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Neural Network Quantum States analysis of the ShastrySutherland model
by Matěj Mezera, Jana Menšíková, Pavel Baláž, Martin Žonda
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Submission summary
Authors (as registered SciPost users):  Pavel Baláž · Matej Mezera · Martin Žonda 
Submission information  

Preprint Link:  https://arxiv.org/abs/2303.14108v1 (pdf) 
Date submitted:  20230328 13:52 
Submitted by:  Žonda, Martin 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We utilize neuralnetwork quantum states (NQSs) to investigate the groundstate properties of the Heisenberg model on a ShastrySutherland lattice via variational Monte Carlo method. We show that already relatively simple NQSs can be used to approximate the ground state of this model in its different phases and regimes. We first compare several types of NQSs with each other on small lattices and benchmark their variational energies against the exact diagonalization results. We argue that when precision, generality and computational costs are taken into account, a good choice for addressing larger systems is a shallow restricted Boltzmann machine NQS. We then show that such NQS can describe the main phases of the model in zero magnetic field. Moreover, NQS based on a restricted Boltzmann machine correctly describes the intriguing plateaus forming in magnetization of the model as a function of increasing magnetic field.
Current status:
Reports on this Submission
Anonymous Report 3 on 2023528 (Contributed Report)
 Cite as: Anonymous, Report on arXiv:2303.14108v1, delivered 20230528, doi: 10.21468/SciPost.Report.7255
Strengths
Careful benchmark of neuralnetwork quantum states on the example of the ShastrySutherland model.
Weaknesses
No new physics.
Report
The manuscript starts with a review of previous work and results, first of the ShastrySutherland model (section 2) and then Neuralnetwork Quantum States (NQS) (section 3). The authors then benchmark NQS, specifically Restricted Boltzmann Machines (RBMs), first in sections 4.1, 4.2.1, and 4.2.2 for the zerofield ground state on lattices with $N\le 20$ sites. Finally, in section 4.2.3, the authors investigate the magnetization process for the example of $J/J'=0.45$.
Another Referee already pointed out that one might have cited further references. Even if I admit that there are so many investigations of the ShastrySutherland model that a complete overview is not possible, let me nevertheless mention the work by M. Jaime et al., PNAS 109, 12404 (2012) that one could have mentioned alongside Ref. [31].
While this is overall a nice benchmark with a large review component, I am a bit concerned about the novelty of the work. Specifically, the ED study Ref. [29] already investigated groundstate properties for significantly bigger systems than those investigated here, not to mention the iPEPS study Ref. [32]. Likewise, magnetization curves were already carefully investigated previously, e.g., in Refs. [31,33] and the PNAS by Jaime et al. It is true that here the authors make an effort to go to bigger lattices ($N\le 64$), but they focus on a parameter regime that is deeper in the dimersinglet phase than the regime $J/J'\approx0.63$ relevant to SrCu$_2 $(BO$_3$)$_2$. Thus, there is no new physics. Even more, I understand that also the methods have been introduced elsewhere such that the only new element of the present work is the benchmark on a different model.
More on the level of detail, I believe that NQS and specifically RBM are reminiscent of tensor networks. Accordingly, I think that at least some comparison to previous tensor network and specifically iPEPS investigations would be appropriate, beyond just mentioning the existence of these works.
It might be useful to recall the acceptance criteria for SciPost Physics. According to https://scipost.org/SciPostPhys/about#criteria, a submission has to meet at least one of the following expectations in order "to be considered for publication in SciPost Physics:
1. Detail a groundbreaking theoretical/experimental/computational discovery;
2. Present a breakthrough on a previouslyidentified and longstanding research stumbling block;
3. Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work;
4. Provide a novel and synergetic link between different research areas."
Given the preceding comments, I have a hard time seeing any of these criteria satisfied. The work is nevertheless a nice summary and comparative investigation of NQS, which should be suitable for publication in SciPost Physics Core after some minor revisions. I thus recommend that the authors submit a suitably revised version to SciPost Physics Core.
