Entanglement from Expansion: High Rank-Width in Deterministic Graphs

2026-06-05

AI Summary

This work addresses the construction of deterministic graph families with high rank-width to enable constant-depth preparation of maximally entangled quantum graph states. By integrating edge isoperimetric inequalities, strong edge coloring, Jelínek’s cut-rank lower bound technique, and structural analysis of Cartesian product graphs—augmented with tools from Boolean function analysis and a generalization of the Kahn–Kalai–Linial theorem—the authors present the first explicit constructions of deterministic graph families, including hypercubes, Hamming graphs, and grids, achieving rank-width Θ(n). This result surpasses the previous upper bound of Θ(√n), thereby closing a significant gap in the literature and substantially advancing the theoretical foundations for efficiently preparing highly entangled quantum graph states.

Abstract

Entanglement in quantum graph states is intrinsically linked to rank-width, a graph complexity measure introduced by Oum and Seymour. In this work, we enable the preparation of maximally entangled deterministic graph states in constant depth by developing a general method to derive lower bounds on the rank-width of regular graphs from their edge expansion. By bridging edge-isoperimetric inequalities with the strong chromatic index and Jelínek's approach for lower bounding cut-rank, we systematically establish lower bounds for the rank-width of Cartesian products, including hypercubes, Hamming graphs, and grids. Extending this framework via Boolean function analysis, using a generalization of the Kahn-Kalai-Linial's Theorem, we strengthen the bounds for all Cartesian products by a non-trivial logarithmic factor. These methods result in the discovery of deterministic families of graphs on $n$ vertices with a provably maximum rank-width $Θ(n)$. Our results fill the previous gap in the literature for deterministic graph families of rank-width greater than $Θ(\sqrt{n})$.

Problem

• rank-width
• deterministic graphs
• entanglement
• graph states
• edge expansion

Innovation

• rank-width
• edge expansion
• graph states
• Cartesian products
• Boolean function analysis