On the exact quantum chromatic number of generalized Johnson graphs
TL;DR
This paper investigates the quantum chromatic number of generalized Johnson graphs and determines the exact quantum chromatic numbers of two infinite families of generalized Johnson graphs, which exhibit an exponential separation between the classical and quantum chromatic numbers.
Abstract
The quantum chromatic number is a fundamental parameter in the study of nonlocal games, capturing the extent to which entanglement can improve performance in distributed tasks. In this paper, we investigate the quantum chromatic number of generalized Johnson graphs. By constructing modulus-one orthogonal representations, we obtain general upper bounds on their quantum chromatic numbers. We further analyze the smallest eigenvalue of these graphs. Combining the resulting Hoffman-type lower bounds with the upper bounds obtained from orthogonal representations, we determine the exact quantum chromatic numbers of two infinite families of generalized Johnson graphs. Finally, applying a forbidden-distance theorem for binary codes, we show that the classical chromatic numbers of these families grow exponentially with $n$, whereas their quantum chromatic numbers grow linearly. These families exhibit an exponential separation between the classical and quantum chromatic numbers.