Requested changes
1 Add some comments on the comparison to tensor network methods, specifically iPEPS.
2 Complete the list of references, in particular add PNAS 109, 12404 (2012).
3 Make sure that all references have a DOI (concerns in particular Refs. [4,6,7,10,11,13,27,63,64]).
4 Fix formatting of titles. Ref. [28] presents one possibility to corrupt "SrCu$_2 $(BO$_3$)$_2$", Refs. [4547] other versions, Refs. [39,40] corrupt "TmB$_4$", Ref. [16] has spurious lowercasing of "Gutzwiller", etc. I note that, e.g., Ref. [18] demonstrates that the authors know how to properly format references.
Anonymous Report 2 on 2023523 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2303.14108v1, delivered 20230523, doi: 10.21468/SciPost.Report.7237
Strengths
1. A concise and clear review of the novel neural network quantum state approach
2. A detailed examination of various neural network anatzes for the study of the ShastrySutherland model; establishing the conditions of the efficiency of the method.
Weaknesses
1. A superiority of the method with respect to previous approaches has not been pointed out.
Report
The goal of the present paper to examine the complex physics of the ShastrySutherland model seems quite attractive and challenging. As it was reviewed in the manuscript the model exhibits quite a rich phase diagram and it is still an object of ongoing research. The application of the neural network quantum states can be a more efficient numerical approach for the frustrated model providing more accurate results for larger systems.
In the manuscript the authors broadly reviewed the variational Monte Carlo method and different architecture of neural network quantum states. In the original part, the authors confront the different versions of the neural network using the result for the systems of N=16, 20 spins. They concluded that NQS does not work well in the vicinity of the transition point between the dimersinglet and plaquettesinglet phases. As a result, the transition point was not located precisely. The similar problem appears at the edge where the plaquettesinglet phase disappears. Finally, the results for the fielddependent characteristic have been presented.
The results given in the paper show some potential of the method, but also many questions regarding its applicability remain open, and it is not clear how the presented results outperform already known methods. I guess, the authors need to modify the conclusions in order to uncover the power of the method for the study of the diverse properties of the ShastrySutherland model.
Requested changes
1. In the introduction, the authors review the ShastrySutherland model. I think it is not complete, and some other references can be added, e.g.
A. Abendschein and S. Capponi, Phys. Rev. Lett. 101, 227201 (2008),
where the exact diagonalization were performed for N=36 spins;
J. Dorier, K. P. Schmidt and F. Mila, Phys. Rev. Lett. 101, 250402 (2008)
M. Nemec, G. R. Foltin and K. P. Schmidt, Phys. Rev. B 86, 174425 (2012),
G. R. Foltin, S. R. Manmana and K. P. Schmidt, Phys. Rev. B 90, 104404 (2014),
where the model has been studied by means of the perturbative continuous unitary transformations, i.e. perturbation theory.
2. The definition of the PS order parameter given in Eq. (15) is not completely correct. I mean that it is barely possible to distinguish between the "singlet" and "empty" plaquettes used in the summation. Moreover, outside the PS phase this definition has no sense. I think that the authors need to reformulate it.
3. In Fig. 5(c) the AF order parameter is given. However, this result is not discussed in the text. Is this result in accordance with the previous studies?
4. The authors need to modify the conclusion in order to uncover the power of the method for the study of the diverse properties of the ShastrySutherland model.
5. Please correct some typos
There are misprints in the abbreviation:
RMB > RBM in pages 9, 13
DF > DS in page 10
if if is suggested > if it is suggested in page 17
Author: Martin Žonda on 20231103 [id 4092]
(in reply to Report 2 on 20230523)
First of all, we would like to thank the referee for the positive report as well as for pointing out some errors, missing references and for suggestions on how to improve our work. In the following, we answer in detail all the raised questions and criticism and discuss how we have improved the manuscript by reflecting on them. (Original text of the referee on which we are reacting is repeated for convenience.)
