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

The authors derive mathematically rigorous bounds for
dynamic decoupling for models with bosonic baths. The
models are fairly general for which the bounds apply.
This is remarkable because the bosonic Hamiltonians are
unbounded by construction. Thus, I am in favor of publication
as SciPost Core or SciPost Physics.
Still, there are a number of points which can be improved.

In the Introduction, I find it appropriate to mention
also the previous estimates obtained for finite Hamiltonians,
in particular for non-equidistant pulses, see e.g.
author={G. S. Uhrig and D. A. Lidar}, title = {Rigorous
Bounds for Optimized Dynamical Decoupling},
journal= Phys. Rev. A, volume=82, pages=012301, year = 2010
or
author = {Y. Xia and G. S. Uhrig and D. A. Lidar}, title = {
Rigorous performance bounds for quadratic and nested dynamical decoupling},
journal = Phys. Rev. A, volume = 84, pages = 062332, year = 2011.

Similarly, the authors need not criticize the filter function
approach as strongly as they do. It is a very useful tool
to design experiments, see e.g.
author= {M. J. Biercuk and H. Uys}, title =
{Dynamical decoupling sequence construction as a filter-design problem},
journal=J. Phys. B, volume=44, pages= 154002, year=2011
even if does not provide mathematically rigorous statements.

The authors should discuss the physics behind their formulae wherever
this is possible. In particular, the existence of non-existence
of a UV cutoff is crucial, see e.g. the condition Eq. (56).
A Lorentzian resonance would not comply with this condition, but
still the dynamic decoupling works. This can indicate directions
for improvement of the bounds.

For the qualitative understanding why the Liouvillean is bounded
even for an unbounded Hamiltonian a very simple example
would be very helpful. At least, this crucial point should be exposed
clearly for the general reader.

Generally, the bounds are still quite loose (generically two
orders of magnitude) which is not uncommon in mathematical physics.
The authors should at least discuss why this is so?
Which effects are so strongly simplified in the estimates?
Would another norm be advantageous?

In all estimates presented, the error scales like 1/N. This property
is inherited from the Trotter formular. The authors also highlight
that the properties of the Trotter formulae are important by themselves.
Wouldn't it be an intriguing to improve the scaling to 1/N^2 or even
higher? This should be possible with non-equistant pieces of Hamiltonians
as in iterated universal dynamic decoupling, see
author = {G. S. Uhrig}, title = {Exact Results on Dynamical
Decoupling by $\pi$-Pulses in Quantum Information Processes},
journal= New J. Phys., volume=10, pages=083024, year = 2008
Of course, I am not demanding to include such a derivation
in the present manuscript. But it could be briefly mentioned and
discussed in the outlook.

It seems that on page 28 there is an error in the second mentioned
inequality, please have a look.

Each paragraph of the report given above suggests a certain change.

Ask for minor revision

Efficiency of Dynamical Decoupling for (Almost) Any Spin–Boson Model by Alexander Hahn, Daniel Burgarth, Davide Lonigro

The manuscript investigates dynamical decoupling schemes to suppress dissipative environmental influence in the dynamics of quantum systems. Specifically, a two-level system coupled to a bath is studied. The authors derive analytically a bound on the convergence speed with the number of decoupling pulses within a given time and compare their bound with numerical results. As important technical steps several known results for pure states have here been extended to mixed states.

The results are interesting and the manuscript well written. Before a final recommendation to publish I would ask the authors to address the following points:

1. Discussing Hamiltonian (1) one might want to refer also to -- U. Weiss, Quantum Dissipative Systems, 4th ed. (World Scientific, Singapore, 2012). -- A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys. 59, 1 (1987).

2. Eq. (19) means that the spectral density can not go analytically down to zero (even with value zero at w=0). That is unusual given that typically Ohmic spectra are investigated. This unusual choice should be mentioned / discussed a bit clearer.

3. page 10: Remark: '...the general case can always be recovered via a shift of the modes ωk .' Excitation energies can only be shifted as long one never touches zero.

4. For proof of (35) is a recent reference given. The fact (35) is, however, already the base of the (discrete) path integral formalism. Thus, I would expect earlier investigations.

5. Assumption 3.1(1) is different from the one stated in Table 1. Both seem very different. Which one is correct ?

Ask for minor revision