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arXiv:2609.04677

On the exact quantum chromatic number of generalized Johnson graphs

Gaojun Luo, Xiwang Cao, Shitao Li, Yang Li

Published Sep 4, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2609.04677

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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 nnn, whereas their quantum chromatic numbers grow linearly. These families exhibit an exponential separation between the classical and quantum chromatic numbers.
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.
We compute the exact value of the quantum chromatic numbers of Hadamard graphs of order n=2Nn=2^Nn=2N for NNN a multiple of 444 using the upper bound derived by Avis, Hasegawa, Kikuchi, and Sasaki, as well as an application of the Hoffman-like lower bound of Elphick and Wocjan that was generalized by Ganesan for quantum graphs. As opposed to prior computations for the lower bound, our approach uses Ito's results on conjugacy class graphs allowing us to also find bounds on the quantum chromatic numbers of products of Hadamard graphs. In particular, we also compute the exact quantum chromatic number of the categorical product of Hadamard graphs.
We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince a referee that they have a colouring of the graph.After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 222, nor if it is 333 in a restricted quantum model; on the other hand, we exhibit a graph on 181818 vertices and 444444 edges with chromatic number 555 and quantum chromatic number 444.
The quantum chromatic number of a graph G is sandwiched between its chromatic number and its clique number, which are well-known NP-hard quantities. We restrict our attention to the rank-1 quantum chromatic number χq(1)(G), which upper bounds the quantum chromatic number, but is defined under stronger constraints. We study its relation with the chromatic number χ(G) and the minimum dimension of orthogonal representations ξ(G). It is known that ξ(G) ≤ χq(1)(G) ≤ χ(G). We answer three open questions about these relations: we give a necessary and sufficient condition to have ξ(G) = χq(1)(G), we exhibit a class of graphs such that ξ(G) ≤ χq(1)(G), and we give a necessary and sufficient condition to have χq(1)(G) ≤ χ(G). Our main tools are Kochen-Specker sets, collections of vectors with a traditionally important role in the study of contextuality of physical theories and, more recently, in the quantification of quantum zero-error capacities. Finally, as a corollary of our results and a result by Avis et al on the quantum chromatic number, we give a family of Kochen-Specker sets of growing dimension.
We make two contributions pertaining to the study of the quantum chromatic numbers of small graphs. Firstly, in an elegant paper, Mančinska and Roberson [Baltic Journal on Modern Computing, 4(4), 846-859, 2016] gave an example of a graph G14G_{14}G14​ 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-rrr quantum chromatic numbers. While it can now be shown that for every rrr, χq\chi_qχq​ and χq(r)\chi^{(r)}_qχq(r)​ 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 G21G_{21}G21​ on 21 vertices with χq(G21)=χq(2)(G21)=4\chi_q(G_{21}) = \chi^{(2)}_q(G_{21}) = 4χq​(G21​)=χq(2)​(G21​)=4 and $ \xi(G_{21}) = \chi^{(1)}_q(G_{21}) = \chi(G_{21}) = 5$. The previous record was held by a graph GmsgG_{msg}Gmsg​ on 57 vertices that was first considered in the aforementioned paper of Mančinska and Roberson and which satisfies χq(Gmsg)=3\chi_q(G_{msg}) = 3χq​(Gmsg​)=3 and χq(1)(Gmsg)=4\chi^{(1)}_q(G_{msg}) = 4χq(1)​(Gmsg​)=4. In addition, G21G_{21}G21​ provides the first provable separation between the parameters χq(1)\chi^{(1)}_qχq(1)​ and χq(2)\chi^{(2)}_qχq(2)​. We believe that our techniques for constructing G21G_{21}G21​ and lower bounding its orthogonal rank could be of independent interest.
We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 2, nor if it is 3 in a restricted quantum model; on the other hand, we exhibit a graph on 18 vertices and 44 edges with chromatic number 5 and quantum chromatic number 4.