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Measuring Renyi Entropy in Neural Network Quantum States
by HanQing Shi, HaiQing Zhang
Submission summary
Authors (as registered SciPost users):  HaiQing Zhang 
Submission information  

Preprint Link:  https://arxiv.org/abs/2308.05513v1 (pdf) 
Date submitted:  20230904 04:55 
Submitted by:  Zhang, HaiQing 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We compute the Renyi entropy in a onedimensional transversefield quantum Ising model by employing a swapping operator acting on the states which are prepared from the neural network methods. In the static ground state, Renyi entropy can uncover the critical point of the quantum phase transition from paramagnetic to ferromagnetic. At the critical point, the relation between the Renyi entropy and the subsystem size satisfies the predictions from conformal field theory. In the dynamical case, we find coherent oscillations of the Renyi entropy after the end of the linear quench. These oscillations have universal frequencies which may come from the superpositions of excited states. The asymptotic form of the Renyi entropy implies a new length scale away from the critical point. This length scale is also verified by the overlap of the reduced Renyi entropy against the dimensionless subsystem size.
Current status:
Submission & Refereeing History
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Reports on this Submission
Strengths
 as far as I can check, the numerical calculations look correct
Weaknesses
 Insufficient understanding of the subject and of the existing literature;
 No new approaches or techniques implemented, but insufficient credit for the original work that introduced them;
 Unclear significance of the presented calculation and of their results;
 Questionable analysis of the behavior of the 2Renyi entropy at the end of the quench.
Report
This manuscript describes a neural network approach to the study of the Renyi2 entanglement entropy for the transverse field Ising chain in the static and linear quench cases.
Its main contribution is to show how neural networks can help in studying the entanglement under a linear ramp of the magnetic field, a problem that is hardly tractable through other approaches. However, the only novelty in this paper is the use of the swap operator to calculate the Renyi2. This latter method is already known and all other elements in the calculations have already appeared in a way or another in the literature. Thus, this manuscript does not detail any new approach and its value should rely purely on the results presented. However, these are not discussed in a way that highlights some important general behavior (see below).
Thus, it is absolutely clear that this work does not remotely reach the high standards for acceptance in SciPost Physics and I am not favorable toward its publication on SciPost Physics Core either. One of the central problems is that the authors do not respect the standard scholar practices and certain sentences indicate a rather poor understanding of the concepts underlying this work.
The former problem manifests itself in the lack of citations for the relevant papers in the literature. For instance, the use of restricted Boltzmann machines to describe the TFIM was pioneered in a work by Carleo & Troyer ( Science 355, 602 (2017) ) and further developed in other works. Instead, this literature is completely ignored and the authors only cite a recent (yet unpublished) paper of them (posted on the arXiv, and which indeed contains most of the relevant citations). Similarly, the main interest of this work is in the outofequilibrium dynamics induced by a linear change in the amplitude of the external magnetic field. The literature on this type of problems is huge and, although it is not possible to cover it exhaustively, the number of papers cited by the authors is ridiculous and unacceptable. It is true that most studies have focused on sudden quenches in the parameters, but there have been studies on linear ramps and this paper never discussed the qualitative understanding of this outofequilibrium problem reached by the community, beyond the KibbleZurek mechanism of defect creation, and the differences between the sudden and linear cases. There is also an understanding regarding the general features of the entanglement entropy after quenches, which could have been an interesting guidance to analyze the results on the present work, but I will not spoil the opportunity of researching and reading the literature on behalf of the authors to impose my personal point of view on the matter.
