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Non-Markovian noise that cannot be dynamically decoupled

by Daniel Burgarth, Paolo Facchi, Martin Fraas, Robin Hillier

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Submission summary

Authors (as registered SciPost users): Daniel Burgarth · Paolo Facchi
Submission information
Preprint Link: https://arxiv.org/abs/1904.03627v4  (pdf)
Date submitted: 2021-04-09 01:35
Submitted by: Burgarth, Daniel
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Mathematical Physics
  • Quantum Physics
Approach: Theoretical

Abstract

Dynamical decoupling is the leading technique to remove unwanted interactions in a vast range of quantum systems through fast rotations. But what determines the time-scale of such rotations in order to achieve good decoupling? By providing an explicit counterexample of a qubit coupled to a charged particle and magnetic monopole, we show that such time-scales cannot be decided by the decay profile induced by the noise: even though the system shows a quadratic decay (a Zeno region revealing non-Markovian noise), it cannot be decoupled, no matter how fast the rotations.

Author comments upon resubmission

Reply to Report 1:
The referee appears to be mostly happy with the paper. We removed Fig. 2 as requested. Regarding the statement that our model is a toy model, see our reply to Report 2.


We provide a line-by-line rebuttal of the critique in Report 2.

“Looks like the numerical analysis has been included to make up for the lack of a solid mathematical result”. We would like to point out that numerical analysis is a recognised and solid mathematical discipline. It is common practice in theoretical physics to use numerics whenever exact proofs appear to be out of reach, and that is exactly what we did: we went analytically as far as possible and then continued numerically. We also included the source code so that the sceptical reader may verify the numerics for themselves. This makes the paper mathematically complete and solid.

“The counter-example is highly artificial.” Counter-examples to folklore statements tend to be less obvious, less natural and less intuitive - otherwise the folklore would not have arisen in the first place. We agree that the statement in consideration is probably true for a number of common models but our aim here is to show that it is not always true; hence one will have to be careful when using the statement and it will make sense to determine more refined conditions in future as to when it does hold. That said, our model is a mathematical toy model that fulfills all requirements of a quantum mechanical model and it is hence sufficient to contradict the folklore statement. Whether a model is relevant or physical is a rather subjective question and the task of “finding a relevant counter-example” is not at all well-defined. We are not in any way claiming that one will find our counterexample “in general” in the lab, although we showed in the magnetic picture that the Hamiltonian shares common features with those encountered, and we hope that our discussion in Section 5 about the possible addition of local bath and system terms will make you more convinced of its physicality. The singular features of our Hamiltonian are also shared amongst most of the models used in quantum open systems. So our counterexample can stand as a warning: here’s a new feature which people were previously unaware of why DD can fail. Here’s an example which shows that one has to be careful with lots of bread-and-butter techniques in open system and control (spectral density, perturbative treatments, bath-system decomposition) when there are unbounded baths. It also should be considered as a starting point and guidance - what features (additional hypothesis) would future proofs have to have so that the folk-knowledge is restored?


“The behavior displayed is not what practitioners using dynamical decoupling schemes would ever worry about. In fact, they would never consider the infinitely fast limit scheme which is the main mathematical focus here.” Practitioners in dynamical decoupling are interested in high decoupling pulse frequencies - typically, the higher the frequency the smaller the decoupling error, and the fundamental idea of dynamical decoupling is that the decoupling error should converge to 0 as the pulse frequency tends to infinity. If decoupling does not work in this limit then it won’t work for arbitrary finite frequencies either, so a statement about the limit is actually stronger and mathematically more elegant. Also, we would like to point out that in practice when working numerically, we always deal with finite frequencies, which is exactly what would happen in the lab. Hence the behaviour we study is highly relevant.

“The discussion over the counter-example and its significance is convoluted.” The referee does not provide any justification for this statement and it is unclear what could be changed about the exposition to address this concern.

“No evidence that this is actually a "folklore".” This is not true. On p.2 of the original version we introduced the statement and provided references [13,14]. [14] is a review article. In order to strengthen this point, we added further exemplary references [15,16,17] in the revised version. We have also replaced “folklore” with the more appropriate term “mechanism” in the manuscript.

“The non-Markovian model considered here is trivially a single particle”. Notice that despite its simplicity, the Lee Friedrichs model is fully Markovian even in the single excitation sector. Moreover, our model is also infinite dimensional (although not with infinitely many modes). The referee writes that we chose a non-Markovian model due to it being “flashy”. This is incorrect. It is already known that Markovian models cannot be decoupled, so to go beyond the state of the art we had to consider non-Markovian ones. We agree with the referee that Markovian dilations tend to be mathematically much more complex than non-Markovian ones. It is exactly in this spirit that it is surprising that even simple models such as ours cannot be decoupled. The purpose of the paper is to emphasise this to the community which believes otherwise (see above).

“The authors mention non-markovianity, but it is not clear what they mean by this.” This is not true, we did define it (comment [9]). This is clearly non-Markovian according to most measures, although some authors (including ourselves in other contexts) would also consider certain models (e.g, the Shallow Pocket model) with exponential decay non-Markovian.

“I am particularly puzzled by the following bizarre phrase in the conclusion...” What we meant by our original sentence is that our model should not be classified as pathological and hence be dismissed but that there may be additional meaningful hypotheses which could be added in order to exclude counterexamples such as ours, and that this is an important future line of studies. We have now changed that sentence in order to be less contentious.

