Correlations and commuting transfer matrices in integrable unitary circuits

We would like to thank both referees for their careful reading of our manuscript, their useful suggestions, and their positive assessment. In the revised version we have addressed all requested changes and made some minor revisions which we hope makes our paper acceptable for publication in SciPost Physics Core.

For the authors,
Pieter W. Claeys

The main changes in the manuscript can be found below.

- The notation of transfer matrix and monodromy matrix has been made consistent with the literature throughout the manuscript.
- We have made clear that we are working with the $\check{R}$-matrix rather than the $R$-matrix, and explicitly mentioned the connection between the two when introducing the braiding relation in Eq. (19).
- The connection with the fusion procedure of Kulish, Resehtikhin and Sklyanin, Lett. Matt. Phys. 5, 393 (1981) has been explicitly mentioned at the end of Section 2.4.
- The BAE for arbitrary spin s have been given in Eq. (43), making it easy to identify our obtained BAE [Eq. (42)] as those for the integrable spin-1 chain.
- In the conclusion we now explicitly mention the importance of nonabelian symmetries for KPZ.
- In Table 1 the dimensions of the Jordan blocks associated with the eigenvalue $\lambda^2/(1+\lambda^2)$ have been corrected: rather than three 6x6 blocks, this should have been three 5x5 and three 1x1 blocks. At the end of Section 2.5 we also provide a reference and short discussion of the work of the added Ref. [59], which allowed us to spot this mistake and presents a useful way of constructing the matrix elements of the transfer matrix in the generalized eigenbasis, but unfortunately does not allow for a direct calculation of the dimension of the Jordan blocks in general.
- Minor typographical changes have been made throughout.

We have chosen to keep Section 2.4 on the inhomogeneous model as is. We agree that the section is technical, but we also believe that the results in there are important for establishing the completeness of the Bethe ansatz in the inhomogeneous case, as well as further clarifying in which ways the presented two-component transfer matrices differ from the one-component transfer matrices usually studied. Parts of this section had also already been moved to Appendix before submission. To accommodate this remark we now mention at the beginning of Section 2.4 that readers only interested in the homogeneous model can skip forward to Section 2.5 for a discussion of the eigenvalues and dimensions of the Jordan blocks.

The suggestion to present all calculations in the folded representation is a useful one, but when writing the manuscript we made the decision to keep all calculations as explicit as possible. While the folded representation would definitely simplify some expressions, we thought it might also obscure some calculations and make the connection with some of the established literature on integrability less clear, which is why we have chosen to present these results in this way.