Measurement-induced criticality in extended and long-range unitary circuits

The authors have included all the requested changes of the referees (especially report 4) and I stand by my previous recommendation that this paper is a worthy contribution to scipost physics core.
The paper provides thorough numerical study of long range Clifford dynamics under measurement. While I partly agree that a stronger statement could be made in conjunction with analytics, I also believe that this is a highly non-trivial endeavor and well beyond the scope and intention of this article.

same as in my previous report

1 - The description of the CHRC model in Sec. 2.1 is still somewhat misleading

The authors have done a good job addressing the referees' remarks from the first round. However, the formal description of the CHRC model in Sec. 2.1 remains a problem, despite the authors' efforts to improve it. Indeed, Eq. (3) appears to be a product of elementary two-body gates U_i,i+r,t. Here, t is the elementary time-step (the vertical size of the smallest rectangular element in the left panel of Fig. 1b). On the other hand, the first sentence of Sec. 2.1 says: "The unitary evolution is a sequence of M-body cluster unitary gates Ui,j,t, each of which is build stacking two body unitary gates". This implies that Ui,j,t is a composite (cluster) gate comprising three two-body elementary gates in Fig. 1b, so that time step t is the vertical size of such a cluster, which is 3 elementary layers in the figure. If I use this definition of the quantity Ui,j,t in Eq. (3), I would get a nonsensical result. Let us, therefore, assume that Ui,j,t is a two-body gate. Figure 1b suggests that, with this definition, at each time t there is only one two-body gate. On the contrary, Eq. (3) tells us that at each time t we should multiply many two-body gates. In particular, for L=6 (with periodic boundary conditions) and M=4, as in Fig. 1b, one gets the following product, if one literally follows the prescription of Eq. (3): U_1,2,t*U_1,3,t*U_1,4,t*U_2,3,t*U_2,4,t*U_2,5,t*... This is not what I see in Fig. 1b, where, for example, for the bottom layer (t=1) I see only U_1,2,1, while all other two-body gates in this product are clearly trivial, i.e., U=1. Note that in the product of two-body operators, as written in Eq. (3), again, the ordering of operators is not specified explicitly, in contrast to what is written in the footnote. In order to have a formal correspondence between Eq. (3) and Fig. 1b, the authors should (i) modify the first sentence of Sec. 2.1; (ii) write explicitly that, for a given value of t, only a single factor in the product in Eq. (3) is not equal to unity in the example of Fig. 1b (this would automatically resolve the problem of the operators' ordering); (iii) to give an explicit expression (perhaps, in terms of a Kronecker delta-symbol involving i, r, and t) specifying the indexes of the non-trivial two-body gate in the product (3) for the shoulder-like arrangement of Fig. 1b.

Improve the mathematical definition of the CHRC model

1- The topic is very relevant and the numerical investigation of the two-different long-range circuit models is comprehensive to a set of other recently appeared long-range models.
2- The paper is very accessible and provides a short and crisp summary of the obtained results. I find it particularly well written.
3- The summary of the entanglement-based observables and the numerical procedure are very precise and helpful.

1- After the revision by the authors, the paper still has a strong focus on the numerical results and less on providing a physical picture of the underlying dynamics.

The authors have considered the suggestions from my previous report and revised the manuscript accordingly. While I agree that an analytical understanding of the presented results is probably not straightforwardly obtained, I also think it is a chance missed by the authors to significantly improve their work and the impact it will have in the future.
However, there are plenty of new and interesting results on the topic of measurement-induced phase transitions presented in the manuscript. It is well written and very accessible and I believe it is a good paper that should be published in SciPost Physics Core.