Symmetric Reduction Techniques for Quantum Graph Colouring

TL;DR

This work tackles the $K$-coloring problem on specially structured graphs by introducing symmetry-based quantum reductions that lower the effective node count from $N$ to $(N+m)/2$ for an axis through $m$ nodes. It classifies reductions into cases based on whether the symmetry axis passes through nodes, edges, or both, and provides Algorithms 1–4 to perform the reductions and a quantum coloring circuit $Q$ with mirroring and amplitude amplification. Key contributions include closed-form improvements in qubit complexity, $\delta C_q$, and substantial reductions in the number of coloring matrices, gates, and iteration counts, lowering runtimes from $O(1.9575^{N})$ to $O(1.9575^{(N+m)/2})$ in symmetric cases. This yields practical advantages for implementing quantum graph coloring on graphs with symmetry, enabling larger instances to be tackled with fewer quantum resources.

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.

Figures (8)

• Figure 1: (a) Symmetric graph with axis passing through nodes only; Cutting the graph along FC (b) Lower half (c) Upper half are obtained
• Figure 2: Graph of type 2.a: Line of symmetry passing through the edges.
• Figure 3: Two parts after division along the symmetric line of type 2.a excluding the critical edges.
• Figure 4: Non-symmetric axis passing through node.
• Figure 5: Non-symmetric axis passing through an edge connecting $c_3$ and $c_5$.