The main purpose of this work is to collect some interesting facts related to quantum graph coloring.We consider the orthogonality graph of the three-dimensional vectors with entries from the set {-1, 0, 1}.If we ignore the overall sign of the vectors, this yields a graph on thirteen vertices, which we denote by G13.It turns out that G13 (or the graph obtained from this by adding an apex vertex) is the smallest known example which exhibits separations between several graph parameters.We are particularly interested in using G13 to separate quantum graph parameters from their classical counterparts.We give a proof that G13 is not quantum 3-colorable, which shows that its orthogonal rank is strictly less than its quantum chromatic number.This also proves, along with other previously known results, that adding an apex vertex to G13 does not change its quantum chromatic number.The graph G13 is the first and only known graph with either of these properties.Lastly, we investigate a graph construction used for taking instances of 3-SAT to instances of 3-COLORING.We show that any graph constructed in this manner has an orthogonal representation in dimension three if and only if it is 3-colorable.Since it has been previously shown that there are such graphs with quantum 3-colorings but no classical 3-colorings, it follows that quantum chromatic number can be strictly less than orthogonal rank.Together with the above, this proves that orthogonal rank and quantum chromatic number are not comparable as graph parameters.
The main purpose of this work is to collect some interesting facts related to quantum graph coloring.We consider the orthogonality graph of the three-dimensional vectors with entries from the set {-1, 0, 1}.If we ignore the overall sign of the vectors, this yields a graph on thirteen vertices, which we denote by G13.It turns out that G13 (or the graph obtained from this by adding an apex vertex) is the smallest known example which exhibits separations between several graph parameters.We are particularly interested in using G13 to separate quantum graph parameters from their classical counterparts.We give a proof that G13 is not quantum 3-colorable, which shows that its orthogonal rank is strictly less than its quantum chromatic number.This also proves, along with other previously known results, that adding an apex vertex to G13 does not change its quantum chromatic number.The graph G13 is the first and only known graph with either of these properties.Lastly, we investigate a graph construction used for taking instances of 3-SAT to instances of 3-COLORING.We show that any graph constructed in this manner has an orthogonal representation in dimension three if and only if it is 3-colorable.Since it has been previously shown that there are such graphs with quantum 3-colorings but no classical 3-colorings, it follows that quantum chromatic number can be strictly less than orthogonal rank.Together with the above, this proves that orthogonal rank and quantum chromatic number are not comparable as graph parameters.
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.
Cameron, Montanaro, Newman, Severini and Winter gave a construction which shows that, for n∈{2,4,8}, any graph G which admits a real n-dimensional orthogonal representation is quantumly n-colorable. This result can be recast as the statement that the real sphere Sn−1 is quantumly n-colorable for these values of n. We investigate possible extensions of their construction. We first show that their hypothesis that the orthogonal representation be real-valued is required by proving that there is no analogue of this for the complex spheres, which all have quantum chromatic number strictly bigger than the dimension except in two dimensions. We also provide candidate finitary witnesses of this and show for the first time that the real and complex orthogonal ranks are distinct as a byproduct. For the real case, we show that if Sn−1 is quantumly n-colorable, then either n=2 or n is a multiple of 4, and show that the converse holds whenever a Hadamard matrix of order n exists. Hence, assuming the Hadamard conjecture, this completely classifies the dimensions to which the CMNSW construction can be extended. Our method of proof involves showing the equivalence between the existence of such a construction and the existence of a maximal code space for Clifford-algebraic errors given a clean ancilla, and we believe that the representation-theoretic techniques we use for tackling the latter problem could be of independent interest. It also follows from this equivalence that Sn−1 admits a rank-one quantum n-coloring if and only if n∈{2,4,8}, thereby settling a conjecture of Zeng and Zhang, as does the fact that for all m≥1, there exists a catalytic zero-error remote state preparation protocol for real m-qubit states with m bits of communication and which consumes m ebits.
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 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 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.
Quantum graphs have been introduced by Duan, Severini, and Winter to describe the zero-error behaviour of quantum channels. Since then, quantum graph theory has become a field of study in its own right. A substantial source of difficulty in working with quantum graphs compared to classical graphs stems from the fact that they are no longer discrete objects. This makes it generally difficult to construct insightful, non-trivial examples. We present a collection of non-trivial quantum graphs that can be thought of in discrete terms, and that can be expressed in the diagrammatic formalism introduced by Musto, Reutter, and Verdon. The examples arise as the quantum graphs acted on by increasingly smaller classical matrix groups, and are parametrised by triples of matrices (A,B,C). The parametrisation reveals a clean decomposition of quantum graph structure into classical and genuinely quantum components: A and C are described by a classical weighted graph called the strange graph, while B provides a purely quantum contribution with no classical analogue. Based on this model, we give exact formulas or establish bounds for quantum graph parameters, such as the number of connected components, the chromatic number, the independence number, and the clique number. Our results provide the first large, parametric families of quantum graphs for which standard graph parameters can be computed analytically.