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

TL;DR

This paper introduces a graph-theoretic framework for understanding information propagation in 2D quantum circuits, demonstrating that certain lattice structures allow for exactly-solvable chaotic quantum models with complex correlations.

Contribution

It establishes that the 2D integer lattice and all regular tilings are geodesically directable, enabling the design of solvable chaotic quantum circuits on these structures.

Findings

$ ext{Z}^2$ is geodesically directable.

All 2D regular tilings are geodesically directable.

Provides a graph-theoretic foundation for solvable chaotic quantum circuits.

Abstract

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 $uv\in G$. 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.