Counterdiabatic Hamiltonians for multistate Landau-Zener problem

1. Analyses are clear.
2. Presentation is good.
3. The result is generic and potentially useful.

1. The main claim is not clear.
2. The statement with respect to Eq. (45) is in question. See Report below.

The Landau-Zener problem is a well-known problem in a two-level system. It was solved exactly and applied widely in chemistry as well as physics. The generalization of the Landau-Zener problem has drawn a lot of interests so far. The authors of this manuscript analyzed multi-level Landau-Zener problems using the method of auxiliary Hamiltonians developed by Sinitsyn et al., and gave approximate but potentially useful formulas to compute the transition probabilities, including the transition from the ground state to ground state, for a generic Landau-Zener type of time dependent Hamiltonian with multi levels.

I would like to give one comment and one question on this manuscript as follows.

Comment. The claim of this manuscript sounds vague. To my understanding, the main point of this manuscript should be the invention of an approximate computation method for the transition probability between the initial and final ground states in a generic multi-level Landau-Zener-type model. However, the manuscript stresses auxiliary Hamiltonians rather than transition probabilities. I suggest that Introduction should be revised so that the letter is stressed more.

Question. It is stated that Eq. (45) is exactly the same as the Brundobler-Elser (BE) formula in Sec. 4. This statement is surprising and mysterious to me. This is because, while the BE formula is exact, Equation (45) is obtained after the zeroth order approximation in g of Z_1(tau) and the second-order cumulant expansion. The statement implies that, although the approximation used here should be valid for small g and delta, the resultant formula is exact for any g and delta. Why does this happen? I would like to ask the authors to solve this mystery.

Apart from the above comment and question, this manuscript is technically sound. The analysis is sufficiently clear. After a minor revision, this manuscript will deserve to be accepted for publication.

1. Revise Introduction so as to stress the approximate computation method for the transition probability between the initial and final ground states in a generic multi-level Landau-Zener-type model.

2. Add an explanation to resolve the question on Eq. (45).

1. The paper contains a set of very instructive calculations, which might also be useful for people from other fields.

2. The presentation is rigorous and clear.

3. The language (English) probably needs a revision.

1. The content lacks novelty. There is no result that adds to the general scenario known already.

In this work, the authors study the multi-level version of the Landau Zener problem in the light of short-cut to adiabaticity. An exact expression for a counter-adiabatic term has been calculated, and it has been shown that the counter-adiabatic Hamiltonians satisfy the zero curvature condition.

The work, though not particularly novel, contains nice and rigorous calculations that might be useful in other contexts. I hence recommend publication of the work in SciPost.

1. The language has to be revised thoroughly.

1. Technical correctness
2. Making interesting connections between close research areas and posing some interesting questions for further study
3. Good presentation
4. Language

1. The approach used is not very original.
2. This work presents a rather small incremental progress.
3. Partial list of references

The authors employ the method of auxiliary Hamiltonians for studying multiple Landau-Zener transitions. Among other results, they prove an equivalence between such Hamiltonians and the counterdiabatic terms appearing in the family of methods known as shortcuts to adiabaticity. Although I believe that the main ideas behind this work were already presented in references [19]-[22], the paper is very clear, scientifically rigorous and well-written. In fact, it really excels in these parameters. For that, I am glad to recommend its acceptance upon the following:
1. Since the novelty here is not very clear, I'd suggest the authors to denote it explicitly and possibly think of other means to enhance the main message in comparison to previous works in this area.
2. In the introduction the authors write: "The same type of the Hamiltonian is used in the method of quantum annealing [13–17]". This sentence is very general and not entirely precise. I'd encourage the authors to elaborate more on this topic.
3. With respect to the latter point, but also in general, I'd like to recommend the authors to use "Quantum Spin Glasses, Annealing and Computation" by Tanaka, Tamura and Chakrabarti (e.g. Chs. 5-7). Additional references to the literature would also be appreciated.

1. Since the novelty here is not very clear, I'd suggest the authors to denote it explicitly and possibly think of other means to enhance the main message in comparison to previous works in this area.
2. In the introduction the authors write: "The same type of the Hamiltonian is used in the method of quantum annealing [13–17]". This sentence is very general and not entirely precise. I'd encourage the authors to elaborate more on this topic.
3. With respect to the latter point, but also in general, I'd like to recommend the authors to use "Quantum Spin Glasses, Annealing and Computation" by Tanaka, Tamura and Chakrabarti (e.g. Chs. 5-7). Additional references to the literature would also be appreciated.