Measuring Renyi Entropy in Neural Network Quantum States

- as far as I can check, the numerical calculations look correct

- 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 2-Renyi entropy at the end of the quench.

This manuscript describes a neural network approach to the study of the Renyi-2 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 Renyi-2. 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 Instead, this literature is completely ignored and the authors only cite a recent (yet unpublished) paper of them (posted on the arXiv, and which indeed contains most of the relevant citations).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 out-of-equilibrium 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 out-of-equilibrium problem reached by the community, beyond the Kibble-Zurek 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 zero-field 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 symmetry-preserving 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 x-direction": 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 x-direction". 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 (non-complete) 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 Fubini-Study 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 Renyi-2;
- 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...

Complete rethinking of this work

1- The results seems to be correct and well explained
2- The manuscript is accessible to a broad audience

1- There is no apparent novelty in the techniques used, nor in the studied physical model

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 many-body 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 Kibble-Zurek 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, 447-450 (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/quant-ph/0211074] and with time-dependent modulations [23].

Finally, suppose the authors would consider instead a non-integrable model, then the neural network method would show an advantage over tensor-network methods if and only if the level of entanglement is such that it cannot be represented by a Matrix-Product-State. 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