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

Dear Editor,

We are re-submitting the manuscript “Generalized Gibbs ensembles in weakly interacting dissipative systems and digital quantum computers”, which has been updated according to the referees' comments and suggestions. Primarily, we complement our analysis with numerical simulations of the exact dynamics, which offer a reliable benchmark for the time-dependent GGE approach. We have made some additional changes, which we list at the end of the reply. The modified text in the manuscript is highlighted in blue.

Below, we address referees’ comments point by point.

Sincerely, Iris Ulčakar and Zala Lenarčič

Referee 1 We thank the referee for their work and positive evaluation: “The calculations seem all correct and the idea is interesting”.
Below we answer their comments and requests, which have been implemented in the manuscript.

• provide numerical simulations to benchmark at least qualitatively the claims Reply: In Fig.2 and the paragraph starting with ”While the GGE Ansatz for…”, we added a comparison of GGE results and tensor network simulations of exact dynamics for the continuous-time setup at $L=80$, showing that they coincide in the $\epsilon\to 0$ limit. In App. F we perform the same comparison also for the Trotterized setup, reaching the same conclusion as in the continuous-time case. Namely, as the coupling strength between system and ancilla qubits is reduced, exact expectation values of local observables converge to the GGE ones.

• comment more on the relations with https://arxiv.org/abs/2404.12175 Reply: Contentwise, the arXiv:2404.12175 focuses on the theoretical aspects of modeling the Mi. et al Science 383 (2024) paper, co-authored by the same people. They upgrade the experimental realization by proposing circuits that would bring the system close to a thermal state by implementing system-ancilla coupling which satisfies detailed balance conditions (approximately). So unlike in our case, where we highlight generic GGE stabilization, their focus is on preparation of mixed states that are close to the ground state. Similar to us, they develop a scattering theory, which in their case involves some further approximations due to a different choice of system-ancilla coupling gates. Completely orthogonal to arXiv:2404.12175, our App.A contains comparison of different methods for calculation of steady states in general weakly interacting open systems. Most importantly, our message is different, pointing out that for an integrable system’s dynamics coupled to ancillary qubits, generically, non-thermal GGE states are stabilized. We can add that the intellectual relation to arXiv:2404.12175 is less clear. At the Marko Medenjak symposium (May 2023), ZL mentioned the GGE interpretation of Mi et al. Science paper to D. Abanin (a co-author on that paper). We further discussed the details of possible scattering theory with J. Lloyd at ICTP School (August 2023). Since no further developments were foreseen, we decided to study the problem ourselves. While their publication was no doubt earlier than ours, it is to us unclear how much our open communication of ideas contributed to that.

Referee 2 We thank the referee for their work and publication recommendation: “this work contains novel and exciting results, and thus I recommend publication in SciPost Physics.” Below we answer their comments and requests, which have been implemented in the manuscript.

• Potentially include simulation data on a small and noisy systems. Reply: We agree that this is a natural suggestion and we started working in this direction. However, implementational challenges are actually not trivial: (a) Unlike circuits optimized by Google AI or IBM for a particular publication, general free or commercial quantum computers apparently still suffer from strong rather than weak noise. (b) In order to measure \sigma^y_i \sigma^z_{i+1}...\sigma^z_{k-1}\sigma^y_{k} Pauli strings, one has to employ tricks that reduce the measurement error and give access to longer strings. Since we believe our manuscript already contains quite a lot of material (generalized scattering theory, comparison of the continuous time and Trotterized setup, comparison of different approaches to steady state calculation, comparison of GGE and full evolution by tensor network, etc.) we would prefer to report on an actual implementation and its challenges in detail in a separate paper.

• Maybe add a legend to Fig. 1a and Fig. 3 to see the time scales it takes to reach the steady state. Reply: Added.

Referee 3 We thank the referee for their work and positive evaluation: “article … deals with the interesting problem of developing simple and effective theories for interacting open quantum systems” and “In general, the topic and the study that are presented are suitable for a publication in SciPost Phys.” Below we answer their comments and requests, which have been implemented in the manuscript.

• 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 calculations 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. Reply: In Fig.2 and paragraph starting with ”While GGE Ansatz for…”, we added a comparison of GGE results and tensor network simulations of full dynamics at $L=80$, showing that they coincide in the $\epsilon\to 0$ limit. In App. F we perform the same comparison also for the Trotterized setup, reaching the same conclusion as in the continuous-time case. Namely, as the coupling strength between system and ancilla qubits is reduced, exact expectation values of local observables converge to the GGE ones.

