Efficiency of Dynamical Decoupling for (Almost) Any Spin–Boson Model

We thank the referees for their positive feedback and their insightful suggestions to improve the manuscript and its presentation. In the following, we address each point mentioned by the referees following the same paragraph structure.

Referee 1

As suggested by the referee, we included the two references on error estimates for dynamical decoupling of finite-dimensional Hamiltonians with non-equidistant pulses in the introduction.

We agree with the referee that the filter function approach is a useful approach in practice to design suitable dynamical decoupling sequences. We purposely emphasized the weaknesses of this approach in the paper to contrast it with the Trotterization approach that we employ in the paper. Both methods have their own advantages and disadvantages. While Trotterization provides rigorous error estimates, it does not help with comparing different decoupling sequences or engineering optimal decoupling sequences for a given noise profile. This is where the filter function approach is very valuable. Nevertheless, from a purely mathematical perspective, the filter function approach is not well-defined in many cases. As it can still give valuable insights in certain regimes, we added two sentences to the introduction highlighting the usefulness of this approach:
- “On the other hand, filter functions are a useful tool to compare different or design optimal decoupling strategies in the perturbative regime [19].”
- “Furthermore, they might help to identify the perturbative regime, in which the filter function approach is favourable.”

Indeed, a physical discussion of Assumption 5.1 would be valuable. We added such a discussion below the statement of the assumption, in which we also refer to the concluding remarks where we point out an avenue of how to potentially relax this assumption.

In general, the Liouvillian will be an unbounded operator on Liouville space if the corresponding Hamiltonian is unbounded. In fact, the situation is even worse: It can happen that the Liouvillian is doubly unbounded even if the Hamiltonian is only semi-unbounded. The reason why the quantities in our error bounds remain finite is because the Gibbs state acts as a regularizer. Under our assumptions, at least the first four moments of the Liouvillian and its rotated versions in the Gibbs state are well-defined and finite. To clarify this point, we added a comment at the end of Section 4.1.

It is true that our bounds are quite loose in their absolute value when compared to the numerical simulations performed in this paper. Indeed, this shows that there is potential to improve our results. This is not surprising given the proof method we use, and we also expect a similar behaviour for other norms. We now explain where in the proof the overestimation of the error comes from, see the newly added paragraph at the beginning of Section 6. Furthermore, we added Remark 4.4 at the end of Section 4 with an abstract result on the somehow more natural trace norm error that also motivates using the Hilbert—Schmidt norm for the explicit decoupling bounds. In addition, we found a small inconsistency in the normalization we used for the Trotter error bound and the decoupling error bound. This led to a slightly larger pre-factor in Theorem 5.3 and our model-specific bounds than necessary. More concretely, the bound was missing a pre-factor of $1/L^2$ coming from Trotterizing between operators $(1/L ad_{H_j})$ instead of just $ad_{H_j}$. We corrected this inconsistency in all affected analytical expressions. Furthermore, we corrected this in the plots that compare the bounds with a numerical simulation. Here, the simulation results were correct but the shown bounds were unnecessarily large. To make the changes clearer, we added further explanation to the proof outline of Theorem 5.3. Through these changes, the bounds became tighter than before.

We thank the referee for pointing out another interesting avenue for extending our results. We added this as a separate point (point (v)) in the Concluding remarks, where we discuss potential directions for generalizing and improving our results. Here, we also comment on a possible way how to obtain the result mentioned by the referee.

Indeed, there was a typo in the inequality. This is corrected now.

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Referee 2

1. We included the mentioned references in the introduction after Equation (1). We thank the referee for pointing out these references.

2. It is true that the assumption in Equation (19) seems to be unnatural when considering Ohmic spectral densities. However, this is not a problem. In fact, as discussed in the paragraph between Equation (22) and (23), the assumption in Equation (19) is equivalent to $m>-\infty$ after shifting the zero-point energy. W.l.o.g., this can always be done (also see the reply to the third comment by the referee) so that Ohmic spectra can be tackled through our method. For clarity, we added a comment on Ohmic spectra in this paragraph.

3. The bath Hamiltonian is bounded from below, and by assumption, we are working in a model with discrete bath modes. This means that there can only be finitely many negative $\omega_k$. Therefore, shifting the excitation energies is always possible w.l.o.g. We are grateful to the referee for pointing out that this has not been clearly presented and added a comment between Equation (22) and (23).

4. We thank the referee for pointing out the existence of earlier work on the statement of Theorem 4.1. Indeed, this statement was first proved by Kato in 1978 in the context of the path integral. Here, one is interested in the Trotter splitting between kinetic and potential energy, i.e., in Trotterizing between two Hamiltonians. In dynamical decoupling, one naturally requires Trotterization between more than two components. This is why, to the best of our knowledge, a generalization of Kato’s result to more than two terms has only recently been obtained in the context of dynamical decoupling (in 2018 in the reference we cited). Nevertheless, it is definitely worth citing Kato’s original work here, and we included this reference.

5. We thank the referee for pointing out that the presentation of Assumption 3(i) might have been confusing. In fact, both assumptions are equivalent after shifting the zero-point energy, which is always possible (again, see our reply to comment 3 by the referee). We made this clear now in the caption of Table 1, also indicating how our error bounds would change through the energy shift.