List of changes

Changes we made to the manuscript:
* We added more references in Section 1 to show that the Zeno effect is indeed used as a mechanism to explain dynamical decoupling and have replaced “folklore” with “mechanism”.

* We rephrased the sentence in Section 5 that Referee 2 had flagged up.

* We removed Fig. 2 as requested by Referee 1.

Current status:
Has been resubmitted

Reports on this Submission

Report #3 by Anonymous (Referee 1) on 2021-6-26 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1904.03627v4, delivered 2021-06-26, doi: 10.21468/SciPost.Report.3127

Strengths

1. The results are intriguing
2. The topic is of timely interest.
3. The result is surprising (at least at the first sight)

Weaknesses

1. The model is pathological

Report

This paper presents an interesting result on dynamical decoupling. Using a artificially defined model as a counterexample, the authors show that it is not always true that the dynamical decoupling works if the initial decoherence of a qubit exhibits the "Zeno" behavior, an indication of non-Markovian quantum baths. The math looks rigorous and the calculated results are reasonable and consistent with the analysis given by the authors.

The effect should come essentially from the Dirichlet boundary condition, which amounts to an infinitely high potential for x<0. With such a potential, the backaction of the dynamical decoupling at higher frequencies would excite higher states in the "bath", which is evident in the wavefunctions shown in Fig. 3. One can argue that if the energy spectrum of the bath is finite or if the potential is smooth, the pathological behaviors of the the dynamical decoupling on the special model would disappear. The authors may want to comment on such an aspect.

Nonetheless, the results in this paper are interesting, in that they show the importance of backaction of dynamical decoupling and may have applications in non-pathological models. For example, to dynamically decouple an interacting bath prepared in a low-energy state, one should design the dynamical decoupling such that it would not cause the excitations of high-energy modes or the DD scheme could be spoiled.

Requested changes

Add comments on the picture of excitation of high energy states by the backaction of the fast DD.

  • validity: good
  • significance: good
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Report #2 by Anonymous (Referee 2) on 2021-6-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1904.03627v4, delivered 2021-06-22, doi: 10.21468/SciPost.Report.3095

Strengths

1) Dynamical decoupling is a timely topic.
2) The work appears to be mathematically sound.
3) The paper is easy to read.

Weaknesses

1) The example considered is artificial.
2) The title and abstract are misleading and overstate the actual results.

Report

The authors present a simple model of a qubit coupled to a particle confined to a half-line. They show that a basic dynamical decoupling protocol consisting of a periodic train of instantaneous pi pulses fails to decouple the qubit from the particle in a certain parameter regime. They support this finding with a combination of analytical arguments and numerical calculations.

I agree with the previous referees that the example given is highly artificial and will likely not change the course of anyone’s research, even of those directly working on dynamical decoupling, as it is far from clear whether it has any consequences for physically relevant systems. The authors point out that the half-line model can be reconstrued as a model of a particle on a full line interacting with a magnetic monopole. However, given that the existence of magnetic monopoles remains dubious, I don’t think this makes the model any more physically relevant. This being said, the result could still be of some value if it eventually leads to insights about dynamical decoupling applied to more relevant systems.

Aside from the artificial nature of the example presented, another issue that concerns me is that the title and abstract are misleading. For example, the title reads “Non-Markovian noise that cannot be dynamically decoupled”. Even if one accepts the definition of non-Markovian noise used in this work, it cannot be claimed that this noise “cannot be dynamically decoupled” based on the results presented. The authors only show that one type of dynamical decoupling fails to do the job; as far as I can tell, they do not show that all possible DD schemes will also fail for this model. I appreciate that the authors are aiming for mathematical rigor in their analysis; I would encourage them to also aim for rigor in their language.

Another statement that I found troubling is the following sentence near the top of page 3: “The example also shows, as a byproduct, that even in 1D, quantum magnetism can be interesting.” This simple and unnecessary sentence dismisses a vast literature on magnetism in one dimension. See for example this old article in Physics Today: Physics Today 31, 12, 32 (1978). The sentence should be removed.

In summary, I recommend publication after the above changes are implemented.

Requested changes

1. Change the title and abstract to better reflect the actual findings.

2. Remove the sentence about 1D magnetism on page 3.

  • validity: ok
  • significance: low
  • originality: ok
  • clarity: good
  • formatting: good
  • grammar: good

Report #1 by Anonymous (Referee 3) on 2021-6-6 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1904.03627v4, delivered 2021-06-06, doi: 10.21468/SciPost.Report.3027

Strengths

I basically confirm my viewpoints as in my previous report, namely:

1) timeliness of the topic
2) mathematical rigour
3) it challenges common wisdom with a specific counterexample

Weaknesses

1) And I still think that the model analysed is somehow a toy model, most likely it can be physically simulated but it does not map straightforwardly to a ready in the lab setup.

Report

In this revised version I tend to side with the authors. Although I agree with my fellow referee on the fact that the model analysed is rather artificial, its study is very much in the style of mathematical physics, where the search of counterexamples of standard views is a well-established area of research. Furthermore, I do not think that numerical analysis in this work masks a lack of rigor. Therefore, although I think that the manuscript will appeal more to the community of mathematical physicists than to theorists or experimentalists working on dynamical decoupling I am in favor of its publication.

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: excellent
  • grammar: excellent

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