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Nonlocal correlations in noisy multiqubit systems simulated using matrix product operators
by H. Landa, G. Misguich
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Submission summary
Authors (as registered SciPost users):  Haggai Landa 
Submission information  

Preprint Link:  https://arxiv.org/abs/2203.05871v2 (pdf) 
Date submitted:  20220812 07:02 
Submitted by:  Landa, Haggai 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
We introduce an opensource solver for the Lindblad master equation, based on matrix product states and matrix product operators. Using this solver we study the dynamics of tens of interacting qubits with different connectivities, focusing on a problem where an edge qubit is being continuously driven on resonance, which is a fundamental operation in quantum devices. Because of the driving, induced quasiparticles propagate through the qubits until the system reaches a steady state due to the incoherent terms. We find that with alternatingfrequency qubits whose interactions with their offresonant neighbors appear weak, the tunneling quasiparticles lead to large correlations between distant qubits in the system. Some twoqubit correlation functions are found to increase as a function of distance in the system (in contrast to the typical decay with distance), peaking on the two edge qubits farthest apart from each other.
Author comments upon resubmission
We would like to thank the referees for reading the manuscript and providing useful feedback.
Following the referees' suggestions we have improved the clarity of the manuscript and the analysis of the results, and added new numerical simulations that solidify our results.
We believe that the novelty of the obtained results and the importance of the techniques to a multidisciplinary audience in the field of open quantum system dynamics merit publication in SciPost Physics.
We provide a list of changes and we reply to each referee's report separately.
Kind regards,
the authors.
List of changes
 We have emphasized in the introduction and conclusion sections the relevance of the models studied in the work as being a first step towards simulating many body dynamics of quantum devices. We have clarified the discussion of the physical motivation and the main results in these sections.
 We have added simulations of ZZ interaction (together with the XY coupling) to the manuscript. Perturbative ZZ interactions are a significant source of uncontrolled dynamics in superconduting devices we chose realistic coupling strengths accordingly. Thosee results are discussed in Sec. 3.4 (towards the end) and in Sec. 4, and presented in detail in App. G.
 We added in Appendix C a study of the quantum Ising model on a ring with strong dissipation. The results coincide with those of earlier literature on this setup.
Current status:
Reports on this Submission
Anonymous Report 2 on 2022923 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2203.05871v2, delivered 20220923, doi: 10.21468/SciPost.Report.5749
Strengths
1 Link to a well written open source code
2 Some appendices are quite pedagogical and generally the paper is well written
Weaknesses
1 No new surprising discovery
2Structure is a bit confusing
Report
Unfortunately I don't see any of the acceptance criteria for SciPost Physics fulfilled.
While it is nice that the paper introduces an accessible open code, I agree with the comment of a previous report: The paper demonstrates the applicability of the code for computing dynamics in two toymodel setups, but it does not present/analyze any real surprising physics discovery. I find this also to be true after the revision.
The discussion on the longdistance correlation (which could be an interesting lead) is too short and superficial, and it seems a bit lost after the "warmup" simulations. Interesting questions would have been for example to analyze this correlation more deeply. Is there entanglement between the distant qubits? Probably this is a classical correlation given the large entropy and the plateau of the OSEE. I find an emergence of such a correlation for a specific mesoscopic setup not very surprising. Generally, I also think that the paper does not put these findings clearly into the context of the many known results of the vast literature on e.g. spreading of correlations.
Furthermore the connection to the IBM device is only vague ("a first step" towards a simulation as the authors describe it themselves). For an analysis of "unexpected correlations" in a real device, finite temperature would need to be included or the validity of the Lindblad approach would need to be tested.
Finally, also in terms of the numerical method there are no new algorithmic advances.
In conclusion, I don't think the paper has enough novelty for SciPost Physics and I don't see any potential for it to open up a new research direction. Nevertheless it is solid research and deserves a publication e.g. in SciPost Physics Core.
