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.

翻译：许多重要的科学和工程问题可转化为NP完全问题，这类问题在计算机科学和数学领域具有重要研究价值。目前，现有经典算法和量子算法均无法在多项式时间内求解此类问题。针对这一难题，本文提出一种量子可行性标记算法，用于标记顶点着色问题（一种著名的NP完全问题）的所有可能解。该算法将顶点着色问题转化为对无序数据库的搜索问题，其中标记了优良元素与劣质元素。最新提出的变分量子搜索算法已在电路深度上实现指数级加速，可在26量子比特规模下从无序数据库中高效找出优良元素。通过将QFL生成的标记及其关联的可能解作为输入，VQS算法能够找出顶点着色问题的所有可行解。QFL所需的量子比特数和电路深度均为顶点数、边数和颜色数的多项式函数。我们已在IBM Qiskit模拟器上实现了QFL，成功求解了一个4色4顶点3边的着色问题。