Quantum Chromatic Number of Subgraphs of Orthogonality Graphs and the Distance-2 Hamming Graph

TL;DR

This work advances the understanding of quantum chromatic numbers by exactly determining χ_q for several subgraphs of orthogonality graphs and for the distance-2 Hamming graph in infinite families. It combines spectral graph theory with combinatorial designs, using eigenvalue analysis of Cayley graphs to obtain lower bounds and leveraging flat orthogonal representations from balanced pair-separating families to achieve matching upper bounds. The results illustrate concrete quantum advantages in coloring problems and provide a unified framework for transferring χ_q values through graph embeddings. The study also highlights open directions, including extending methods to broader classes of graphs and exploring design-based constructions beyond symmetric BIBDs.

Abstract

The determination of the quantum chromatic number of graphs has attracted considerable attention recently. However, there are few families of graphs whose quantum chromatic numbers are determined. A notable exception is the family of orthogonality graphs, whose quantum chromatic numbers are fully determined. In this paper, we extend these results by determining the exact quantum chromatic number of several subgraphs of the orthogonality graphs. Using the technique of combinatorial designs, we also determine the quantum chromatic number of the distance-2 Hamming graph, whose edges consist of binary vectors of Hamming distance 2, for infinitely many length.

Theorems & Definitions (29)

• Lemma 1
• proof
• Theorem 2
• proof
• Claim 3
• proof
• Corollary 4
• Theorem 5
• Theorem 6
• proof