On the growth of operator entanglement in brickwork circuits with Yang--Baxter gates

This is a preprint version of a manuscript, which will be submitted to publication.

Abstract:

We study the operator entanglement of local operators in one-dimensional brickwork circuits whose two-site gate satisfies the braid relation; throughout this work, we call such a gate a Yang--Baxter gate.  We establish upper bounds for several structured, overlapping classes of Yang--Baxter gates.  We show that the operator Schmidt rank remains uniformly bounded in time for all qubit Yang--Baxter gates and, in arbitrary local dimension, for permutation gates obtained from non-degenerate Yang--Baxter maps.  We also show that it grows at most polynomially for involutive dual-unitary Yang--Baxter gates and for arbitrary phase dressings of permutation gates obtained from non-degenerate Yang--Baxter maps. These results imply, respectively, constant and logarithmic upper bounds on the operator entanglement.  Conversely, we construct a seven-state involutive Yang--Baxter gate without dual unitarity and a one-site operator whose exact operator Schmidt rank grows exponentially, although the corresponding operator entropies remain undetermined.  Entanglement growth in the general Yang--Baxter case remains open. All proofs and selected examples were constructed by ChatGPT 5.6 Sol.