Quantum Feasibility Labeling for NP-Complete Vertex Coloring Problem

Many important science and engineering problems can be converted into NP-complete problems which are of significant importance in computer science and mathematics. Currently, neither existing classical nor quantum algorithms can solve these problems in polynomial time. To address this difficulty, this paper proposes a quantum feasibility labeling (QFL) algorithm to label all possible solutions to the vertex coloring problem, which is a well-known NP-complete problem. The QFL algorithm converts the vertex coloring problem into the problem of searching an unstructured database where good and bad elements are labeled. The recently proposed variational quantum search (VQS) algorithm was demonstrated to achieve an exponential speedup, in circuit depth, up to 26 qubits in finding good element(s) from an unstructured database. Using the labels and the associated possible solutions as input, the VQS can find all feasible solutions to the vertex coloring problem. The number of qubits and the circuit depth required by the QFL each is a polynomial function of the number of vertices, the number of edges, and the number of colors of a vertex coloring problem. We have implemented the QFL on an IBM Qiskit simulator to solve a 4-colorable 4-vertex 3-edge coloring problem.