On the second major problem:
 repeatedly, the authors claim that the ln 2 for the entanglement entropy at zero external field is related to the exact double degeneracy there. This is not correct. The ground state manifold without external field is spanned by two factorized states, each of them ferromagnetically ordered in the positive/negative direction. Being factorized, these states have no entanglement. However, at finite field and finite sizes, the ground state has a definite parity for the transverse magnetization (along the magnetic field), this being a conserved charge of the Hamiltonian, and the zerofield state in continuity is an even/odd superposition of the two ferromagnetic state, thus developing a ln 2 contribution to the entanglement entropy. Thus, the ln 2 observed by the authors in their studies is a consequence of the fact that they use a symmetrypreserving state in a symmetry broken phase. It is clear that the authors ignore these concepts and their interplay by the way they interpret and explain their work;
 in the first paragraph of page 2 the authors write that "the system is in a paramagnetic phase with spins all pointing along xdirection": this is correct only for $h \to \infty$. What they should have written is "the system is in a paramagnetic phase with a finite expectation value for the spins only in the xdirection". Similarly, in the following sentence they claim that for $h=0$ the system is completely ferromagnetic, while if this was the case the entanglement entropy would be zero;
 The fact that the entanglement entropy can serve as an efficient detector of quantum phase transitions is an already established fact, arguably one of the main reason for its success in the community over the past couple of decades, and the fact that the author repeatedly state this as a central result of their work is at least worrisome regarding their understanding of the literature.
Although I recommend that the author completely rethink and improve this work before resubmitting to publication, here is a (noncomplete) list of other problems with this manuscript:
 as explained, it is not acceptable to cite only Ref. [5] regarding machine learning methods for the TFIM;
 the discussion around eq. (2) for the Renyi entropy is poor: one should first specify that the system is taken in a pure state $\Psi>$ and is bipartite into a subsystem A and its complement B. Then $\rho_A = \tr_B \Psi><\Psi$ and so on...
 the static behavior of the Renyi entropy for the TFMI for large system sizes has been analytically worked out in J. Phys. A: Math. Theor. 41 (2008) 025302, which should be used for comparison in this work;
 before eq. (4) it would be appropriate to state that there are M variables $h_i$ and after eq (4) the authors claim that the ground state function is obtained from random parameters, without explaining that this is the starting point for a convergence procedure;
 before eq. (5) the FubiniStudy distance should be properly referenced and the author should discuss why they choose this quantity to guide the minimization;
 the manipulations after (6) are new to me: if they appeared already in the literature they should be referenced, otherwise the level of details provided is not sufficient to understand exactly what the authors actually did to follow the evolution;
 the discussion around eq. (7) is not clear. The existence of the Schmidt decomposition is not an assumption, but a mathematical theorem and should be explained as such with proper reference to its name. Furthermore, the authors should explain that the method involves creating two copies of the same system and operating the swap between the degrees of freedom of just one subsystem. If the two systems are copy of each other, the Schmidt coefficients in (7) should be the same (C and C, not C and D). Otherwise, the authors can defined the swap operator in generality for two different systems and then specialize for the calculation of the Renyi2;
 the caption of Fig. 1 should report that panel a) has N=100;
 toward the end of page 3, when the authors notice that the period is compatible with $\pi/2$, being the latter a pure number and the former a dimensionful one, a scale of time/energy should be provided;
 in the caption of fig. 3 the author should write $\tau_Q \gg 1$ instead of $\tau_Q \to \infty$;
 on page 3 the authors remark that the period of oscillation is virtually independent from $\tau_Q$ and then on page 4 they proceed on studying the evident dependence of the oscillations on \tau_Q... As a matter of fact, on page 4 I think they study the amplitude dependence on $\tau_Q$, but that should be explained more clearly;
 panel b) of fig. 4 is used by the authors to prove that the entanglement entropy follows eq. (10), but to me it shows that, beside for small values, it is independent from the RHS of eq. (10). This is because eq. (10) at best should provide the scaling of S_{saturation}...
 in discussing fig. 4, panel a), the authors write "we can still see the symmetry under $N_A → 100 − N_A$.". This symmetry is always present if the total ground state of the system is pure...
Requested changes
Complete rethinking of this work
Strengths
1 The results seems to be correct and well explained
2 The manuscript is accessible to a broad audience
Weaknesses
1 There is no apparent novelty in the techniques used, nor in the studied physical model
Report
The manuscript "Measuring Renyi Entropy in Neural Network Quantum States" investigates the time evolution of Renyi entropies in Spin Models. This is a numerical work that relies on the use of restricted Boltzmann machines to model the manybody quantum state. It is explained in particular how to extract the Renyi entropy from the neural network using the replica trick.
The numerical observations are in agreement with Conformal Field Theory at the critical point of the Ising Chain. In the quench case, the KibbleZurek physics is explored, in relation with Ref [23]
I am a bit confused about the actual aim of the paper.
