**1. Introduction**

With the development of modern technology, artificial intelligence is becoming more and more important in society, and more and more mature, and can be used to deal with many problems that cannot be solved by traditional methods, such as wind energy decision system (WEDS) [1] and social cognitive radio network (SCRN) [2]. There are many kinds of classification methods; the simplest classification method is divided into the traditional method and modern intelligent method [3]. The traditional optimization algorithms generally deal with structured problems. The algorithms are deterministic and have only one global optimal solution. Meanwhile, the intelligent optimization algorithms generally deal with a more general description of the problems, which is heuristic, and have multiple extreme values. Intelligent optimization algorithms require certain strategies to prevent falling into the local optimum and try to find the global optimum, such as scheduling [4–7], image [8–10], feature selection [11–13] and detection [14,15], path planning [16,17], cyber-physical social system [18,19], texture discrimination [20], factor evaluation [21], saliency detection [22], classification [23,24], object extraction [25], gesture segmentation [26], economic load dispatch [27,28], shape design [29], big data and large-scale optimization [30,31], signal processing [32], multi-objective optimization [33,34], big data optimization [30,31], unit commitment [35], vehicle routing [36], knapsack problem [37–39], fault diagnosis [40–42], and test-sheet composition [43]. Because intelligent algorithms are not strictly dependent on the mathematical relationship, and because the problems that need to be solved are becoming more and more complex, intelligent algorithms are widely used to solve various complex optimization problems. As a result, the frequency of intelligent algorithms has exceeded that of traditional algorithms. At present, some main intelligent algorithms are widely used, such as genetic algorithm (GA) [44,45], particle swarm optimization algorithm (PSO) [46–48], krill herd (KH) [49–54], ant colony optimization algorithm (ACO) [55–57], differential evolution algorithm (DE) [58,59], adaptive island evolutionary algorithm (AIE) [60], and delayed start parallel evolutionary algorithm (DSPE) [61].

The swarm intelligence algorithms mainly come from the bionic idea, which is a set of methods summarized by referring to the laws of life and predatory behavior of animals in the biological world. A swarm intelligence algorithm is a kind of algorithm that works by studying the collective behavior of animals, and the most famous swarm intelligence algorithms are particle swarm optimization algorithm (PSO) [46] and ant colony optimization algorithm (ACO) [55]. An evolutionary algorithm is mainly a kind of method obtained by studying the process of biological genetic change, and these include genetic algorithm (GA) [44], genetic programming (GP) [62], evolutionary strategy (ES) [63,64], differential evolution (DE) [58] and other algorithms are based on genes. The krill herd (KH) [49] and quantum-behaved particle swarm optimization (QPSO) [65] algorithms mentioned in this paper belong to the swarm intelligence algorithm. The KH algorithm has the advantages of simplicity, flexibility, and high computational efficiency, and the QPSO algorithm has a relatively fast convergence speed, but when they are used to solve the 100-Digit Challenge problems, they do not do well. Owing to the poor local optimization ability of QPSO, it is easy to fall into the local optimal value, so it does not solve this problem well. In order to study an algorithm with very high accuracy, as well as to simultaneously optimize the exploitation and exploration and improve the accuracy of annealing krill quantum particle swarm optimization (AKQPSO), we studied the KH algorithm with strong exploitation and the QPSO algorithm with strong exploration. By combining their advantages, the new algorithm overcomes their original shortcomings and has a strong ability in exploration and exploitation. We combine the advantages of KH and QPSO, and optimize them, forming the annealing krill quantum particle swarm optimization (AKQPSO) algorithm. The new algorithm solved the 100-Digit Challenge better. Moreover, the study solved the problem of the poor accuracy of general algorithms, and exploitation and exploration cannot achieve the optimal at the same time.

In this paper, the 100-Digit Challenge problem is difficult to solve by traditional methods, and we can solve this problem better by using a swarm intelligence algorithm. In 2002, academician Nick Trefethen of Oxford University and the Society for Industrial and Applied Mathematics (SIAM) jointly developed the 100-Digit Challenge problem, which was proposed mainly to test high-precision computing [18]. The challenge consists of ten problems, each of which needs to be accurate to ten decimal places, with a total of 100 digits, so the problem is named the 100-Digit Challenge. However, traditional methods need a lot of computation to solve this challenge, and they cannot ge<sup>t</sup> good results, so we use swarm intelligence algorithm to solve these problems, and ge<sup>t</sup> satisfactory results.

The rest of paper is organized as follows. Most representative studies regarding the KH and QPSO algorithm are reviewed in Section 2, and these two algorithms are introduced in Section 3. The proposed algorithm is described in Section 4 and the experimental results of AKQPSO are presented in Section 5. Section 6 provides the main conclusions and highlights future work.
