus, I would leave this decision to the other referees and the editor in charge.

In any case, I would still have a few minor, cosmetic remarks which are listed below.

1- Regarding point 8 of the last report (the claims in the introduction of Sec. 5), the authors replied that these claims are discussed in Sec. 5.2 and 5.3 . This is correct and completely satisfying (especially with the now expanded sections 5.2 and 5.3). However, for the sake of clarity, I would recommend changing the wording from "It turns out the scaling functions ..." to something like "In Sec. 5.2 and 5.3, we discuss how the scaling functions..." or "In the following, we demonstrate that the scaling functions...". Just to tell the reader that this claim is yet to be discussed in the manuscript.

2- In the abstract, the authors write "We give a general algorithm [...] and prove an approximation result that...". The phrasing of "proving an approximation result" sounds strange to me. I would suggest "provide an approximation theory", "provide an approximation result" or "introduce an approximation result/theory".

3- In Fig. 4, the authors use the parameters L and K, which are not introduced in the context of the figure but rather in the appendix. I would suggest to (i) either include a short description of the parameters in the main text near Fig. 4 or (ii) to move the figure to the appendix and referring to the appendix in the main text.

Further, the figure shows the computed correlations for different modes of the harmonic oscillator and the exact values. Since they seem to overlap nicely, it could be more expressive to plot here the actual deviation of the computations with respect to the exact results, i.e. |<p_0 p_n>_{K,L} - <p_0 p_n>_{exact}| rather then |<p_0 p_n>_{K,L}|.