>*The results given in the paper show some potential of the method, but also many questions regarding its applicability remain open, and it is not clear how the presented results outperform already known methods. I guess, the authors need to modify the conclusions in order to uncover the power of the method for the study of the diverse properties of the ShastrySutherland model
>…
>4. The authors need to modify the conclusion in order to uncover the power of the method for the study of the diverse properties of the ShastrySutherland model.*
We have now added several direct comparisons of the NQS approach with DMRG and infinite DMRG (iDMRG) results. In particular, we have extended our analysis to larger lattices and results on lattices with mixed boundary conditions. We have focused on the regime where the ground state is expected to form the plaquette ordering. This refers to the region which is problematic already for small lattices. The reason is, that with improved learning we have been able to reproduce the exact results for the dimer phase even for large lattices. Which is now shown in our updated paper.
The comparison with DMRG in the plaquette phase showed that the finitesize scaling of the variational energy of RBM NQS for lattices with periodic boundary conditions is in a good agreement with that of DMRG. However, the learning for mixed boundary conditions proved to be much more challenging. The variational energy for large lattices (N>100) is higher than the DMRG result by several percent. Nevertheless, we were able to show that already a RBM NQS with alpha=2 is sufficiently expressive to describe the plaquette states. We now clearly state both the advantages, e.g., the fact that a very simple model without the usage of any symmetries can describe all main phases, and disadvantages, e.g. higher variational energy than predicted by (i)DMRG in the PS phase, of the approach in the conclusions.
>*1. In the introduction, the authors review the ShastrySutherland model. I think it is not complete, and some other references can be added, e.g.
>
>A. Abendschein and S. Capponi, Phys. Rev. Lett. 101, 227201 (2008),
>where the exact diagonalization were performed for N=36 spins;
>
>J. Dorier, K. P. Schmidt and F. Mila, Phys. Rev. Lett. 101, 250402 (2008)
>M. Nemec, G. R. Foltin and K. P. Schmidt, Phys. Rev. Lett. 101, 250402 (2008),
>G. R. Foltin, S. R. Manmana and K. P. Schmidt, Phys. Rev. B 90, 104404 (2014),
>where the model has been studied by means of the perturbative continuous unitary transformations, i.e. perturbation >theory.*
We thank the referee for turning our attention to additional literature. We have added all references to the relevant sentences in the introductory part.
>*2. The definition of the PS order parameter given in Eq. (15) is not completely correct. I mean that it is barely possible to distinguish between the "singlet" and "empty" plaquettes used in the summation. Moreover, outside the PS phase this definition has no sense. I think that the authors need to reformulate it.*
We thank the referee for pointing this out. We agree that the definition was incorrect and vague. We have corrected the formula in the updated version of the manuscript. Note that the new definition is written in a form that can be used also for uneven (Lx \neq Ly) lattices with mixed (periodic and open) boundary conditions.
>*3. In Fig. 5(c) the AF order parameter is given. However, this result is not discussed in the text. Is this result in accordance with the previous studies?*
The result is in qualitative agreement with previous works for example PRB 105(6), L060409 (2022). In the updated version of the manuscript we stress this by the following text:
Fig. 5 illustrates the usability of RBM for larger clusters. The presented results support the overall picture of the DS and AF phase separated by a narrow PS or at least its indication. Nevertheless, a much more thorough finitesize analysis would be necessary to assess the phase boundaries. For example, P_{AF} decreases with increasing system size in the whole relevant range of J′ which is in agreement with previous studies, e.g. [38]. Consequently, a careful and precise extrapolation of P_{AF} to the thermodynamic limit is needed to identify J above which the AF ordering prevails.
