Brick Wall Quantum Circuits with Global Fermionic Symmetry

1. Analytical derivation of the spectral structure of the circuit operator U_F is far from trivial.
2. Using insights from the global Fermionic symmetry allows us to clarify the presence of gapless regions.

1. Presentation: The presentation is very dry and technical, which might limit accessibility to a broader audience.
2. Entanglement computation and quantum processor analysis: The Authors include some discussion about entanglement dynamics and SWAP test demonstrations on IBM (classical) simulators. But the whole analysis seem poorly connected to the text. (What does the analytical solution give as insights to the entanglement dynamics is not clearly stated. )
3. Numerical Examples of Entanglement dynamics: Being matchgates, the numerics on free fermions can be pushed to extensive system sizes.

In this paper, A. Zorzato and collaborators investigate a brick-work circuit (BWC) built out of nearest-neighboring 2-qubit matchgates (or free fermionic gates). The gates are tailored to reproduce the 2-particle scattering matrix in integrable, supersymmetric quantum field theory (QFT) in 1+1 dimensions. This fact allows for the analytic solution of the BWC's spectral properties.

Additionally, the Authors explore the entanglement dynamics generated by the $U_F$ for small system sizes, and benchmark the SWAP test calculations with an IBM quantum processor.

Overall, the manuscript topic is interesting, and the analytical insights on the $U_F$ under consideration are far from trivial. For this reason, I believe that a properly revised manuscript is suitable for publication in Scipost Physics.

The manuscript's main problems are twofold. First, the presentation is dry and challenging for non-experts to read. I strongly advise the Authors to rephrase part of the manuscript in a more accessible way. Some suggestion include:
1. Please write a proper introduction. What is the state of the art? (Please add citations. E.g. in random circuits, they are mentioned but not cited. How should I know what is the open problem you tackled if you mostly don't mention it?)
Why is the problem of interest? What are the quantities that you study, and why these quantities may have some interest? (E.g. It is not reasonable that you discuss entanglement after 20 pages of text, without properly motivating this at the introduction stage).
All these points are *required* (cf. https://scipost.org/SciPostPhys/about, the Acceptance Criteria, in particular point 3, 5, 6).

At a more substantial level, I find the analysis of the entanglement entropy confusing and limited. Limited because, being free fermionic gates, the numerics I expect from this type of system should be much larger (cf. for instance, the classic reference, https://arxiv.org/abs/0906.1663).
Confusing because I don't understand how well is tied to the whole analysis of $U_F$. Why not simply studing the evolution of an observable (like the magnetization), instead of the entanglement? What am I learning from this quantity in relation to the previous discussion about $U_F$?
Also, what is the numerical implementation? What is the IBM QASM (no link or reference is provided)?

1. Improve presentation: Write proper Introduction, including state-of-the-art and main results; Include proper bibliography; Clarify the numerical technical side.

2. Improve numerics on the entanglement entropy, and reach decent system sizes (say $N\le 128$). This is simply doable, as these are free fermions.

Ask for major revision

The Authors consider quantum unitary circuits composed of two-body unitary gates that correspond to the scattering matrices of integrable supersymmetric QFT in 1+1 dimensions. The supersymmetry of the unitary gate induces two supercharges of the circuit, termed “global fermionic symmetries”, somehow similarly to how a U(1) symmetry of the R-matrix induces magnetization conservation in the Heisenberg model. The Authors demonstrate connections to Kitaev chain and topological phases of matter, which arise in the continuous-time limit when the global fermionic symmetry associated with the supercharges is broken.

While the problem has free-fermionic structure, its solution is far from trivial: the Authors nevertheless tackle it quite well and are able to produce some exact results w.r.t. the model’s spectrum. I believe the results (which are interesting in themselves) also guarantee interesting future investigations. On the one hand, the supercharges contain Jordan-Wigner strings and remind of conserved twist fields (but on a lattice!) which could have interesting implications for the long-time nonequilibrium dynamics in lattice models. On the other hand, from the perspective of equilibrium physics, there are connections to nontrivial topological phases that should be explored in future works.

On the negative side there are two (possibly related) things:



(A) The presentation is somewhat lacking and needs to be thoroughly improved before publication. Mostly the motivation for some of the calculations is not clear and the first sections that mention supersymmetric QFT are somewhat cryptic as if intended for a very restricted target audience. I would sincerely recommend the Authors to expand the target audience to readers that are proficient in quantum circuits (and maybe quantum computation!), but less proficient in SUSY, CFT and QFT: for example, the first thing is to explain better what is going on in the beginning of section 1.2. 


