Circuit Design for $k$-coloring Problem and Its Implementation in Any Dimensional Quantum System

TL;DR

This paper introduces a generalized quantum algorithm for the $k$-coloring problem applicable to any dimensional quantum system, reducing qubit costs and gate counts, and providing an automated framework for implementation on various quantum devices.

Contribution

It presents the first implementation of the $k$-coloring problem in any $d$-dimensional quantum system, with a comparator-based approach that reduces resource requirements and an automated framework for practical deployment.

Findings

Reduced qubit cost compared to binary systems

Significant gate count reduction in ternary oracle circuits

Framework applicable to NISQ and multi-valued quantum devices

Abstract

With the evolution of quantum computing, researchers now-a-days tend to incline to find solutions to NP-complete problems by using quantum algorithms in order to gain asymptotic advantage. In this paper, we solve $k$-coloring problem (NP-complete problem) using Grover's algorithm in any dimensional quantum system or any $d$-ary quantum system for the first time to the best of our knowledge, where $d \ge 2$. A newly proposed comparator-based approach helps to generalize the implementation of the $k$-coloring problem in any dimensional quantum system. Till date, $k$-coloring problem has been implemented only in binary and ternary quantum system, hence, we abide to $d=2$ or $d=3$, that is for binary and ternary quantum system for comparing our proposed work with the state-of-the-art techniques. This proposed approach makes the reduction of the qubit cost possible, compared to the state-of-the-art binary quantum systems. Further, with the help of newly proposed ternary comparator, a substantial reduction in quantum gate count for the ternary oracle circuit of the $k$-coloring problem than the previous approaches has been obtained. An end-to-end automated framework has been put forward for implementing the $k$-coloring problem for any undirected and unweighted graph on any available Near-term quantum devices or Noisy Intermediate-Scale Quantum (NISQ) devices or multi-valued quantum simulator, which helps in generalizing our approach.