If the aim is to show that one can access Renyi entropies from neural networks wavefunctions (as the title suggests), I am afraid this is known already, see for instance the extensive discussion and numerics in [Nature Physics 14, 447450 (2018)]. Typically, one uses indeed the same replica trick as in quantum monte carlo.
So the novelty of the paper may lie in the application of the numerical method to the quantum Ising model. But, as recalled for instance in Refs 19, 23, the model is exactly solvable. All the entropies can be computed exactly in the equilibrium case [https://arxiv.org/abs/quantph/0211074] and with timedependent modulations [23].
Finally, suppose the authors would consider instead a nonintegrable model, then the neural network method would show an advantage over tensornetwork methods if and only if the level of entanglement is such that it cannot be represented by a MatrixProductState. This is not the case for the data presented in this paper with entropy <=2.
In summary, all the results appear to be correct, but I do not see a particular new insight, technical method, numerical observation, that would justify publication in SciPostPhys
Author: HaiQing Zhang on 20231016 [id 4041]
(in reply to Report 1 on 20231004)Dear Editor and Reviewer,
Thank you very much for your correspondence and comments. We do not agree with the reviewer’s comments that “There is no apparent novelty in the techniques used, nor in the studied physical model” . The reasons are included in the replies to the reviewer’s comments in the following.
The reviewer’s comments: “I am a bit confused about the actual aim of the paper. If the aim is to show that one can access Renyi entropies from neural networks wavefunctions (as the title suggests), I am afraid this is known already, see for instance the extensive discussion and numerics in [Nature Physics 14, 447450 (2018)]. Typically, one uses indeed the same replica trick as in quantum monte carlo. So the novelty of the paper may lie in the application of the numerical method to the quantum Ising model. But, as recalled for instance in Refs 19, 23, the model is exactly solvable. All the entropies can be computed exactly in the equilibrium case [https://arxiv.org/abs/quantph/0211074] and with timedependent modulations [23]. Finally, suppose the authors would consider instead a nonintegrable model, then the neural network method would show an advantage over tensornetwork methods if and only if the level of entanglement is such that it cannot be represented by a MatrixProductState. This is not the case for the data presented in this paper with entropy <=2.”
Authors’ reply:
The aim of our paper is to combine the neural networks quantum states and the “Swap” operations to study the Renyi entropy in transversefield quantum Ising model (TFQIM), both at the critical point and away from the critical point. This means we need to study the equilibrium case and dynamical evolution of the Renyi entropy.
We have to admit that in the static case, especially near the critical point, the Renyi entropy of TFQIM was previously studied by Ref.[22] ([Nature Physics 14, 447450 (2018)]) from the neural network methods. However, Ref.[22] did not study the Renyi entropy by quenching the system away from the critical point, which was done in our paper. You will see that the Renyi entropy away from the critical point after the quench is very interesting and comprises most part of our paper, including Fig.2, 3 and 4.
The novelty of our paper lies at the end of the quench, which is away from the critical point. In this case, we found coherent oscillations of the Renyi entropy in the end of the quench, and we explained it as the superposition of the excited states. As far as we know, this was the first time that this oscillations were observed from the Renyi entropy. Besides, from the Renyi entropy away from the critical point, we found a new length scale which was proportional to \sqrt{\tau_Q}\ln(\tau_Q). Of course this new length scale was also found in previous literatures, such as in Ref.[23]. However, Ref.[23] studied the von Neumann entropy (S_1) rather than Renyi entropy (S_2). We need to note that S_1 and S_2 have different definitions and different meanings.
Therefore, as we have clarified, the novelty of our paper lies in studying the Renyi entropy away from the critical point:
1.From the Renyi entropy away from critical point we observed the coherent oscillations which were induced by the superpositions of the excited states. This was reflected in Fig.2 in our paper.
2.From the Renyi entropy away from critical point we confirmed the theoretical formula Eq.(10) in our paper and found a new length scale as \sqrt{\tau_Q}\ln(\tau_Q). This was reflected in Fig.3 and Fig.4 in our paper.
The above two points are first observed and studied from the aspect of Renyi entropy.
In total, we appeal against the reviewer’s comments that our paper is not novel. We hope that our replies can help to clarify the novelties in our paper.
Best Regards,
The authors