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

TL;DR

This paper presents a quantum algorithm for the NP-complete k-coloring problem, utilizing Grover's search and a comparator-based approach, with an automated framework for implementation on NISQ devices.

Contribution

It introduces a qubit-efficient quantum algorithm for k-coloring and an automated framework for implementation on NISQ devices, generalizing the approach for any unweighted, undirected graph.

Findings

Reduced qubit cost compared to existing methods

Successful implementation on NISQ devices

Framework applicable to any unweighted, undirected graph

Abstract

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.