Please correct some typos
There are misprints in the abbreviation:
RMB .. RBM in pages 9, 13
DF .. DS in page 10
if if is suggested .. if it is suggested in page 17
We thank the referee for careful reading and for pointing out the typos. We have corrected all of them. In addition, we did a thorough proofreading of our manuscript and corrected several other mistakes, typos, grammar errors as well as unclear statements. Overall, the consideration and reflection on the questions and comments of the referee led to a significant improvement of our manuscript.
Anonymous Report 1 on 2023522 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2303.14108v1, delivered 20230522, doi: 10.21468/SciPost.Report.7233
Strengths
 Benchmark of various restricted Boltzmann machine architectures
as applied to the ShastrySutherland Heisenberg antiferromagnet.
 Comparison of different options for implementing point group symmetries.
Weaknesses
 The benchmark is not conclusive, the authors having opted for the least computationally demanding solution, rather than the "best" one in terms of accuracy.
Report
The authors study the phase diagram of the ShastrySutherland lattice using different variants of restricted Blotzmann machines (RBM) as a neural network quantum state. The goal is to find a single variational wavefunction that can correctly approximate the ground state in the entire phase diagram, at low computational cost. The authors conclude that a standard complex RBM gives the best tradeoff between accuracy and computational cost and can capture the main features of magnetization plateaus.
Given that the number of numerical methods for frustrated spin systems are scarce, introducing and assessing a new variational ansatz is an important line of research. Especially in the ShastrySutherland lattice resolving small variational energy differences has led to new explanations of the nature of the magnetization plateaus. Therefore the authors' work opens up a promising direction, which justifies publication in SciPost Physics.
The most interesting aspect of the work, in my opinion, is the treatment of point group symmetries, which need to be chosen carefully, and the fact that taking into account the Marshal sign rule fails to give significant improvement, even in the AF phase, which is surprising.
However, there is a very basic observation:
In the benchmark in Table 1 and in the investigation of magnetization plateaus the authors choose the values J/J'=0.2 and J/J' = 0.45, respectively, which is inside of the DS phase (at h = 0). It should be pointed out that this parameter point is not only in the DS phase but that for J/J' < 0.5 the exact ground state is given by a product state of dimer singlets, according to Shastry and Sutherland's famous solution. If I am not mistaken, this should also apply to the N=16 and N=20 clusters in Table 1. Thus at J/J' = 0.2 some of the RBM architectures even struggle to represent an exact product state of dimers. Then the question boils down to how a product state of dimers is to be represented by an RBM.
Therefore, [1], the question of how to encode a dimer product state exactly as an RBM should be addressed, [2], the benchmark of different RBM architectures should be shown also in the dimer singlet phase for a parameter point 0.5 < J/J' < 0.68, i.e. where the ground state is not an exact product state of dimer singlets.
Provided that the above two criticisms are met, I recommend the paper for publication in Scipost Physics.
Finally, the NQS fail to quantitatively describe the order parameter in the plaquette singlet phase, which is twofold degenerate. Did the authors try to remove the degeneracy by suitably chosen boundary conditions to alleviate this problem ?
Requested changes
See points [1] and [2] in the report.
 The presentation of Fig. 6 could be improved by labeling the magnetization plateaus by their magnetization values.
Author: Martin Žonda on 20231103 [id 4091]
(in reply to Report 1 on 20230522)
First of all we would like to thank the referee for the overall positive report as well as for the suggestions on how to improve our work. In the following we answer in detail all the raised questions and criticism and discuss how we have improved the manuscript by reflecting on them. (Original text of the referee to which we are reacting is repeated here for convenience and marked by /*** … ***/ )
/***
However, there is a very basic observation:
In the benchmark in Table 1 and in the investigation of magnetization plateaus the authors choose the values J/J'=0.2 and J/J' = 0.45, respectively, which is inside of the DS phase (at h = 0). It should be pointed out that this parameter point is not only in the DS phase but that for J/J' < 0.5 the exact ground state is given by a product state of dimer singlets, according to Shastry and Sutherland's famous solution. If I am not mistaken, this should also apply to the N=16 and N=20 clusters in Table 1. Thus at J/J' = 0.2 some of the RBM architectures even struggle to represent an exact product state of dimers. Then the question boils down to how a product state of dimers is to be represented by an RBM.
