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Measurementinduced criticality in extended and longrange unitary circuits
by Shraddha Sharma, Xhek Turkeshi, Rosario Fazio, Marcello Dalmonte
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Submission summary
Authors (as registered SciPost users):  Marcello Dalmonte · Shraddha Sharma · Xhek Turkeshi 
Submission information  

Preprint Link:  https://arxiv.org/abs/2110.14403v4 (pdf) 
Date submitted:  20220119 16:34 
Submitted by:  Turkeshi, Xhek 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We explore the dynamical phases of unitary Clifford circuits with variablerange interactions, coupled to a monitoring environment. We investigate two classes of models, distinguished by the action of the unitary gates, which either are organized in clusters of finiterange twobody gates, or are pairwise interactions randomly distributed throughout the system with a powerlaw distribution. We find the range of the interactions plays a key role in characterizing both phases and their measurementinduced transitions. For the cluster unitary gates we find a transition between a phase with volumelaw scaling of the entanglement entropy and a phase with arealaw entanglement entropy. Our results indicate that the universality class of the phase transition is compatible to that of short range hybrid Clifford circuits. Oppositely, in the case of powerlaw distributed gates, we find the universality class of the phase transition changes continuously with the parameter controlling the range of interactions. In particular, for intermediate values of the control parameter, we find a nonconformal critical line which separates a phase with volumelaw scaling of the entanglement entropy from one with subextensive scaling. Within this region, we find the entanglement entropy and the logarithmic negativity present a crossover from a phase with algebraic growth of entanglement with system size, and an arealaw phase.
Current status:
Author comments upon resubmission
We thank the Referees for their careful reading of our manuscript and for their constructive comments and criticisms.
We have embodied these discussions in the new version of the manuscript, which we now hope is suitable for publication in Scipost Physics Core.
List of changes
In response of the Referee reports, we have implemented the following changes:
 Corrected typos;
 Added Ref [90] in bibliography;
 Changes in Sec. 2 clarifying the models and the difference with Hamiltonian setups;
 Updated caption of Fig 1;
 Replaced linear scale with logarithmic scale in the xaxis of Fig 8;
 Expanded discussion in Sec 5 .
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 3 on 2022315 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.14403v4, delivered 20220315, doi: 10.21468/SciPost.Report.4688
Report
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 nontrivial endeavor and well beyond the scope and intention of this article.
Anonymous Report 2 on 202226 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.14403v4, delivered 20220206, doi: 10.21468/SciPost.Report.4327
Strengths
same as in my previous report
Weaknesses
1  The description of the CHRC model in Sec. 2.1 is still somewhat misleading
Report
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 twobody gates U_i,i+r,t. Here, t is the elementary timestep (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 Mbody 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 twobody 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 twobody gate. Figure 1b suggests that, with this definition, at each time t there is only one twobody gate. On the contrary, Eq. (3) tells us that at each time t we should multiply many twobody 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 twobody gates in this product are clearly trivial, i.e., U=1. Note that in the product of twobody 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 deltasymbol involving i, r, and t) specifying the indexes of the nontrivial twobody gate in the product (3) for the shoulderlike arrangement of Fig. 1b.
Requested changes
Improve the mathematical definition of the CHRC model
Anonymous Report 1 on 2022120 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.14403v4, delivered 20220120, doi: 10.21468/SciPost.Report.4207
Strengths
1 The topic is very relevant and the numerical investigation of the twodifferent longrange circuit models is comprehensive to a set of other recently appeared longrange 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 entanglementbased observables and the numerical procedure are very precise and helpful.
Weaknesses
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.
Report
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 measurementinduced 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.
Author: Marcello Dalmonte on 20220319 [id 2304]
(in reply to Report 2 on 20220206)We thank the Referee of their reading of our revised version. We have changed the text before and after Eq. 3, to accommodate extra explanations following the points in the report. We have also revised Fig. 1b caption accordingly."