Integrable deformations of superintegrable quantum circuits

We are thankful to the Referees for the careful reading of the text and for the various comments and requests. Here are our replies.

Referee 1

1. Indeed this was not meaningful, we deleted it.

2. The referee is right that the commutation of the additional charges with the Floquet operator does not follow from our argument in Section 4. For the model of Section 4, We are content to prove that the Floquet operator commutes with an integrable Hamiltonian. For Rule 54, we have proved precisely that the additional charges commutate with the update rule, and this method could be used in section 4, but it would require more modifications. Based on the above, we have modified the paragraph following Equation 4.10.

3. We deleted the exclamation marks. Indeed we were very excited, because this was unexpected, and whenever we tell this to other researchers they are also surprised and they like this peculiar behaviour. Nevertheless it is indeed not necessary to keep so many exclamation marks.

4. This is a good question. At this moment we don't know, whether the two properties (integrable deformations of superintegrable circuits, and the possibility to define the evolution in space) are related. They might be. Also, we think that the evolution in space of [17] is likely related, but we have not worked on it, and this would require longer computations than what is reasonably expected from a minor revision.

Referee 2

1. Done.
2. We deleted the word completely'' and kept onlyintegrable'' and ``integrability''.
3. We added a footnote on page 1. summerizing ing the superintegrability of the Kepler problem.
4. We added a footnote on page 2. explaining the key difference between the classical an the quantum case.
5. We performed the required modifications.
6. We applied a unified convention, now it is indeed ``six-site'', etc.
7. We exchanged "original classical model" to "original superintegrable model".
8. In the Introduction we say that ``However, the algebraic integrability of the resulting model was not clarified in [12].'' We still think that this is a true statement. With this sentence we do not say that the authors of [12] failed to solve the problem, we do not say that they attempted to solve the problem. But we want to say that the problem exists, and in that work it was simply not solved. So here we did not make any changes, and we hope the referee can agree with this.
9. We tried to clarify the paragraph starting with ``We find a somewhat unexpected phenomenon: '' We can not say anything about all superintegrable cellular automata, but we are claiming the statements about the examples that we consider in the work.
10. We added a clarification to the very last sentence of the Introduction.
11. We clarified this.
12. We added "and the structure of $\VV$ defines the notion of a brickwork circuit." to the sentence after (2.2).
13. We added an extra explanation.
14. We added also some other dynamical possibilities.
15. We added a footnote about this on page 3.
16. We replaced generic'' withstandard'', however, following this it is not clear to us what the referee requests.
17. We put here modulo 2pi''. But afterwards, we don't understand the comment of the referee. Our sentence starts with Whereas the concept of a ground state is missing in such models''. We are just explaining that the level spacing statistics can be defined even in this case, when there is no natural beginning'' andend'' of the spectrum.
18. We clarified that statement. The existence of one glider is in fact enough to guarantee exponential increase of the number of conserved charges, in the limit of large sizes.
19. We deleted it.
20. We deleted it.
21. We replaced this.
22. We performed the requested changes.
23. We added a new equation (2.7) clarifying this.
24. It is not clear to us which sentences should we delete. The very last two sentences of Section 2 are explaining the plan of what we intend to do. It is not clear why we should delete those.
25. We added the definition of the permutation operator.
26. We corrected this.
27. We replaced "obvious that" with "straightforward to check that", but it is not clear why should we put (3.5) and (3.6) into the same equation.
28. We added an explanatory paragraph before eq. (3.6).
29. We changed this.
30. We added the signs.
31. We still think that the Appendices should be presented separately.