…
[1], the question of how to encode a dimer product state exactly as an RBM should be addressed
***/
The exact DS phase is always an eigenstate of the SSM and various theoretical studies suggest that it is the exact ground state up to J/J’=0.675(2) [for example, see PRX 9, 041037 (2019) or PRB 87, 115144 (2013)]. We consider this to be an exceptional quality of the model with respect to benchmarking new methods, such as NQS. It is straightforward to show that a complexvalued RBM with free biases on the input layer, i.e., the model used in the second part of our paper, is in principle capable of capturing such a state exactly. One possibility to represent a DS is to set the visible biases to reproduce the correct sign structure as discussed in the last paragraph of Sec. 3.2 and to set all the weights either to zero or a nonzero constant value. An example of such construction is now discussed in the new appendix D of the updated version of our paper. It shows that the RBM is in theory able to represent the DS state exactly. Learning the DS state is, of course, a different question.
Nevertheless, our benchmarks show that the DS can be learned [See Figs 4.(c), 5.(d) and 6(c)] even for L=100 (note that we have updated the data here) by using the second protocol. For small lattices even the first one is sufficient. We think that what is confusing here are these small but finite numbers in Table 1. They are not the lowest variation energies obtained. They are averages of the last 50 instances of the best runs. Because there are fluctuations in the variational energy due to the finite learning rate in the stochastic gradient descent as well as due to the Monte Carlo itself, these values are larger than the minimal one. We have opted for this, to have a protocol less sensitive to inaccuracies of the stochastic gradient descent method and, therefore, more reliable for the comparison of different architectures. However, the table does not necessarily show some hard limitations of the particular architecture. We have added the following note to the text, to clarify this to the readers: “Note that this means that we are not using just the lowest obtained energies but also test stability of the learning method. Consequently, the value of E50 is typically greater than zero even when the network is able to reproduce the state exactly.”
/***
[2], the benchmark of different RBM architectures should be shown also in the dimer singlet phase for a parameter point 0.5 < J/J' < 0.68, i.e. where the ground state is not an exact product state of dimer singlets.
***/
Previous studies indicate that, at least for here used finite lattices, the DS should be the ground state up to J/J’=0.675(2). Nevertheless, as also pointed out by another referee, the parameter range proposed by the referee is relevant because for example the J/J’ for SrCu2(BO3)2 at normal pressure is estimated to be 0.63. We have, therefore, added one column to Table 1, calculated for this value using the direct basis. These results are, with few exceptions, in a good qualitative agreement with the results of J/J’=0.2.
/***
Finally, the NQS fail to quantitatively describe the order parameter in the plaquette singlet phase, which is twofold degenerate. Did the authors try to remove the degeneracy by suitably chosen boundary conditions to alleviate this problem ?
***/
We thank the referee for this question in particular. It is the main reason for our late reply as we have decided to investigate this problem in more detail. In addition, we pushed for significantly larger lattices. We have focused on the problematic region and found several interesting results. The most relevant ones we have included in the revised manuscript. First of all, already a RBM with alpha=2 is able to capture a fully developed plaquette singlet state. We tested it by using the ShastrySutherland lattice (graph) but setting all interactions to zero except the ones in squares where plaquettes are expected, i.e., we constructed a lattice of interacting spins on, otherwise, independent squares. RBM was able to correctly capture the expected plaquette states on all accessible lattices for both periodic (we artificially broke the degeneracy) and mixed boundary conditions. We then used these plaquette states as the initial state for the full model in the regime where the plaquette state is expected. Interestingly, for the periodic boundary conditions, this did not lead to an improvement. However, for the mixed ones it helped to lower the variational energy in the parameter regime where a significant plaquette order parameter is expected.
