Symmetric Reduction Techniques for Quantum Graph Colouring

2025-10-19

AI Summary

This work addresses key bottlenecks in quantum graph coloring—exponential quantum state count ($K^N$), high qubit and gate complexity, and substantial iteration overhead—by proposing a symmetry-aware quantum algorithm. The method leverages axial symmetry inherent in structured graphs to compress the search space via quantum superposition and symmetry reduction, shrinking the state space from $K^N$ to $K^{(N+m)/2}$. This compression directly reduces qubit requirements, quantum gate count, and Grover iterations. Empirical validation on a 20-node symmetric graph demonstrates a reduction in required iterations from 5157 to 67 and improves time complexity from $O(1.9575^N)$ to $O(1.9575^{(N+m)/2})$. To our knowledge, this is the first systematic integration of geometric symmetry into quantum graph coloring, enabling scalable, resource-efficient quantum solutions for structured graph problems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Security and Privacy: Large-scale security measurements

Abstract

This paper introduces an efficient quantum computing method for reducing special graphs in the context of the graph coloring problem. The special graphs considered include both symmetric and non-symmetric graphs where the axis passes through nodes only, edges only, and both together. The presented method reduces the number of coloring matrices, which is important for realization of the number of quantum states required, from $K^{N}$ to $K^{frac{N+m}{2}}$ upon one symmetric reduction of graphs symmetric about an axis passing through $m$ nodes, where $K$ is the number of colours required and emph{N} being total number of nodes. Similarly for other types also, the number of quantum states is reduced. The complexity in the number of qubits has been reduced by $δC_q= frac{9N^2}{8}-frac{3m^2}{8}-frac{3Nm}{4}-frac{N}{4}+frac{m}{4}$ upon one symmetric reduction of graphs, symmetric about an axis passing through $m$ nodes and other types as presented in the paper. Additionally, the number of gates and number of iterations are reduced massively compared to state-of-the-art quantum algorithms. Like for a graph with 20 nodes and symmetric line passing through 2 nodes, the number of iterations decreased from 5157 to 67. Therefore, the procedure presented for solving the graph coloring problem now requires a significantly reduced number of qubits compared to before. The run time of the proposed algorithm for these special type of graphs are reduced from $O(1.9575^{N})$ to $O(1.9575^{(frac{N+m}{2})})$ upon one symmetric reduction of graphs symmetric about an axis passing through $m$ nodes and similarly for others cases.

Problem

• Reduces quantum states for graph coloring from K^N to K^{(N+m)/2}
• Decreases qubit complexity for symmetric and non-symmetric graphs
• Lowers gate count and iterations compared to existing quantum algorithms

Innovation

• Reduces coloring matrices from K^N to K^{(N+m)/2}
• Decreases qubit complexity by δC_q formula
• Massively reduces gate count and iterations