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.
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
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.
The quantum chromatic number,
χq(G), of a graph
G was originally defined as the minimal number of colors necessary in a quantum protocol in which two provers that cannot communicate with each other but share an entangled state can convince an interrogator with certainty that they have a coloring of the graph. We use an equivalent purely combinatorial definition of
χq(G) to prove that many spectral lower bounds for the chromatic number,
χ(G), are also lower bounds for
χq(G). This is achieved using techniques from linear algebra called pinching and twirling. We illustrate our results with some examples.
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
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-
r quantum chromatic numbers. While it can now be shown that for every
r,
χq and
χq(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
G21 on 21 vertices with
χq(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
Gmsg on 57 vertices that was first considered in the aforementioned paper of Mančinska and Roberson and which satisfies
χq(Gmsg)=3 and
χq(1)(Gmsg)=4. In addition,
G21 provides the first provable separation between the parameters
χq(1) and
χq(2). We believe that our techniques for constructing
G21 and lower bounding its orthogonal rank could be of independent interest.
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.
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.
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 noncontextuality 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, Hasegawa, Kikuchi, and Sasaki on the quantum chromatic number, we give a family of Kochen-Specker sets of growing dimension.