A Quantum Framework for K Coloring of Graphs

Abstract:Graph coloring is a well-known NP-complete problem with applications in scheduling, register allocation, frequency assignment, and network optimization. Quantum computing offers the potential for polynomial or even super-polynomial speedups in certain problem instances, yet practical quantum graph coloring methods must carefully balance qubit resources, circuit depth, and constraint enforcement. We proposed a solver agnostic quantum framework for $K$-coloring. Along with a novel encoding and efficient constraint implementation for its exact coloring exploiting Grover search and almost optimal coloring by the Quantum Approximate Optimization Algorithm \emph{(QAOA)}, and \emph{Quantum Annealing}. Unlike using $O(NK)$ qubits as in \emph{SOTA}, we reduced the qubit requirements to $O(N \log_2 K)$, along with optimized comparator circuits enforcing adjacency constraints. Further symmetry-based graph reduction is incorporated as an optional preprocessing step to further reduce the instance size before quantum execution.

arXiv:2609.32348 [quant-ph]

(or arXiv:2609.32348v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.32348 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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