Generalized Gibbs ensembles in weakly interacting dissipative systems and digital quantum computers

- Tackles a difficult problem: development of effective theories for non-integrable open quantum systems.
- It develops connections with state-of-the-art experimental platforms (digital quantum computers).

See the report and the list of main issues.

Dear Editor,

the article "Generalized Gibbs ensembles in weakly interacting dissipative systems and digital quantum computers" by I.Ulcakar and Z.Lenarcic deals with the interesting problem of developing simple and effective theories for interacting open quantum systems. In particular, the authors discuss two specific examples that they argue can be efficiently described with the formalism of time-dependent generalized Gibbs ensembles. The latter has been introduced by one of the authors (Lenarcic) in an early paper (current Ref. 30, but see also the accompanying works 28 and 29). In general, the topic and the study that are presented are suitable for a publication in SciPost Phys. Nonetheless, I find the presentation improvable and I think that several issues (most of them admittedly minor) should be addressed before I can give a recommendation concerning the publication of the article in this journal.

Main issues.

- The authors present two studies based on the tGGE. However, this approximation scheme is not completely under control and I believe that the authors should benchmark their calculatiosn against an exact simulation of the model. Of course, a direct calculation is doable only for short lattices, likely of L=14, nonetheless I think that this is very important if they want the tGGE method to become widely accepted by the community. With ready-to-use packages like qutip it will not take much to do that and I strongly recommend the authors to do it.

- To the best of my knowledge, the application of tGGE methods to Floquet systems as it is done in Sec. 4 is very innovative. On the other hand, discussions like the one in Sec. 3 have already appeared in the literature, as for instance in Refs. 35 or 37 (but many more exist, even cited by the authors). I think the authors should add a sentence in Sec. 3 writing that similar calculations have already been presented, or that similar techniques have already been employed, or an equivalent reformulation.

- Following up on the previous point, I am unhappy with the current formulation of the conclusions. First sentence: "We derived an effective description of non-interacting integrable many-body systems": this is too generic, like I said this had already been done, even by one of the two authors. The authors should write explicitly "for two specific non-integrable Lindblad evolutions". In the second sentence they write: "we show that generalized Gibbs ensembles". This is not correct because the authors have not benchmarked the results against other more controllable techniques. Also the sentence "A digital quantum computer... non-thermal baths" makes me unhappy as other people have proposed other setups where to study these effects, like quantum reaction-diffusion systems or lossy gases.

- In Section 2, I do not agree with the statement: "we showed that the zeroth order approximation... has the form of a GGE". Now, what is explicitly written in the article by one of the authors is that *any* density matrix that is diagonal in the basis of energy eigenvectors would satisfy the zero-th order equation but that one should take the GGE form for physical reasons (I guess, entropy maximization). This is not "showing", according to me: this is making an hypothesis. I'm very happy with this hypothesis, which I find insightful and useful, and I am not asking to develop another approach, but still it is not a proof (to show). I urge the authors to reformulate.

- One of the advantages of tGGE is that it yealds equations that are simple to integrate, but that are also sufficiently simple to get some analytical insight. I am a bit surprised that the authors do not attempt an analytical manipulation of the equations to derive some properties of the dynamics or of the stationary state. I think they should consider doing that.

- Concerning Sec. 4, the discussion is entirely done for the Floquet Hamiltonian. Could the authors write something about what would be observed in the original problem?

Minor issues.

- Sec.2, the general theory is presented in a very generic fashion that at the beginning is a bit confusing. Is Hamiltonian (1) bosonic or feermionic? Is it a one-dimensional or d-dimensional problem? A bit of care in introducing the concepts that are studied would be appreciated.

- in bulk -> in the bulk; around site i -> at site i? in a small region around site i?

- Sec. 3 "Continuous model". Well, it's on a lattice... I suggest "Continuous-time model".

- Improve Eq. 9, many different things are displayed in the same way.

- Between eq. 12 and 13: "preserve the parity" -> "preserve the parity of the number of fermions"

See the report.

Ask for major revision

1- Well written paper containing exciting results on a timely topic.
2- Combination of analytical and numerical results.

The authors compare different methods for non-interacting, integrable many-body systems that are weakly coupled to baths and explore their potential implementation on digital quantum computers, including platforms like superconducting circuits and trapped ions. The dissipation is realized by coupling the system to ancillas that are periodically reset, and it is demonstrated that highly non-thermal generalized Gibbs ensembles can be stabilized.

Given the interest of the work to the realization on actual NISQ devices, it might be interesting to show some data for small and noisy systems to get an idea about how challenging it would be to observe the non-thermal on actual devices (e.g., using the Qiskit or Cirq simulator that allows including noise models).

The manuscript is well written, and the main results are presented clearly. In summary, this work contains novel and exciting results, and thus I recommend publication in SciPost Physics.

1- Potentially include simulation data on a small and noisy systems.
2- Maybe add a legend to Fig. 1a and Fig. 3 to see the time scales it takes to reach the steady state.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

- interesting application of GGE and dissipation
- could be relevant for digital quantum experiments

- too much overlap with https://arxiv.org/abs/2404.12175
- no numerical confirmations of the main claims

The authors consider a slight generalisation of the circuit implemented in the recent experiments Science 383(6689), 1332 (2024), where a many body system is cooled to low temperature using interaction with a spin ancilla bath that is constantly resetted to its ground state. Here the authors want to show that this cooling if the many-body system is integrable can lead to a GGE and not to a simple low-temperature state. Their approach uses the fact that if the interaction with the ancilla is weak, then one can write a kinetic equation for the quasiparticle of the many-body model (mappable to free fermions).
The calculations seem all correct and the idea is interesting, but it has a large overlap with another publication that came a few months before their work - https://arxiv.org/abs/2404.12175.

Moreover, most of their claims are never substantiated by numerical simulations. Given that the authors want to say that their work is relevant for experiments, they should be able to do a simple numerical simulation other with exact dynamics or MPS for 10 or 20 sites and show the approach to a GGE instead of a GE. Without this, there are too many assumptions (namely that quasiparticles remain stable on the scale where the cooling is done) that make the main results not fully trustable.

- provide numerical simulations to benchmark at least qualitatively the claims
- comment more on the relations with https://arxiv.org/abs/2404.12175

Publish (meets expectations and criteria for this Journal)