On the Quantum Chromatic Numbers of Small Graphs

TL;DR

This paper proves the smallest known graph with a quantum-classical chromatic number separation and introduces a new example demonstrating the difference between rank-based quantum chromatic parameters.

Contribution

It provides a computer-assisted proof of the smallest graph separating quantum and classical chromatic numbers and presents the smallest known graph separating rank-1 and rank-2 quantum chromatic numbers.

Findings

Proved the minimal graph with quantum-classical chromatic separation is on 14 vertices.

Discovered the smallest graph with rank-1 and rank-2 quantum chromatic separation on 21 vertices.

Developed techniques for constructing and lower bounding orthogonal rank of graphs.

Abstract

We make two contributions pertaining to the study of the quantum chromatic numbers of small graphs. Firstly, in an elegant paper, Man\v{c}inska and Roberson [\textit{Baltic Journal on Modern Computing}, 4(4), 846-859, 2016] gave an example of a graph $G_{14}$ on 14 vertices with quantum chromatic number 4 and classical chromatic number 5, and conjectured that this is the smallest graph exhibiting a separation between the two parameters. We describe a computer-assisted proof of this conjecture, thereby resolving a longstanding open problem in quantum graph theory. Our second contribution pertains to the study of the rank-$r$ quantum chromatic numbers. While it can now be shown that for every $r$, $\chi_q$ and $\chi^{(r)}_q$ are distinct, few small examples of separations between these parameters are known. We give the smallest known example of such a separation in the form of a graph $G_{21}$ on 21 vertices with $\chi_q(G_{21}) = \chi^{(2)}_q(G_{21}) = 4$ and $ \xi(G_{21}) = \chi^{(1)}_q(G_{21}) = \chi(G_{21}) = 5$. The previous record was held by a graph $G_{msg}$ on 57 vertices that was first considered in the aforementioned paper of Man\v{c}inska and Roberson and which satisfies $\chi_q(G_{msg}) = 3$ and $\chi^{(1)}_q(G_{msg}) = 4$. In addition, $G_{21}$ provides the first provable separation between the parameters $\chi^{(1)}_q$ and $\chi^{(2)}_q$. We believe that our techniques for constructing $G_{21}$ and lower bounding its orthogonal rank could be of independent interest.