• 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. Reply: We added a paragraph in Sec.3 (starting with “We should note that a similar simplification…”) and a sentence in the introduction, which mentions that similar approaches have been used before. We also comment that our example is instructive as our steady states are non-trivial and have structure in the particle occupation. Our examples show that dissipation can be used for non-thermal state engineering, for example in nearly integrable materials as highlighted in the Conclusions.

• 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. Reply: In the first sentence we now specify that we consider a system “that relax to highly non-thermal steady states due to the interplay of integrability and weak coupling to structured baths considered”. In the third sentence, we now highlight that our claim about usage of GGE Ansatz is “Numerically supported by a comparison to the exact dynamics performed with tensor network TEBD simulation”. In the last sentence of the Conclusions section "A digital quantum computer…” we try to give a broader perspective on the relevance of such quantum computer realization in the context of quantum materials. Following the suggestion of the Referee, we (1) omitted that this would “be the first” realization to reveal the peculiar nature of nearly integrable models, (2) we added to the references in that sentence other works considering structured steady state GGEs, and (3) stress that we are referring here to non-trivial steady state properties that are expected also in nearly integrable materials.

• 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. Reply: We replaced “showed” with “argued”. Later, in Fig.2, we show comparison between the GGE and exact results obtained with tensor network simulations, giving evidence that the hypothesis is well supported for the continuous-time setup. As already said above, in App. F we perform the same comparison also for the Trotterized setup. We summarize that our study provides a numerical support for the GGE approximation also in the Conclusions.

• One of the advantages of tGGE is that it yields 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. Reply: The focus of this manuscript is (a) highlighting the stabilization of structured quasiparticle population, (b) proposing how this physics can be realized with digital quantum computers, and (c) comparing different approaches to weakly interacting open systems. All this is achieved already by numerically obtained results on thermodynamically large systems. The main analytical insight we obtained is in formulating the dynamics via a scattering theory which exposes what kind of quasiparticle processes are happening for our choice of Lindblad / system-ancilla coupling. While in some cases, analytical expressions (when possible under further approximations) can yield deeper insight, we currently unfortunately do not see what we would gain specifically by doing so for our case.

• 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? Reply: Fig.4(a) for the Floquet setup is the equivalent of the Fig.1(a) for the continuous-time setup. They both illustrate the main message that structured dissipation can stabilize structured and non-thermal quasiparticle occupation. In Sec.4 we take that message further and highlight how non-thermal nature can be enhanced and displayed with local observables. Same could be done in Sec.2 by plotting the correlators $\langleS^{yy}_l \rangle$ in the continuous-time case. Since the main emphasis of our paper is on the discussion of physical realization with the reset protocol, we believe it is advantageous for the narrative to spare some of the highlights for the corresponding Sec.4.

• 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 fermionic? Is it a one-dimensional or d-dimensional problem? A bit of care in introducing the concepts that are studied would be appreciated. Reply: Done. Since the Hamiltonian could in principle be bosonic or fermionic, we did not specify that in Sec.2.

• in bulk -> in the bulk; around site i -> at site i? in a small region around site i? Reply: Done.

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

• Improve Eq. 9, many different things are displayed in the same way. Reply: Done.

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

Additional Changes:

• In the original version, the compact form of the scattering equations presented in Eqs. 15, 16, 17 and 29, 30 was not expressed with sufficient care (Eqs. 14, 15, 16, 28, 29 in the previous version). Specifically, Eqs. 15–17 were overly simplified, and Eq. 30 contained a typo. These issues have been corrected in the revised manuscript. Importantly, all numerical simulations shown in Fig. 1 and Fig. 4 were based on the correct, non-simplified forms of the scattering equations, even in the original submission. Therefore, the results presented in these figures remain unchanged.

• We have revised the wording related to the reset protocol in Appendix C, to avoid phrasing it as a projection onto a state.

• We have rewritten the paragraph on the symmetries of the scattering equations in App. E and have updated Fig.6 to include an additional symmetry.

- Sec 1: More explicit referring to other works using GGE in the introduction.
- Sec 1: Updated outline of the paper.
- Sec 2: Clarified introduction of the Setup
- Sec 3: Eq. (9) rewritten in Eqs. (9,10)
- Sec 3: Correction to Eqs. (15-17)
- Sec 3: Comparison to full dynamics obtained with tensor networks added, added Fig.2
- Sec 3: Reference to other works using related scattering equations added
- Sec. 4: Correction to Eq. 29-30
- Sec 4: Extended discussion on the additional integrability breaking sources
- Sec. 4: Reference to the comparison with the exact circuit dynamics in App. F added
- Sec 5: Updated Conclusions with reference to the exact calculations
- App. C: Revised wording on the reset protocol
- App. E: Rewritten paragraph on the symmetries of the scattering equations. Updated Fig.6.
- App. F: Comparison of GGE and exact circuit dynamics