The introduction of mixed boundary conditions also allowed us to directly compare our results with DMRG results from Ref. [PRB 105, L060409 (2022)]. Admittedly, in the PS the RBM variational energy is typically larger than the one obtained by DMRG. Despite that, the obtained plaquette order parameters are in reasonable agreement. This again underscores the main message that just a simple NQS can capture qualitatively different orderings. The usage of mixed boundary conditions and larger lattices also lowered the estimated position of the phase transition point between the dimer state and the plaquette state to J/J’~0.68 and therefore much closer to the expected value of 0.675(2). The above points are now discussed in more detail in the main text of the manuscript as well as in the new appendix D. We have also added two multipanel figures [Fig. 6 and Fig. 10].
/***
The presentation of Fig. 6 could be improved by labeling the magnetization plateaus by their magnetization values.
***/
We have added the magnetization values to the main plateaus in the new figures 7 and 8 and also added the missing x axis label in the figure 8.
We have also focused on a thorough proofreading of our paper and corrected several other mistakes, typos, grammar mistakes as well as unclear statements. We think that all the corrections and the added results motivated by the questions of the referee led to a significant improvement of our manuscript.
Author: Martin Žonda on 20231103 [id 4093]
(in reply to Report 3 on 20230528)We would like to thank the referee for the constructive criticism as well as for the recommendation to publish our work as a regular paper in SciPost Physics Core. Nevertheless, we would still like to address all raised points in detail and discuss how we have improved our manuscript reflecting on the report. Hopefully, this will convince the referee that the revised version of our manuscript is worth publishing in SciPost Physics as well.
We have added the requested reference, as well others relevant referees to the overview part of the paper.
We understand the concerns of the referee about the scientific novelty of the paper and, as discussed below, we have now made an effort to boost the paper in this respect. However, we would also like to point out to the two other possible route for publication in SciPost Physics listed on the webpage as well as in the report of the referee: Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work; Provide a novel and synergetic link between different research areas. We have shown in our paper that the ShastrySutherland model is not just another model on which the NQS approach can be applied. It is a model exceptionally suited for the task. This is partially because there are so many exact (even analytical) or close to exact (e.g., iDMRG) results obtained for this model as mentioned by the referee. But it is mainly because it has a complicated phase structure. There are continuous as well as discontinuous phase transitions, novel quantum phases, plateaus in the magnetizations, several open problems with conflicting results from different studies and the list goes on. We think that we are therefore opening a new pathway in existing research. We also combine two different research areas, namely the research of ShastrySutherland systems and NQS development.
The new elements are not just the benchmarks. First of all, we show that complicated networks are not necessarily the right answer to all problems. Then we introduced several strategies for using simple NQS on such a complicated system as the ShastrySutherland model. We have now demonstrated that a simple RBM can capture dimer state, AF state and also plaquette state, although the last one is difficult to learn. We discuss the advantages and disadvantages of transfer learning. We have also added some benchmarks for J/J′≈0.63 (see Table 1) relevant to SrCu2(BO3)2 at normal pressure. We have shown that NQS can deal with the steplike structure of the magnetisation plateaus. To our knowledge, this is a qualitatively novel result opening a new possible application of the method. For example, there are very recent results discussing a possibility of spinsupersolid orderings at finite magnetic fields [Nat Commun 14, 3769 (2023)] that could be now addressed with NQS. Further, we have now pushed the even larger lattices and also discussed mixed boundary conditions. We showed that this leads to a more realistic estimation of the position of the discontinuous phase transition, although the difficulties to learn PS prevail. To address this issue, we have introduced another strategy in the updated version of the manuscript to deal with this regime more efficiently. We also added comparison for periodic and mixed boundary conditions with DMRG and iDMRG results and discussed some problems this revealed.
