Combinatorial foundations for solvable chaotic local Euclidean quantum circuits in two dimensions

We investigate a graph-theoretic problem motivated by questions in quantum computing concerning the propagation of information in quantum circuits. A graph G is said to be a bounded extension of its subgraph L if they share the same vertex set, and the graph distance d_L(u, v) is uniformly bounded for edges uvin G. Given vertices u, v in G and an integer k, the geodesic slice S(u, v, k) denotes the subset of vertices w lying on a geodesic in G between u and v with d_G(u, w) = k. We say that G has bounded geodesic slices if |S(u, v, k)| is uniformly bounded over all u, v, k. We call a graph L geodesically directable if it has a bounded extension G with bounded geodesic slices. Contrary to previous expectations, we prove that mathbb{Z}² is geodesically directable. Physically, this provides a setting in which one could devise exactly-solvable chaotic local quantum circuits with non-trivial correlation patterns on 2D Euclidean lattices. In fact, we show that any bounded extension of mathbb{Z}² is geodesically directable. This further implies that all two-dimensional regular tilings are geodesically directable.