Square-root growth of operator entanglement in an integrable brickwork circuit

UPDATE: A newer version of the intented manuscript was posted meanwhile, see:

https://arxiv.org/abs/2609.39401

This new arxiv version will be sent to peer review. 

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Intro to the earlier version:

This is a preliminary version of a manuscript. The document contains AI generated proofs. The human authors verified those computations. However, the text has not yet reached its final version: we plan to improve it in various places. A future version will be sent to a journal for peer review.

We are releasing the current version due to the closeness with the related paper "On the growth of operator entanglement in brickwork circuits with Yang–Baxter gates" (linked below). 

One of the central statements in the document (a lower sqrt bound for the von Neumann entanglement) was autoformalized in Lean, it is uploaded as a separate zenodo entry and it is linked below. 

Abstract of this document:

We consider the growth of operator entanglement in quantum brickwork circuits. It is generally expected that in integrable quantum many-body systems the von Neumann entanglement of local operators grows at most logarithmically with time. Here we present a counterexample. We construct a concrete integrable brickwork circuit and prove that the entanglement of a selected local operator follows a square-root law. For Rényi entropies of order $\alpha$, we find that the entropy grows linearly for $0\leq \alpha<1$ and logarithmically for $\alpha>1$. The two-site gate of the circuit is involutive and satisfies the braid relation. The model is also associated with a spectral-parameter-dependent solution of the Yang--Baxter equation and consequently has an extensive family of nontrivial conserved quantities. Furthermore, it has an exponentially growing number of gliders and can therefore be regarded as superintegrable. We thus show that logarithmic entanglement growth does not follow from Yang--Baxter integrability. The counterexample was found by ChatGPT 5.6 Sol.