Operator Entanglement in Local Quantum Circuits II: Solitons in Chains of Qubits

We thank Dr Dubail for his thorough reading of our paper, for his positive assessment, and for his relevant comments and suggestions. In the following we address his comments in a point-by-point fashion starting from his main remark.

Referee: "One more important remark is that the most peculiar behavior is found for the class of circuit models that sustain solitons going in one direction only, and this is insufficiently emphasized. I believe the physical discussion of that class of circuits could be expanded, and their phenomenology should be explained more clearly. Also, the authors cautiously conclude that the local operator entanglement grows unbounded in that case (for some subset of operators); however it is unclear whether they think that growth is linear as in Paper I, or if it is sublinear (maybe logarithmic as in Ref. [14]). I understand that, for now, this may be out-of-reach analytically, but the authors also did some numerics (as displayed in Fig. 1) so it would be interesting to have a discussion of their conclusions/conjectures about this peculiar case."

To address this point we expanded the discussion of the circuits with chiral solitons. Furthermore, we included an additional figure (Fig. 2) and additional table (Tab. 1), which show that the numerical results are consistent with the logarithmic growth of the operator entanglement entropy for the case where the growth is not bounded.


Referee: "I think the observation that some operators have unbounded growth in the 'chiral case' (i.e. circuits with solitons moving only in one direction) is important, and it should appear in the abstract."

Thanks for the suggestion: we agree. This point is now mentioned in the revised abstract.


Referee: "In the definition of a 'soliton' in the introduction, the meaning of 'ultra-local' should perhaps be recalled. Also, why is a 'soliton' restricted to live on a single site? In general, can't one imagine a more extended object, created by an operator acting on a few neighboring sites, and still satisfying Eq. (3)?"

It is surely possible con consider circuits with solitons with larger support. Providing a complete analysis of the general case, however, goes beyond the scope of the present paper (it is somewhat similar to considering ultralocal solitons in circuits with higher Hilbert space). Here we aim at studying the simplest possible case: ultralocal solitons in circuits of qbits.

To reflect this, in the revised version we define the solitons in general (for generic local quantum circuits and soliton range) while we clearly state that we will only analyse the case of ultralocal solitons on circuits of qbits.

Referee: "In Eqs. (5b) and (5c) it would probably be clearer to write 'and' rather than put a wedge"

Changed.

Referee: "The discussion below Eq. (7) about Yang-Baxter integrability could be clarified. It is unclear what 'Yang-Baxter' means in the context of unitary circuits, because Yang-Baxter is an equation that involves spectral parameters and there are no such parameters in the circuit models; so the authors need to explain what they mean here. The next sentence about 'this somehow being reminiscent of kinetically constrained models' and of 'scarring' also sounds mysterious to me; perhaps the authors can explain this better."

Integrable circuits have the local gate given by an integrable R-matrix (fulfilling the Yang-Baxter equation) at a specific spectral point. This is explained more in detail in the revised version where we also clarified the subsequent sentence.


Referee: "Eq. (11): what are s and s′"

We thank the referee for spotting this typo: They are s1 and s2 from the previous equation.


Referee: "In the next sentence, 'To find the implications of (15)', the ref. '(15)' is probably wrong, I guess it should be '(10)'. A similar problem seems to have occured several times in the manuscript, with incorrect references to equations, please check this."

Once again, we thank the referee for spotting error in referencing (we carefully checked all other equation referencing in the paper and did not find any other mistake).


Referee: "Short discussion after Eqs. (13): 'nothing can move in such a circuit'. This is similar to a localized phase; then is this an interesting model to study localization? Maybe the authors could quickly comment on that."

We expanded the discussion in the revised version.


Referee: "The discussion in the last 5 lines of Sec. 2.2 about 'operators on integer sites generating a front moving opposite to the solitons' and 'fronts shooting back solitons and other generic operators' is not very clear. Since this class of circuit models plays an important role later in the paper, I think that paragraph could be expanded quite a lot. Perhaps a figure could also make that discussion clearer."

We expanded the discussion and added two pictorial equations, we hope that the point is now clearer.

Referee: "In Eq. (47), how are the two states |∘⟩ and |a⟩ normalized?"

To one. We added this explanation in the parenthesis after Eq. 47.


Referee: "for notational consistency the Renyi index 'α' in Eqs. (72)-(73) should probably be replaced by 'n'"

The alpha in (72)-(73) is a real parameter, whereas the n in Section 3 is integer. Therefore we decided to keep the two symbols different. In the new version we used 'n' in Eqs (74) and (77).


Referee: "The paragraph after 'Property 4.2', which mentions the unbounded growth of operators on half-integer sites, could be expanded. This is an important result of this paper, so it should be discussed more. In particular, do the authors think the growth is linear? or sublinear? maybe logarithmic? This does not look obvious from Fig. 1."

See our above response to the referee's main point.


Referee: "The conclusion of Sec. 4.3 would probably be clearer if the final result (95) was given directly for the Renyi entropy (instead of the present form of Eq. (95), where the reader needs to recall what tr[(~B[a])2] is)."

We agree. Accordingly, in the revised version we rewrote the equation in terms of the Renyi entropy.


Referee: "In the conclusion, I don't quite understand the sentence 'for qubit chains, circuits with solitons (integrable or not) seem to be the only case where the operator complexity does not grow'. What about qubit chains that map to free fermions (XX chain, Ising chain, etc.)? Those have Hamiltonian evolution and the operator complexity is also known to remain constant for a subset of operators (see e.g. Ref. [22]). In what sense is that different, in terms of operator complexity?"

With that sentence we wanted to stress that here the complexity growth of all ultra-local operators is bounded, whereas in the examples mentioned by the referee some operators (namely the order parameters) have logarithmically growing complexity.


Referee: "some typos:
page 2: 'In Section 3 recall' -> 'In Section 3 we recall'
page 13: Eq. (83) is an empty line
page 14: 'originates a similarity' -> 'originates from a similarity'"

Fixed, thanks!

1- Sentence about chiral circuits added to the abstract;

2- Definition of solitons updated;

3- Wedge replaced by "and" in Eqs. (5b) and (5c);

4- Discussion on integrable circuits improved;

5- Discussion after Eq. 13d expanded;

6- Discussion after Eq. 27 expanded;

7- Discussion after Property 4.2 expanded; Figure 2 and Table 1 added;

8- Various typos and errors in references corrected.