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

TL;DR

The paper shows that bounded extensions of two-dimensional lattices can be engineered to have uniformly bounded geodesic slices, enabling exactly-solvable chaotic local quantum circuits on 2D lattices. It reduces the problem to constructing a finite-weighted graph $H$ on $\mathbb{Z}^2$ with bounded geodesic slices, then builds a multiscale fractal graph $H_{a,b}$ using a $p$-adic structure to bound geodesic slices across scales. Key contributions include proving $\mathbb{Z}^2$ is geodesically directable, extending this to all bounded extensions of $\mathbb{Z}^2$ and to all 2D Euclidean tilings, and providing explicit constructions. The results have implications for quantum circuit design and benchmarking by offering a structurally controlled setting where correlation patterns along geodesics can be exactly analyzed.

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.

Figures (5)

• Figure 1: Illustration of an $n$-block (left, shaded grey) with its sides coloured black and a vertical $n$-strip (right, shaded grey) drawn on $\mathbb{Z}^2$ for $n = 3$. Vertices in $(n\mathbb{Z})^2$ are marked in grey.
• Figure 2: Illustration of a pair of horizontally aligned vertices drawn in black for $p = 3$. Vertices in $(p\mathbb{Z})^2$ are coloured grey.
• Figure 3: A bounded extension of the hexagonal lattice which is isomorphic to $\mathbb{Z}^2$
• Figure 4: The triangular lattice as a bounded extension of $\mathbb{Z}^2$
• Figure 5: Illustration of the bounded extension $G$ of $\mathbb{Z}^2$, drawn over $[0, 36]\times [0, 36]$. Edges of the base lattice $\mathbb{Z}^2$ are drawn in light grey. New edges (edges in $G$ but not $\mathbb{Z}^2$, i.e. quantum gates) with length two are drawn in dark grey, and new edges with length four are drawn in black. Endpoints of new edges are marked.

Theorems & Definitions (49)

• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Proposition 2.4
• Lemma 2.19
• proof
• Lemma 2.20
• proof
• Lemma 2.21
• proof