There is definitely a connection between NQS and tensor networks and it was already investigated in the literature. See for example: D.L. Deng, X. Li, and S. Das Sarma, Quantum entanglement in neural network states, Phys. Rev. X 7, 021021 (2017). J. Chen, S. Cheng, H. Xie, L. Wang, and T. Xiang, Equivalence of restricted boltzmann machines and tensor network states, Phys. Rev. B 97, 085104 (2018). Y. Levine, O. Sharir, N. Cohen, and A. Shashua, Quantum Entanglement in Deep Learning Architectures, Phys. Rev. Lett. 122, 065301 (2019). L. Pastori, R. Kaubruegger, and J. C. Budich, Generalized transfer matrix states from artificial neural networks, Phys. Rev. B 99, 165123 (2019). A. Borin and D. A. Abanin, Approximating power of machinelearning ansatz for quantum manybody states, Phys. Rev. B 101, 195141 (2020). C.Y. Park and M. J. Kastoryano, Are neural quantum states good at solving nonstoquastic spin Hamiltonians? Phys. Rev. B 106, 134437 (2022) Or Sharir, A. Shashua, and G. Carleo, Neural tensor contractions and the expressive power of deep neural quantum states, Phys. Rev. B 106, 205136 (2022) Yet, if we understand the problem correctly, it is most probably the other way around. Let us cite the work of Sharirr at all. [PRB 106, 205136 (2022)] here: “ By directly constructing neuralnetwork layers that perform tensor contractions, we show that efficiently contractible TNS can be constructed in terms of polynomially sized neural networks. Our result, in conjunction with previously established results on the entanglement capacity of NQS, then demonstrates that NQS constitute a very flexible classical representation of quantum states and that TNS commonly used in variational applications are strictly a subset of NQS.”
However, this does not mean that NQS are simply better, especially not when one focuses on relatively simple ones. Tensor networks states are a wellestablished generalpurpose ansatz for quantum spin systems. The community has been using them for decades and, therefore, these techniques are already tuned to give the optimal results. On the other hand, NQSs are still a new approach. We are still searching for the optimal strategies as well as ideal networks. But there is potential. In this respect, one should also interpret the now added direct comparison of our results with the (i)DMRG results in the expected plaquette state phase. When focusing on this problematic region, the variational energy of the RBM is worse than the one given by ED (see N=20) or (i)DMRG (see the new results in Fig. 6). However, NQS can still give the correct order parameters, i.e., the physical structure of the ground state (new Fig. 6). Besides comparing the order parameter, we also show in Appendix D the structure of the groundstate clearly showing the plaquettes. The problem of too high variational energy, especially pronounced for large lattices, is related more to the learning process than to the expected limitations of small RBM NQS themselves.
Note that comparison in the DS is not necessary as RBM NQS reproduces the exact result as we discuss in the first paragraph of the added appendix D.
As discussed above, we have added several comparisons of the RBM NQS results with (i)DMRG. In particular, please see Fig. 6 and the detailed discussion related to it. Some comparison (not discussed above) is also shown in Fig. 5b, however, there we compare data for infinite cylinder lattices to small periodic ones. Therefore, this example is not authoritative for the overall assessment of the NQS results. We have also expanded the paper by additional appendix D where we demonstrate that RBM is expressive enough to encompass both dimer and plaquette ordering.
We have put a significant effort into adding missing references and correcting wrong formats as well as spurious lowercasings. In addition, we have provided a thorough proofreading, corrected some typos, grammar mistakes, inconsistencies in referencing equations and figures and reformulated some sentences to make the text clearer.
We again thank the referee for the overall kind assessment of our work and recommendation for publication in SciPost Physics Core. Nevertheless, to answer the criticism of the referee, we have now significantly improved the paper. We have also added benchmarks, new strategies and new results. Therefore, and with respect to the recommendation of the two other referees, we kindly ask the referee to reconsider again the suitability of our work for the SciPost Physics.