Circuit Design for $k$-coloring Problem and Its Implementation on Near-term Quantum Devices

Nowadays in Quantum Computing, the implementation of quantum algorithm has created a stir since Noisy Intermediate-Scale Quantum (NISQ) devices are out in the market. Researchers are mostly interested in solving NP-complete problems with the help of quantum algorithms for its speed-up. As per the work on computational complexity by Karp \cite{karp}, if any of the NP-complete problem can be solved then any other NP-complete problem can be reduced to that problem in polynomial time. In this Paper, $k$-coloring problem (NP-complete problem) has been considered to solve using Grover's search. A comparator-based approach has been used to implement $k$-coloring problem which enables the reduction of the qubit cost compared to the state-of-the-art. An end-to-end automated framework has been proposed to implement the $k$-coloring problem for any unweighted and undirected graph on any available Noisy Intermediate-Scale Quantum (NISQ) devices, which helps in generalizing our approach.