Quantum dynamics in transverse-field Ising models from classical networks

1- Presents a systematic method, which can be used to study quantum dynamics in a general system.
2- The method can be used also in higher dimensional systems.
3- The method works - in principle - for general quantum systems having short-range interactions.
4- The method provides accurate numerical results for short times.

1- The accuracy of the method is limited to short times.
2- Higher order (3rd, 4th, etc.) expansions seem to be very cumbersome.

In this paper the quantum dynamics of transverse-field Ising models is studied in a formalism of classical networks, which is evaluated by standard Monte-Carlo methods. The method, based on the cumulant expansion seems to be quite general, which can - in principle - be used also in higher dimensions and can be generalised by other quantum systems with short-range interactions. Using this method the authors have calculated the time-dependence of local observables, near-neighbour correlations, two-site entanglement and the rate function of the Loschmidt echo. The results are accurate for short times and in principle can be improved by considering higher order terms of the expansion.

The subject of the paper is of interest of researchers is quantum physics. It is basically well written and merit publication in the SciPost, however some points needs to be clarified.

i) The presented examples in the text are for a small perturbation term: $h/J=0.05$ and mainly in first order of the cumulant expansion. The authors should try to estimate the time-window in which the higher order expansion terms are still accurate.

ii) In two-dimensions the transverse-field Ising model has been studied before in Refs.[20,21]. The authors should try to compare their numerical results with the previous one, if possible.

iii) The results in the paper are about local observables, near-neighbour correlations and two-site entanglement. Can the method also be used to calculate long-range correlations, block-entanglement, etc.?

iv) Fig. 1a appears quite early and needs more explanation to be understandable.

v) The quantity "maximal bond dimension" should be defined in context with the iMPS.

vi) For the transverse-field Ising model in 1d the time-evolution after a quench of several observables in the free-fermion method have been first calculated in Phys. Rev. A 2, 1075 (1970); Phys. Rev. A 3, 786 (1971); Phys. Rev. A 3, 2137 (1971); Phys. Rev. Lett. 85, 3233 (2000) and Phys. Rev. A 69, 053616 (2004). These papers are recommended to be cited.

1- Fig. 1a appears quite early and needs more explanation.
2- The quantity "maximal bond dimension" should be defined in context with the iMPS.
3- For the transverse-field Ising model in 1d the time-evolution after a quench of several observables in the free-fermion method have been first calculated in Phys. Rev. A 2, 1075 (1970); Phys. Rev. A 3, 786 (1971); Phys. Rev. A 3, 2137 (1971); Phys. Rev. Lett. 85, 3233 (2000) and Phys. Rev. A 69, 053616 (2004). These papers are recommended to be cited.

1) Introduces a new ansatz based on classical networks to obtain dynamics of quantum many body systems
2) This approach is more accurate than usual perturbation theory
3) The method works in any dimension and is not restricted to 1D or quasi-2D, as MPS/DMRG methods are.
4) Getting the classical network can help to develop neural network ansatzes for the quantum dynamics of generic many body systems, which recently has been found to work reliably in 2D Ising systems (see Ref. [23] in the manuscript).

1) The results compare well with the exact solution only at small times.
2) As an outlook, the possibility to further develop the method is described, but it is difficult to estimate how well this might work.

The authors introduce a method to compute the time evolution of quantum many body systems via a mapping to classical networks, whose properties are then evaluated using classical Monte Carlo techniques. The results presented are obtained for the transverse field Ising model in 1D, 2D and 3D. Local observables as well as (short-range) correlations are computed, as well as the entanglement entropy and the rate function of the Loschmidt amplitude. The latter allows one to investigate for dynamical quantum phase transitions, which occur as non-analytic behavior in the course of time when going to the thermodynamic limit.
The approach is a very interesting idea and opens the path to further developments. As the authors show, it works reliably at short times, but looses its accuracy on time scales smaller than the typical ones accessible to MPS methods in 1D. However, the approach is more flexible than MPS since it is not restricted to low-D systems. In addition, the results shown most probably have room for improvement by going to higher orders of the cumulant expansion used, or by using other techniques for evaluating the networks. Already on the level presented, the results are more accurate than perturbation theory, which is rather encouraging. The manuscript is well written and I think is ready for publication in SciPost Physics after including the following minor changes:

- Figure 1a) is hard to understand and needs better explanation, either in a legend, or in a much more detailed caption.

- The authors report that in their classical Monte Carlo approach, negative probabilities appear. This sign problem is apparently resolved using the approach introduced in Ref. [61] of the manuscript. However, this is only briefly mentioned in the main text on page 7 and then discussed in some detail in the appendix. I would find it useful to say more about the significance of this problem for the calculations and its solution in the main text.

- I find Sec. 2.5 a little confusing: apparently, a mapping to an artificial neural network (ANN) was done and the time evolutions of the couplings are discussed in Fig. 5, but no results for observables is presented. It is not clear to me if there is a deeper reason for this, or whether the results obtained by an ANN will be identical to the ones already shown before.

- In Fig. 6, probably the black line shows the exact results; for the sake of clarity I suggest to add a legend saying this and to describe the different colors used, or at least mention in the caption what the various colors mean.

- In Appendix A.3, as complementary information the authors discuss the complexity of an equivalent iTEBD calculation. In the Conclusions section, this is referred to, but it is not really discussed how the computational complexity of both ansatzes compares. To be more specific, the authors say that the network ansatz in first order keeps only three couplings, while the iMPS needs 64. However, this does not automatically imply that the approach with the fewer couplings needs smaller computational resources, and one needs to keep in mind, that a good accuracy is obtained only at small times when using the networks. I would find it useful to comment or compare in the appendix the computational resources (e.g., CPU-time, memory) needed to reach a faithful result at short times (e.g., times < 2) using both approaches.

1) improve Fig. 1a and caption
2) better discuss negative probabilities in the main text
3) results for observables using ANNs?
4) improve Fig. 6
5) comment and/or compare resources needed for iMPS and the network ansatzes.