On the quantum chromatic number of Hamming and generalized Hadamard graphs
Xiwang Cao, Keqin Feng, Hexiang Huang, Yulin Yang, Zihao Zhang
TL;DR
The paper analyzes the quantum chromatic number $\chi_Q$ for $q$-ary Hamming graphs $H(n,q,d)$ and a generalized Hadamard family $\varOmega_n^{(\mathbb{G})}$. It develops a linear programming framework over the Hamming scheme to construct modulus-one orthogonal representations, yielding upper bounds on $\chi_Q$, and pairs these with spectral lower bounds based on minimum eigenvalues to bound $\chi_Q$ from below. The authors obtain several concrete results, including exponential quantum-classical separations for certain parameter regimes and partial determinations of $\chi_Q$ for generalized Hadamard graphs (notably $\chi_Q(\varOmega_n^{(\mathbb{Z}_q)})=n$ for large $n$ under parity conditions and $\chi_Q(\varOmega_n^{(\mathbb{F}_q)})=n$ when $n$ and $q$ are prime powers). The work also identifies key open problems, such as tightening the bounds for $\chi_Q(H(n,q,d))$ in the intermediate regime and proving conjectured eigenvalue formulas for all $n$ in the Hadamard generalizations.
Abstract
Quantum coloring finds applications in quantum cryptography and information. In this paper, we study the quantum chromatic numbers of Hamming graphs and a generalization of Hadamard graphs. We investigate the separation between the quantum and classical chromatic numbers of these graphs and determine the quantum chromatic numbers for some of them. For the upper bounds of the quantum chromatic numbers, we develop a linear programming approach over the Hamming scheme to construct modulus-one orthogonal representations. For the lower bounds, we determine the minimum eigenvalues for some of these graphs to derive corresponding spectral lower bounds on their quantum chromatic numbers.