The study of quantum chromatic numbers of graphs is a hot research topic in recent years. However, the infinite family of graphs with known quantum chromatic numbers are rare, as far as we know, the only known such graphs (except for complete graphs, cycles, bipartite graphs and some trivial cases) are the Hadamard graphs $H_n$ with $2^n$ vertices and $n$ a multiple of $4$ . In this paper, we consider the graphs in Hamming schemes, we determined the quantum chromatic numbers of one class of such graphs. Notably, this is the second known family of graphs whose quantum chromatic numbers are explicitly determined except for some cases aforementioned. We also provide some bounds for the quantum chromatic numbers of some other graphs in Hamming schemes. Consequently, we can obtain the quantum chromatic numbers of products of some graphs.The study of quantum chromatic numbers of graphs is a hot research topic in recent years. However, the infinite family of graphs with known quantum chromatic numbers are rare, as far as we know, the only known such graphs (except for complete graphs, cycles, bipartite graphs and some trivial cases) are the Hadamard graphs $H_n$ with $2^n$ vertices and $n$ a multiple of $4$. In this paper, we consider the graphs in Hamming schemes, we determined the quantum chromatic numbers of one class of such graphs. Notably, this is the second known family of graphs whose quantum chromatic numbers are explicitly determined except for some cases aforementioned. We also provide some bounds for the quantum chromatic numbers of some other graphs in Hamming schemes. Consequently, we can obtain the quantum chromatic numbers of products of some graphs.

arXiv:2412.09904Dec 13, 20240 citations

Quantum graphs

Theoretical and mathematical physicsSep 22, 20080 citations

Basics of Quantum Graphs

Sovan Samanta, Antonios Kalampakas, Tofigh Allahviranloo

Studies in computational intelligenceJan 1, 20250 citations

A Graph Isomorphism Algorithm based on Continuous-Time Quantum Walks

Zhan Li, Yang Wang, Xiaogang Qiang

Graph isomorphism is an essential problem in graph theory and widely used in a variety of applications. With graph size increasing, graph isomorphism become difficult for classical computation since there is no polynomial algorithm for general graphs yet. Considering the advantages of quantum computation over classical computation in various tasks, quantum graph isomorphism algorithms have been studied to explore the potentials of quantum computation in graph isomorphism. In this work, we propose a graph isomorphism algorithm based on continuous-time quantum walks, where we construct the graph certificates from the output probabilities of continuous-time quantum walks on the tested graphs with added self-loops to graph vertices, and distinguish graphs by comparing the graph certificates. Numerous tests on diverse graph classes demonstrate the effectiveness of the proposed GI algorithm, which is not limited to a specific topological structure graph. Moreover, we test the proposed algorithm on classes of strongly regular graphs, and the results show that the distinguishing ability of the algorithm can be improved with the increasement of the added self-loops to distinct vertices.Graph isomorphism is an essential problem in graph theory and widely used in a variety of applications. With graph size increasing, graph isomorphism become difficult for classical computation since there is no polynomial algorithm for general graphs yet. Considering the advantages of quantum computation over classical computation in various tasks, quantum graph isomorphism algorithms have been studied to explore the potentials of quantum computation in graph isomorphism. In this work, we propose a graph isomorphism algorithm based on continuous-time quantum walks, where we construct the graph certificates from the output probabilities of continuous-time quantum walks on the tested graphs with added self-loops to graph vertices, and distinguish graphs by comparing the graph certificates. Numerous tests on diverse graph classes demonstrate the effectiveness of the proposed GI algorithm, which is not limited to a specific topological structure graph. Moreover, we test the proposed algorithm on classes of strongly regular graphs, and the results show that the distinguishing ability of the algorithm can be improved with the increasement of the added self-loops to distinct vertices.

International Conference on Computing, Communication, Perception and Quantum Technology (CCPQT)Aug 4, 20232 citations

Quantum Graph Learning: Frontiers and Outlook

Shuo Yu, Ciyuan Peng, Yingbo Wang +3 more

Quantum theory has shown its superiority in enhancing machine learning. However, facilitating quantum theory to enhance graph learning is in its infancy. This survey investigates the current advances in quantum graph learning (QGL) from three perspectives, i.e., underlying theories, methods, and prospects. We first look at QGL and discuss the mutualism of quantum theory and graph learning, the specificity of graph-structured data, and the bottleneck of graph learning, respectively. A new taxonomy of QGL is presented, i.e., quantum computing on graphs, quantum graph representation, and quantum circuits for graph neural networks. Pitfall traps are then highlighted and explained. This survey aims to provide a brief but insightful introduction to this emerging field, along with a detailed discussion of frontiers and outlook yet to be investigated.Quantum theory has shown its superiority in enhancing machine learning. However, facilitating quantum theory to enhance graph learning is in its infancy. This survey investigates the current advances in quantum graph learning (QGL) from three perspectives, i.e., underlying theories, methods, and prospects. We first look at QGL and discuss the mutualism of quantum theory and graph learning, the specificity of graph-structured data, and the bottleneck of graph learning, respectively. A new taxonomy of QGL is presented, i.e., quantum computing on graphs, quantum graph representation, and quantum circuits for graph neural networks. Pitfall traps are then highlighted and explained. This survey aims to provide a brief but insightful introduction to this emerging field, along with a detailed discussion of frontiers and outlook yet to be investigated.

arXiv:2302.00892Feb 2, 20233 citations

Quantum Graph Theory by Example

Gian Luca Spitzer, Ion Nechita

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.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.

math.OACross-listedarXiv:2603.23651QuantumMar 26, 20260 citations

H_n

with

2^n

vertices and

n

a multiple of

4

. In this paper, we consider the graphs in Hamming schemes, we determined the quantum chromatic numbers of one class of such graphs. Notably, this is the second known family of graphs whose quantum chromatic numbers are explicitly determined except for some cases aforementioned. We also provide some bounds for the quantum chromatic numbers of some other graphs in Hamming schemes. Consequently, we can obtain the quantum chromatic numbers of products of some graphs.

(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.