Precision magnetometry exploiting excited state quantum phase transitions

We are very glad that the reviewer considers our manuscript well-written and interesting. We have addressed the issues raised by the reviewer, and listed the changes in the following.

Reviewer:
It is not clear if the different colors have a specific meaning in Figure 1a. If yes, can the authors please explain it?

Answer:
The colours in figure 1(a) do not have a specific meaning, but are intended to make the figure clearer for small magnetic fields when the curves become closer.

Reviewer:
Can please the author comment which techniques they use to compute the spectrum, even in the sector $s=N/2$ with dimension $N+1$ for the considered systems with a big number of spins, e.g., $N=12000$?

Answer:
We have computed the spectrum of the Hamiltonian in the sectors with $s=N/2$ and fixed parity (either even or odd) using exact diagonalization using MatLab 2018b and the command “eig”. The time required for exact diagonalization is of the order of several minutes even for N=12000. We have added the following sentence five lines after equation (1): “These symmetries allow us to compute numerical exact diagonaliziation of the LMG Hamiltonian in the orthogonal subspaces with $s=N/2$ and even or odd parity”.

Reviewer:
Reorganize the citations, avoid multiple citations to facilitate the readability. For example at the end of the second paragraph of page 4, the citation [50, 27, 51, 52, 23, 53, 47, 54, 27] can be rearranged to have [23, 27, 47, 50-54].

Answer:
The document class and the bibtex style in the previous version were responsible of the order of citations. After using the SciPost template, the citations are organized as the reviewer suggested.

Reviewer:
Is it possible to give the analytical expression of the vertical critical lines in the caption of Figure 2?

Answer:
We have added the analytical expression of vertical the lines in the caption of figure 2. These lines correspond to the critical fields $h_c^k$ of different eigenstates ($|E_k\rangle$ and $|\tilde E_k\rangle$), namely the field value such that $E_k=E_c=-Nh_c^k/2$. Therefore, $h_c^k=-2E_k/N$. The analytical expression of the eigenvalues $E_k$ is not known in general.

Reviewer:
I think there is an extra (d) In the second line in the caption of Figure 2, and that the expression critical fields for two adjacent excited states’’ should be explicated.

Answer:
We thank the reviewer for noticing this typo. Indeed, the first of the (d) has been replaced with (b). The expression “critical fields for two adjacent excited states” appeared in the paragraph starting with “We further compare…” at page 6. We have re-written that sentence. We have also replaced the term “adjacent eigenenergy gap $\Delta E=E_{k+1}-E_{k+1}$” with “eigenenergy gap $\Delta E=E_{k+1}-E_{k+1}$” in the caption of figure 2.

Reviewer:
I would suggest to rewrite subsection 4.1.1 about probe preparation, it is less clear than the rest of the manuscript.

Answer:
We have re-written subsection 4.1.1. In particular, we slightly changed the first paragraph in order to be a little bit more explicit concerning the reduction to the subsystem with $s=N/2$, and we have added several details on the phase estimation algorithm used for probe preparation in the subsequent paragraphs.

Reviewer:
I would also like to point out that (ground state and excited states) phase transitions of such type of critical systems are nowadays feasibly simulated on NISQ-hardware, see e.g., Phys. Rev. E 107, 024113 (2023).

Answer:
We thank the referee for bringing the interesting paper Phys. Rev. E 107, 024113 (2023) to our attention. It is indeed a promising study of simultations of the LMG eigenstates with measurements of the corresponding phase transitions on NISQ device. We have introduced very shortly the context and the terminology of NISQ technologies sentence in the introduction. We have then mentioned in the conclusions that the paper Phys. Rev. E 107, 024113 (2023), together with experimental platforms for realizing the LMG model, may inspire new implementations with NISQ devices.

- We have added the following sentence five lines after equation (1): “These symmetries allow us to compute numerical exact diagonaliziation of the LMG Hamiltonian in the orthogonal subspaces with s=N/2 and even or odd parity”.

- We have used the SciPost template, and the citations are organized as the reviewer suggested.

- We have added the analytical expression of vertical the lines in the caption of figure 2.

- We have re-written the sentence starting with “We further compare…” at page 6.

- We have re-written subsection 4.1.1. In particular, we slightly changed the first paragraph in order to be a little bit more explicit concerning the reduction to the subsystem with s=N/2, and we have added several details on the phase estimation algorithm used for probe preparation in the subsequent paragraphs.

- We have added the following sentence in the introduction “It is also desirable to investigate metrological schemes suited for several physical platforms and using different physical phenomena in the search for feasible implementations, so called noisy intermediate-scale quantum (NISQ) technologies [15,16]” in order to introduce the context and the terminology. We have also and added the following sentence in the conclusions “Eingenstates of the LMG model at small size can be simulated also with variational hybrid quantum-classical algorithms with low depth on superconducting transmon qubits, and signatures of their phase transitions can be computed [112]” after mentioning other realizations of the LMG model, and conluded with “The robustness of magnetometric protocols based on the ESQPT and the aforementioned variety of platforms for realizing or simulating the LMG model make these protocols good candidates for NISQ devices”.