Generalized hydrodynamics of integrable quantum circuits

Response to Referee 3

In the following, we present a point-by-point response to the Referee comments.

“The Authors extend the framework of generalized hydrodynamics to integrable quantum circuits. This framework is a »classical« description of large-scale phenomena in integrable models, which possess infinitely many local conservation laws (continuity equations). It allows one to treat exactly the bipartition protocols / Riemann problems in integrable models, for example. The topic is timely due to the advent of quantum computing devices which often implement circuits similar to the ones described by the Authors herein. Although the calculations are mostly generalization of the known results in the continuous-time models, one of the results is surprising: a localized perturbation may result in global change of the system’s state, which propagates (and persists) on ballistic scales."

We thank the Referee for their positive assessment of our work.

“The Authors have assessed that two conditions for SciPost physics are met: (1) Provide a novel and synergetic link between different research areas; (2) Detail a groundbreaking theoretical/experimental/computational discovery. I am not convinced that (2) is true, but I cannot argue that (1) is not (although, as pointed out by Referee 2, the »synergetic link« has essentially been done in ref. [33]). All in all, I would lean more towards the recommendation of Referee 2, that the paper might be more suitable for SciPost Physics Core. However, since my arguments against the claim that criterion (1) is met might not be deemed really strong, I am not strongly opposed to publication in SciPost Physics either. In any case, below I provide three remarks that can slightly improve the otherwise clear and well-written paper."

The Referee is not convinced that our work meets the SciPost acceptance criteria. Their main criticism is the same by Referee 2, namely that this work does not introduce enough novelty compared to Ref. [33]. We strongly disagree on this point, as we explain in detail in our response to Referee 2, to which we refer. In fact, it may be useful to briefly reiterate why we believe that our work meets the SciPost acceptance criteria:
(1) it provides a link between generalized hydrodynamics (an extremely active area of theoretical and cold-atomic research over the past few years) and digital dynamics (an emerging area of research at the intersection of many-body physics and quantum-information theory) which are suitable for implementation in benchmark experiments on quantum platforms; (2) it details an important theoretical discovery: namely, a local defect in a simple discrete dynamical evolution is sufficient to cause a change in the emerging macroscopic stationary state at late times (once again, with implications on benchmark experiments on quantum devices).

“(1) At the end of section 2.1 the Authors remark on two families of conserved quantities which break the single-site translation symmetry. They claim that the latter maps one family into the other, referring to ref. [29]. Looking at eqs. (10), (11) in the latter, their statement does not seem obvious to me. Looking at those two eqs. in the reference, for example, the densities of the first two charges are indeed translated w.r.t. each other for one site, but they have a \pm sign difference in one of the terms. Could the Authors comment on this?"

We thank the Referee for spotting and pointing out this incorrect statement. What we wanted to say is that the operation leaving the set of conserved charges invariant is the combination of the single-site translation and a flip of the staggering parameter x. However, the statement as it was written in our previous version was indeed incorrect (since this statement was never used in this or the previous paper, we did not notice it). We fixed the statement in the most recent version.

"(2) After eq. (59), the Authors mention a global prefactor to the eigenvalue of the propagator. I assume that prefactor is not just a phase (-1)^L, otherwise it could have been written down. Where does is come from? Looking at Faddeev’s notes, eq. (412) in arXiv:hep-th/9605187, there is no L-dependent prefactor in the definition of the quasienergy, for example."

We thank the Referee for pointing this out. In general a prefactor may arise from the choice of normalization of the transfer matrix/evolution operator, but we have checked that in the present case such a prefactor is absent.

“(3) The Authors mention refs. [99-101] as examples of local perturbation evolving into a macroscopic change of the system’s state. Some other examples of such non-dispersing localized perturbations: arXiv:1207.0862, arXiv:1909.02841, arXiv:2111.06325."

We thank the Referee for point out these relevant references that we missed. We have added them in the most recent version of the draft.