(B) I somehow failed to understand what the Authors refer to as the dynamics of entanglement and what their calculation has to do with (perhaps) more relevant notion of entanglement growth.

The above criticisms are specified in the remarks below (Requested changes): please see them as suggestion for improving the manuscript. I think I will recommend publication once the presentation is thoroughly improved and the remarks answered.

(1) In the introduction the Authors separately cite ref. [26] as the one where “Trotterization was studied”. I assume they refer to the study of physical phenomena in integrable Trotterizations. In this regard, I would like to propose more literature. In fact, the physical properties that lead to the conclusions regarding the phase transition in studied in ref. [26] were already studied in [M. Ljubotina, L. Zadnik, T. Prosen, Phys. Rev. Lett. 122 (15), 150605]. Other interesting physical phenomena in Trotterizations were studied in [M. Medenjak, T. Prosen, L. Zadnik, SciPost Phys. 9, 3]. There were also other applications of integrable Trotterizations, for example, in relation to quantum chaos and random circuits [A. J. Friedman, A. Chan, A. De Luca, J. T. Chalker, Phys. Rev. Lett. 123 (21), 210603], in relation to open circuits [L. Sá, P. Ribeiro, T. Prosen, Phys. Rev. B 103, 115132], etc. Moreover, numerous experiments have already been performed in addition to the one reported in ref. [14]: see, for example, [E. Rosenberg, et al., Science 384, 48 (2024)], and [A. Morvan, et al., Nature 612, 240 (2022)].

(2) I suggest the Authors to provide additional explanations on the context of their work in Section 1.2, in particular, on the relevance of supercharges for readers who are not well versed in these concepts. In the Introduction I miss some paragraph that will explain what will be done in the paper and the motivation.

(3) The first thing one notices about the scattering matrix is that it is that of an 8-vertex model — perhaps this could be mentioned before the free-fermion condition is discussed.


(4) Three different symbols Q are used — eqs. (1.1), (1.4), (1.5): I suggest the Authors to add some clarification.

(5) In eq. 1.11, K(\theta) is a 2x2 matrix, while Q seems to be a 4x4 matrix. I assume K denotes then a tensor product with identity on another one-particle Hilbert space. But is it K \otimes 1 or 1 \otimes K? This should be commented on. 



-In addition, under eq. 1.11, the Authors say K commutes with the sum of the supercharges, but the rapidity in Q changes, so is this still a commutation? 



-Also, why is there only one rapidity parameter in Q, while in eq. 1.5 there are two? Is Q in eq. 1.11 a one-particle operator? If that is so, it should be defined — from eq. 1.5 it is not clear at all, how Q^l and Q^r for one particle look like.


(6) I find the very last paragraph on page 4 cryptic (if it is supposed to convey relevant info, it should be rewritten). 



-For example: where is the original observation that S is best understood in fermionic language written?

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Also, what does it mean that “a J-W transformation back to bosonic qubits is *in place*”? And what are bosonic qubits here?


(7) Under eq. 1.13 the Authors refer to qubits again. What are those qubits then; are these the degrees of freedom related via J-W transform to spins 1/2, operated upon by Pauli matrices (say those in eq. 1.4)?

(8) In eq. 2.3, rhs, the Authors probably meant p(v_i) —> p(e_i).

(9) The definition in eq. 2.4 is not clear. For example, what is the meaning of a “row (column) of a subspace”? 



-What is the meaning of parity — with respect to which property of operators is it defined? For example, what is the parity of operator \sigma_j^z, where lower index is the site? What is instead the parity of a half-string \prod_{i<j}\sigma^z_i? Notice, that two half-strings ending at different positions commute. Aren’t half-strings supposed to be there in order to represent anticommuting operators?

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Simple, but clearly explained examples of grading (e.g., J-W transformation) would greatly improve readability.


(10) In eq. 2.10 the involution equations are missing the right-hand side. 


(11) The paragraph above eq. 2.11 is not very clearly written: the off-diagonal elements of what are referred to? If I understand correctly S_{L,1} is special because the strings in the generalized J-W transformation do not cancel out and therefore a full parity operator remains? I think such additional explanations would be welcome: the reader should be aware, that the product in eq. 2.8 (for example), is an ordinary product of operators that are (mostly local), as in standard brickwork circuit. That is, apart from S_{L,1}.



-A question is now the following: what is the advantage of all grading over simply considering the fermionic circuit in odd and even sectors separately, similarly to how we diagonalize free models?



-I know this is probably impossible, but looking at the narrative now, it would probably be better to start with fermionic representation of S and then represent it in terms of operators using graded spaces. At least this should be emphasized more in the beginning (anywhere possible).


