Report - Precision magnetometry exploiting excited state quantum phase transitions
The authors report two novel metrological protocols for the estimation of the magnetic field with a particular critical spin system with an infinite-range interaction. Their approach is based on the exploitation of excited state quantum phase transitions (ESQPT), and considering they focus on finite-size models the criticality enhancing the metrological protocols relies on the crossings of excited levels.
The authors characterize the spectral density of the ESQPTs and found that are around the critical point of the model in the thermodynamical limit. Moreover, within this interval, each excited state has a unique critical field, and at these points, the Quantum Fisher Information (QFI), which measures measurement precision, exhibits broad peaks.These spectral properties of ESQPT are essential for two efficient magnetometry schemes. The first scheme achieves sub-shot-noise accuracy with a small quantum register and a constant running time. The second scheme offers high accuracy with a logarithmic number of register qubits and polynomial running time, outperforming conventional estimations with fewer qubits. Importantly, these ESQPT-based magnetometry protocols are robust against various noise sources due to the characteristics of QFI peaks. While entanglement is usually crucial in quantum metrology, here, the critical behavior of the states plays a pivotal role in achieving enhanced precision magnetometry.
The paper is well written and presents very interesting and new results, so I recommend the publication. Anyway, I have a few suggestions to improve the clarity of the manuscript.
It is not clear if the different colors have a specific meaning in Figure 1a. If yes, can the authors please explain it?
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?
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]
Is it possible to give the analytical expression of the vertical critical lines in the caption of Figure 2?
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.
I would suggest to rewrite subsection 4.1.1 about probe preparation, it is less clear than the rest of the manuscript.
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).
Author: Ugo Marzolino on 2023-10-12 [id 4036]
(in reply to Report 1 on 2023-09-24)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.