Experimental protocol for observing single quantum many-body scars with transmon qubits

I agree with the 3 points listed by Anonymous referee #1 and don't need to repeat them.

It seems to me that the authors have missed an opportunity to argue for the importance of observing a single many-body scarred state where there isn't a tower of scar states.

I agree again with Anonymous referee #1 about the strengths and that this paper merits publication and won't repeat that. Instead, I would like to explain why it may be that the authors may have missed an opportunity to argue for the importance of what they are doing.

First consider simple isolated, bounded, quantum dynamical systems with classical analogs. It's extremely difficult to prove ergodicity for the classical analog. None of the proven examples I am aware of include smooth nonsingular continuous time Hamiltonians. In fact, loose arguments about near tangencies of stable and unstable manifolds seems to make it unlikely there exist such systems. Following Percival's argument in the early 1970's, both regular and irregular eigenstates would coexist in almost all smooth dynamical systems. For systems that are mostly chaotic or dominated by exponentially unstable dynamics, there would still exist the consequences of the Kolmogorov, Arnol'd, Moser (KAM) theorem, which is to say, very small locally stable regions of motion. Bohigas et al. Physics Report shows how that creates nearly evenly spaced sequences of regular states. These result from the quantization of the small stable dynamical regions (torus quantization) and are expected. The associated eigenstates are highly localized. There is nothing surprising about this.

The entire point of Heller's original introduction of quantum eigenstate scarring ('84) is that there is a weak form of eigenstate localization in the complete absence of any stable or effectively stable dynamics, and that this is the thing which is surprising. Furthermore, dynamical recurrences and revivals (not quite the same thing) are both associated with stable motion in such systems. There is a kind of "set of measure zero" exception discussed for chaotic revivals in the 90's, but unless specifically set up, one would not ever expect them.

These sorts of considerations translate fairly straightforwardly to many-body bosonic systems on a lattice, but of course it is much more difficult to translate these ideas to spin or fermionic many-body systems.

Nevertheless, it seems that if the so-called many-body quantum scars come with a tower of scar states and recurrences (i.e. uniformly spaced scar state energies, the distinction is that revivals need uniform 2nd discrete energy differences), then one's first thought is that perhaps there is nothing surprising about that, maybe it is associated with a small, but locally stable or effectively stable dynamics, and the many-body scarring or eigenstate localization in such a case is no more surprising than torus quantization is surprising (i.e. not surprising at all). One would have to demonstrate that there is no possibility of an effective locally stable dynamics for it to be related to scarring in Heller's original sense. However, the absence of a tower of scar states implies that there cannot be any such locally stable effective dynamics, and maybe then the use of the term "scarring" taken from the original context has a true many-body correspondence, and this is an indicator of a weak form of localization in the scarred eigenstate.

I would just request that the authors consider indicating some kind of a reasoning for why it is important to seek many-body quantum scars in a system without a tower of scar states.

Ask for minor revision

1 - The paper provides a very detailed experimental protocol on how to observe quantum scars in a superconducting system.
2 - It establishes a very good link between quantum many-body scars and superconducting circuits and opens a pathway towards new proposals for quantum many-body phenomenon.
3 - The paper is well-organized and structured.

1 - The paper lacks the comparison between single quantum many-body scars and tower of scars states and why the former is interesting and also difficult to observe.

In their manuscript, authors propose a protocol for the experimental observation of a single Quantum Many-Body Scar (QMBS). QMBSs are interesting due to their implications for thermalization theory and potential applications in quantum technologies, however authors claim that the experimental evidence so far has been limited to systems with towers of scar states.

The authors build on the theoretical framework presented in Ref. [10], where a family of 1D spin-1/2 models with single scar states was introduced. In this work, they identify specific Hamiltonians that can be realized experimentally using transmon qubits with fixed nearest neighbor couplings.

A key strength of the paper lies in its methodological innovation. It proposes novel experimental signatures to detect a single quantum scar state, combining trotterized dynamics, local observable measurements, and entanglement entropy quantification. I especially liked Fig. 1 where they describe the experimental procedure visually. Moreover, the paper is well organized, starting from an introduction to quantum many-body scars, setting the stage for the building blocks of the experiments, and to further validation of their proposed protocol through numerical simulations.

Overall, I liked the paper and I think it deserves a publication in SciPost Physics. However, I propose some changes in the manuscript to make this nice paper even better.

1) Introduction to QMBS: I really believe that a more extensive introduction to QMBS would be helpful for the reader. 2) Detailed explanation and comparison between single and tower of scar states: I think for a non-expert, the introduction really lacks a detailed explanation on both what is meant by single quantum scars and tower of scar states. The only relevant reference authors provide is Ref. [10] and this downgrades the paper's generality.

And a final suggestion:

3) Code and data availability: While the authors provide a lot of detail about their numerical simulations, publishing the code they used for the simulations would greatly enhance the impact of the paper.

Ask for minor revision