A numerical approach for calculating exact non-adiabatic terms in quantum dynamics

1) The paper demonstrates a clear understanding of the state of the field and will no doubt provide readers with the necessary context to understand the author's perspective

2) The authors introduce a new perspective on known approaches to computing the AGP which should render the problem far more tractable in a number of practical cases

1) The paper has some structural issues. In particular the exposition of section 3 is both essential to communicating the author's contributions, but is also the weakest part of the paper in terms of presentation. Explaining the structure captured in figure 1 more carefully, perhaps in a fully worked out example, would be a nice change here. As written, I needed to consult other examples and appendices to understand what the authors were saying. A reworking of this section in relation to the rest of the text is probably necessary.

2) The paper is not written in such a way that its novel contributions are obvious. Comparing this paper with, for example, their reference [48], I see that this paper offers clever computational advantages over existing applications of the commutator expansion for the AGP. Is that the extent of the novelty? That the "punchline" contributions of this paper are not entirely clear is certainly a weakness.

If so, in order to meet the criteria of SciPost physics, it is necessary to demonstrate the computational power of these methods more directly

Given weakness (2) listed above, I am not currently convinced that the authors have met the journal's acceptance criteria. I think the authors have done good work and am inclined to give them the benefit of the doubt, but in order to approve the paper for publication I need to see the aforementioned weaknesses addressed. Particularly in regards to point (2), I think their position could be strengthened substantially by providing a direct comparison of some kind between their approach to computing the AGP and previous variational methods. As it stands, their primary contribution seems to be that their approach to the variational problem offers a new computational advantage; to meet the criteria of the journal, this advantage has to be demonstrated to be strong in a concrete way. At the moment, this is somewhat lacking.

1) A serious rework of section 3, which is essential to the paper's exposition, should be done. As a general rule, I would say that if the reader needs to consult the appendix to understand crucial details, more revisions are necessary. The algebraic structure the authors are discussing at the level of equations should be made clearer in section 3.

2) The authors should state more clearly exactly what the takeaway from this paper should be for the reader. My own sense of how computationally advantageous their formalism is is a bit muddled. This could be cleared up particularly by having a comparison with previously existing numerical methods.

3) This overlaps with point (2), but in restating what the key takeaways are, the authors should keep in mind the journal's acceptance criteria. It is not currently clear that they have been met.

Ask for major revision

1. The authors provide sufficient details of their approach.

2. The authors apply their method to the interesting system of Ising model on graphs.

1. The authors do not properly review existing methods that are very closely related to the method they introduce.
2. The application of their method to more general systems is not discussed.

The manuscript is appropriately prepared and the methods are explored properly.
The main advantage of their method over other methods of a similar nature are unclear. The manuscript will be suitable for the Journal once the authors have considered the changes requested.

I request the authors to address the following question in their manuscript:

Q 1: It is known that the choice of any orthonormal basis (Controlling and exploring quantum systems by algebraic expression of adiabatic gauge potential, PhysRevA.103.012220) will minimize the action. Depending on the problem, some bases are analytically tractable while others are not. The authors employ the Pauli basis for their computation. What are the advantages (in general) in using that basis as compared to say, the Krylov basis?

For context, the Krylov basis approach was developed for the AGP in
1. A Lanczos approach to the Adiabatic Gauge Potential (arXiv:2302.07228)
2. Shortcuts to adiabaticity in krylov space (arXiv:2302.05460)
and applied to a large class of problems. It is also known that the Krylov basis in general has lesser number of elements than the Pauli basis. Therefore it seems as if expressing the AGP in the Krylov basis is advantageous over expressing the same in the Pauli basis.

Q 2: The authors say a few words about the presence of a natural truncation point in one of their examples (due to their method). It was shown (arXiv:2302.07228) that the truncation of the Krylov chain connects to the chaoticity of the system.
Can the authors comment on the generic nature of the truncation point in their method vis-à-vis the Krylov method?

The manuscript is clearly organized and well written.
Providing a numerical algorithm helps people who want to use counterdiabatic driving.

Simple examples (the ferromagnetic Ising model with a transverse field on some graphs) are only studied.
Source code is not available.

In this manuscript, the authors provide a numerical algorithm for calculating the adiabatic gauge potential. The adiabatic gauge potential is a key idea of nonadiabatic transitions and it can also be used in assisted adiabatic passage, or in other words, shortcuts to adiabaticity by counterdiabatic driving. Their algorithm is based on some previous results, i.e., the variational approach (Ref.25), the algebraic approach (Ref.40), and an idea of nested commutators (Ref.48). They use the above results in an algorithmic way. I believe that the present manuscript is interesting, clearly organized, and well written, and thus it deserves to be published in SciPost Physics.

Optional comments:
1. n->N in Eq.24.
2. The model in Sec.4.1 is the transverse Ising chain with the periodic boundary condition, and the adiabatic gauge potential of it is well studied in the literature. The authors should cite some reference there.
3. The absence of a peak in the adiabatic gauge potential of the Lipkin-Meshkov-Glick model sounds very interesting because we have believed that it shows a significant peak at the critical point. I recommend emphasizing this finding.
4. The present paper is useful for people who want to use counterdiabatic driving. It becomes more useful if the source code (or package code) is available.