(12) In eq. 3.1, is \theta missing in the linear term on the rhs?


(13) After eq. 3.7 the Authors state that topological phases require breaking of the global fermionic symmetry. Is that because breaking of the latter opens the gap? If so, the Authors should refer to the previous statement about breaking criticality, above eq. 3.6, for a better readability.

(14) On page 12, abbreviation BDI is introduced. It seems what it stands for is never written.


(15) In the paragraph after eq. 3.18, some background on the relevance of real/imaginary \Delta would be appreciated.

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Also, the sentence that introduces matrix G is not really clear, neither is its relevance. I suggest the Authors to be more explicit about what they specifically mean there.



-Regarding the chiral symmetry — which chiral symmetry specifically is meant there? In particular, suppose that in the model with nonequal masses, considered in that paragraph, one takes the limiting case of the gates S which are simply graded permutations (I assume this is possible, otherwise I stand corrected). If I am not mistaken, in that case the circuit drives the masses m_1 and m_2 in opposite directions. Clearly the dynamics is then not chiral. How does this reconcile with the chirality C defined in the mentioned paragraph?

(16) The deformed coefficients in eq. 3.22 are only defined in eq. B.1 — the latter eq. should be then referred in the main text.


(17) In the discussion of the Renyi entropy growth, after eq. 5.2 the Authors assign the role of time to the parameter \theta of the unitary gate. Do they mean time step? 



-As far as I understand circuits, time should correspond to the depth of the circuits, namely, it should be proportional to the number of applied layers in the circuit. As far as I see, the latter number has nothing to do with \theta. If one considers the Trotter limit, then \theta needs to be small, and only in that case it actually corresponds to an infinitesimal time increment, if I understand 3.1 correctly. Could the Authors be more clear in what is meant in that paragraph after eq. 5.2?



-In fact, I do not understand what is the relevance of RE(\theta): it is not obvious to me that this is really time-dependence of entanglement entropy. Instead, looking at figure 9, only panel c) seems really relevant from the point of entanglement growth — if I am not wrong m=number of layers then corresponds to time. The only problem here is that L=4, so boundary effects immediately take place (plotted m-s are mostly of the same order or larger) and the relevance of such a calculation again becomes questionable. In any case, even if I have misunderstood the discussion, the Authors should be more precise with what they mean by dynamical evolution of entanglement and what is the relevance of the discussion referring to figs. 9 from the point of view of entanglement growth.

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I think the full discussion around eq. 5.3 and on page 28 should be more to the point and its relevance should be made clear.

Ask for minor revision

1- clear and detailed study of the spectrum and (absence of) topological features

1- the physical motivation for such circuits with fermionic symmetry is not clearly explained
2- the study of the spectrum of the circuit dynamics does not bring about a huge amount of novelty with respect to that of the Hamiltonian limit
3- analysis of entanglement based on numerics for very small system sizes (10 sites at most), hence no strong conclusions can be reached

This paper conducts an in-depth study of a new sort of brickwork quantum circuits, characterized by a fermionic symmetry inspired from previous studies in quantum field theory.
From the statistical mechanics point of view those are equivalent to special regimes of previously known eight vertex model, however the fermionic symmetry brings about new features : first, it introduces a graded tensor product structure; second, and more importantly, imposing the fermionic symmetry imposes stronger restrictions which pin the model to a gapless phase. This is clearly seen in the Hamiltonian limit, connecting to the well-known Kitaev chain, but the authors further extend their study to the circit geometry.
This a potentially interesting paper, however the physical motivation for the study of such circuits should be better explained in my opinion.
Why, from the condensed matter point of view, study such circuits ? If my understanding is correct that the difference between these and circuits constructed directly from the eight vertex model (that is, without the graded tensor product structure) boils down to a question of boundary conditions, then are there important physical differences between the two settings ? And should we expect to see one more naturally than the other in experiments or physical materials ?
I would appreciate if the authors could elaborate more on these issues, as well as on the more specific points addressed in the "Weaknesses" and "Requested changes" sections.

Besides this, this is a well-written and technically solid paper, which may be suitable for publication in SciPost.

1- Section 1 : could the authors give more physical motivation for the study of such circuits ? (see Report section)
2-Section 1.3: when mentioning statistical mechanics, the name "eight vertex model" is never used. Would seem relevant though.
3- At the end of Sec. 2.4, it would be useful to see the expression of the fermionic symmetry generators $Q^l$, $Q^r$ in terms of the fermion creation/annihilation operators
4- Before eq. (3.13) and eq. (4.9) : notation $|- \rangle$ should be explained
5- p. 15: one or several references would be appreciated when discussing the class BDI

Ask for minor revision