Non-Stabilizerness of Sachdev-Ye-Kitaev Model

• Timely topic
• Accurate presentation of the work and of the results
• Scientifically sound

• It is a purely numerical work and does not attempt to improve me methodology, nor to interpret (analytically) the results

• On page 4, the sentence " On the other hand, odd-parity strings can describe as logical operations." should be corrected for English;
• After eq. (12): can the author comment if they check that another filling fraction produces similar results or has $N_p=N/2$ been choosen for a particular reason to be clearly stated?
• Eqs. (17) and (18) contains delta-function contributions not visible in the plots of Figs . 1: can the authors explain why?
• Does the choice of the initial state in eq. (20) matter for the evolution or other separable, stabilizer states behave similarly?
• I think that a little more details should be given in the conclusion on the relation between this work and those in ref. [72,73]

Accept in alternative Journal (see Report)

1. Timely topic
2. Potential interest in different communities

1. Results are only presented for relatively small sizes even for SYK2 which is non-interacting.

The authors study the Stabilizer Renyi Entropy and the distribution of the Majorana spectrum, measurements of magic, for both the SYK2 and the SYK4 model for complex fermions. Both have random couplings though the former is non-interacting and the latter is strongly interacting. The main results are that the SRE has the same linear scaling with N in both models though in the interacting one is much larger. The distribution is different in both cases, exponential in the non-interacting model while it is a Gaussian in the interacting case.

I think this is an interesting addition to the literature because these results potentially provide a way to characterize the impact of interaction on magic. Moreover, these results could also be of interest in the low dimensional quantum gravity community due to the duality of the interacting SYK model to JT gravity. Moreover, this community is currently interested in quantum information concepts and idea.

I am glad to accept the paper for publication after address the following comments: 1. I am surprised that computationally the SYK_2 does not allow a short-cut to reach much larger sizes. Even numerically, the application of the Wick theorem should make possible to reach much larger sizes. Could the author comment on this? 2. I would also ask the authors to provide a more detailed description of similar calculations in the recent literature. They mention overlap with 72,73. They should elaborate. More specifically, they should comment to what extent their results are universal at least in the context of Fermionic theories. Is expected a linear scaling with N of the SRE in all fermionic systems? Is a Gaussian and an exponential distribution generic for interacting and integrable fermions? 3. Why do the author study the SYK with Dirac fermions instead of the flavor with Majoranas? I would expect that the Majorana one is more interesting because larger sizes are available and because it is more directly related to Pauli spins. 4. The SYK references require some tweaking. When referring to SYK the first time, [45] is irrelevant and [49] should be cited and Kitaev talk should likely be first. SYK models for Dirac fermions in the context of RMT were studied much earlier than [43] but I do not want to be too intrusive with respect to citations. When referring to black holes, the authors should refer to seminal papers and not to what look like a review. The relevance to gravity/black holes is already in Kitaev talk and also in [49]. There were several early papers that pointed out this relation. For instance, https://arxiv.org/abs/1611.04650 https://arxiv.org/abs/1610.03816 https://link.springer.com/article/10.1007/JHEP08(2017)136 https://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.041025 The authors could take a look a choose a few.

I ask the authors to modify the manuscript according to the question raised in the report

Ask for minor revision

Quantum entanglement and magic are two distinct probes of a quantum system’s capacity for quantum information processing. In recent years, there have been various breakthroughs in understanding the entanglement properties of many-body quantum systems, while the study of magic has only recently begun to attract attention. In this manuscript, authors study the magic by considering the Stabilizer Renyi Entropy in both the chaotic SYK4 model and the SYK2 model that describes random hopping Majorana fermions. The main observation is the qualitatively different distribution for the Majorana spectrum: SYK4 shows a Gaussian spectrum, as proposed in previous works while SYK2 exhibits an exponential Laplace distribution.

I believe these results are of particular interests to the field of quantum dynamics and SYK-like models: They provide valuable examples for the behavior of Majorana spectrum/stablizer Renyi entropy in concrete many-body systems. In addition, given the close relationship between SYK models and gravity, this may inspire the study of stablizer Renyi entropy in holography. Therefore, I'm happy to suggest the publication of this work in SciPost Physics.

My minor comments to considered by authors include:

1. The Majorana spectrum is auctually the expectation of a Majorana string. For the SYK2 model, the expectation should satisfies the Wick's theorem, which relates a generic Majorana spectrum to two-point correlators. Does this directly lead to the observed exponential Laplace distribution directly?

2. In the main discussion, authors consider the complex SYK model. Could authors comment on this choice? In particular, is there any difference when considering the Majorana SYK model? Does charge conservation play any role?

Publish (meets expectations and criteria for this Journal)