Requested changes
1 I'm a bit confused about the presentation of the Lindblad terms: rates g_0, g_1, and g_2 are introduced, but I think only g_0 and g_2 are used in the main text? Furthermore, when talking about "energy relaxation", I suppose they refer to spontaneous emission. The latter would be connected to g_1, so did the authors confuse g_0 and g_1, or is it just an unconventional definition of the spinlowering/raising operators? This should be written more clearly.
2 Some text pieces are too vague, in particular:
Section 2.5: The part with the "1D path" geometry is quite hard to understand for someone not familiar with the method. A little sketch would help? Furthermore, here it would be more useful to say a sentence on the type of densitymatrix linearization used, e.g. referring to Eq. (40).
Section 3.4, bottom paragraph on page 15: The descriptions here (about the OSEE) are very vague and handwavy. E.g. it's not clear how the authors distinguish classical and quantum correlations. Also the connection to the bond dimension does not become clear and is not entirely precise. I suggest to be either more mathematical or to cut pieces and replace them with references.
4 I suggest to modify the structure of the paper. There are a lot of appendices, some with interesting results on correlations (which I find more interesting than the chain results in Sec. 3.2). I would propose to make one clear section with the correlation results for the IBMtype setup, and one for other "benchmarktype" simulations, both in the main text.
Report
The authors implemented the suggestions I pointed out in my first report. In my opinion the manuscript is now suitable for the publication in SciPost Physics in its present form. In particular the manuscript satisfies the criteria 3 : Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work;
Author: Haggai Landa on 20221125 [id 3071]
(in reply to Report 2 on 20220923)We would like to thank the referee for her/his reading of the manuscript and valuable feedback.
Following the referee's suggestion to quantify entanglement in the system, we have looked at the concurrence as a measure of nonseparability of the edge qubits and added a plot of its dynamics for different system sizes, together with a discussion of this point (in Sec. 4.2). The suggestion raised by the referee that the correlation is classical is correct in the steady state. This is an expression of the fragility of the entanglement, which is initially mediated by the propagating excitations that manifest rich dynamics. The eventual state, though not entangled, is nontrivial and correlated, and we think that this setup merits further exploration. Our results exemplify a stateoftheart tool allowing to study the dynamics of correlation spreading in systems relevant in various subfields of open quantum systems.
For the referee's comment that "finite temperature would need to be included or the validity of the Lindblad approach would need to be tested", we note that energy exchange with a finite temperature Markovian bath is already supported in the model (since the Lindblad equation includes relaxation and excitation operators). In qubit devices the environment temperature is typically significantly lower than the qubit's energy gap, and hence keeping the spontaneous emission terms only (and dephasing of course) is typically an accurate approximation. We have added a comment regarding this point together with a reference.
For the referee's requested changes: 1. Indeed with superconducting qubits the Hamlitonian is often taken with a negative sign for the singlequbit \sigma^z (energy terms), as it appears in Sec. 3.1 of the paper, making up> the qubit's ground state, and hence the spontaneous emission rate is g_0. We have clarified this source of confusion in the text. 2. About Section 2.5: we have improved the explanation of the "1D path". We have in particular made connection with Eqs. 4142 (product of matrices) in App. A, and we are now also referring to a graphical illustration presented in Ref. 35. 3. About Section 4.1: The OSEE indeed does not tell if the correlations are mostly classical or quantum. In other words, it is not an entanglement measure. We added this remark as a footnote in Sec. 4.1. Regarding the lack of details concerning the connection between the OSEE and the bond dimension, this connection is in fact not entirely direct and making a comprehensive mathematical statement is not easy, going beyond the scope of the current discussion. We have added a reference that discusses this question (in the closely related framework of MPS). 4. We have implemented the referee's suggestion and created separate sections for the chain dynamics and for the plaquette dynamics. We have shortened the former and have incorporated some correlation results from the appendix into the latter.
To conclude, we wish to thank the referee again, and resubmit the amended manuscript to SciPost Physics Core.