# A Quantum-Based Beetle Swarm Optimization Algorithm for Numerical Optimization

^{1}

^{2}

^{*}

## Abstract

**:**

## Featured Application

**The algorithm proposed in this paper can be widely used in many fields, such as combinatorial optimization, parameter tuning, path planning, etc.**

## Abstract

## 1. Introduction

- We solved the shortcoming of the BAS algorithm in that it cannot handle high-dimensional optimization problems, and the designed QBSO algorithm has an excellent performance in solving 30-dimensional CEC benchmark functions.
- We used quantum representation to deal well with the balance between the population size in terms of the exploratory power and the algorithmic speed, using fewer individuals to represent more information about the population.

## 2. Related Work

## 3. Algorithm

#### 3.1. Principle of the BAS Algorithm

#### 3.2. Principle of the QBSO Algorithm

#### 3.2.1. Quantum Representation

#### 3.2.2. Quantum Rotation Gate

#### 3.3. Computational Complexity Analysis

## 4. Experiment

#### 4.1. Unimodal Unconstrained Optimization

#### 4.2. Multimodal Unconstrained Optimization

#### 4.3. Population Diversity Study

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Wu, Q.; Shen, X.; Jin, Y.; Chen, Z.; Li, S.; Khan, A.H.; Chen, D. Intelligent beetle antennae search for uav sensing and avoidance of obstacles. Sensors
**2019**, 19, 1758. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wu, Q.; Lin, H.; Jin, Y.; Chen, Z.; Li, S.; Chen, D. A new fallback beetle antennae search algorithm for path planning of mobile robots with collision-free capability. Soft Comput.
**2019**, 24, 2369–2380. [Google Scholar] [CrossRef] - Jiang, X.; Lin, Z.; He, T.; Ma, X.; Ma, S.; Li, S. Optimal path finding with beetle antennae search algorithm by using ant colony optimization initialization and different searching strategies. IEEE Access
**2020**, 8, 15459–15471. [Google Scholar] [CrossRef] - Zhu, Z.; Zhang, Z.; Man, W.; Tong, X.; Qiu, J.; Li, F. A new beetle antennae search algorithm for multi-objective energy management in microgrid. In Proceedings of the 2018 13th IEEE Conference on Industrial Electronics and Applications (ICIEA), Wuhan, China, 31 May–2 June 2018; pp. 1599–1603. [Google Scholar]
- Jiang, X.Y.; Li, S. Beetle Antennae Search without Parameter Tuning (BAS-WPT) for Multi-objective Optimization. FILOMAT
**2020**, 34, 5113–5119. [Google Scholar] [CrossRef] - Zhao, Y.X.; Li, S.H.; Jin, F. Overlapping community detection in complex networks using multi-objective evolutionary algorithm. Comput. Appl. Math.
**2017**, 36, 749–768. [Google Scholar] - Pizzuti, C. A multiobjective genetic algorithm to find communities in complex networks. IEEE Trans. Evol. Comput.
**2012**, 16, 418–430. [Google Scholar] [CrossRef] - Dhiman, G.; Kumar, V. Seagull optimization algorithm: Theory and its applications for large-scale industrial engineering problems. Knowl. Based Syst.
**2019**, 165, 169–196. [Google Scholar] [CrossRef] - Karaboga, D. An Idea Based on Honey Bee Swarm for Numerical Optimization; Technical Report TR06; Erciyes University: Kayseri, Türkiye, 2005. [Google Scholar]
- Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey Wolf Optimizer. Adv. Eng. Softw.
**2014**, 69, 46–61. [Google Scholar] [CrossRef] [Green Version] - Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the ICNN’95—International Conference on Neural Networks, Perth, WA, Australia, 27 November–1 December 1995; Volume 4, pp. 1942–1948. [Google Scholar]
- Holland, J. Adaptation in Natural and Artificial Systems; University of Michigan Press: Ann Arbor, MI, USA, 1975. [Google Scholar]
- Dorigo, M.; Gambardella, L.M. Ant colony system: A cooperative learning approach to the traveling salesman problem. IEEE Trans. Evol. Comput.
**1997**, 1, 53–66. [Google Scholar] [CrossRef] [Green Version] - Zamani, H.; Nadimi-Shahraki, M.; Gandomi, A. Starling murmuration optimizer: A novel bio-inspired algorithm for global and engineering optimization. Comput. Methods Appl. Mech. Eng.
**2022**, 392, 114616. [Google Scholar] [CrossRef] - Kirkpatrick, S.; Gelatt, C.D.; Vecchi, M.P. Optimization by simulated annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] [PubMed] - Jiang, X.; Shuang, L. BAS: Beetle Antennae Search Algorithm for Optimization Problems. Available online: www.hhtp://arXiv:1710.10724 (accessed on 30 October 2017).
- Khan, A.T.; Cao, X.W.; Li, S. Enhanced Beetle Antennae Search with Zeroing Neural Network for online solution of constrained optimization. Neurocomputing
**2021**, 447, 294–306. [Google Scholar] [CrossRef] - Sabahat, E.; Eslaminejad, M.; Ashoormahani, E. A new localization method in internet of things by improving beetle antenna search algorithm. Wirel. Netw.
**2022**, 28, 1067–1078. [Google Scholar] [CrossRef] - Khan, A.; Li, S.; Zhou, X. Trajectory optimization of 5-link biped robot using beetle antennae search. IEEE Trans. Circuits Syst. II-Express Briefs
**2021**, 68, 3276–3280. [Google Scholar] [CrossRef] - Jiang, X.; Lin, Z.; Li, S. Dynamical attitude configuration with wearable wireless body sensor networks through beetle antennae search strategy. Measurement
**2020**, 167, 108–128. [Google Scholar] [CrossRef] - Khan, A.H.; Cao, X.; Xu, B.; Li, S. A model-free approach for online optimization of nonlinear systems. IEEE Trans. Circuits Syst. II: Express Briefs
**2022**, 69, 109–113. [Google Scholar] [CrossRef] - Han, K.H.; Kim, J. Quantum-inspired evolutionary algorithm for a class of combinatorial optimization. IEEE Trans. Comput.
**2002**, 6, 580–593. [Google Scholar] - Wang, J.; Chen, H. BSAS: Beetle Swarm Antennae Search Algorithm for Optimization Problems. 2018. Available online: https://arxiv.org/abs/1807.10470 (accessed on 27 July 2018).
- Khan, A.H.; Cao, X.; Li, S.; Katsikis, V.N.; Liao, L. BAS-ADAM: An ADAM based approach to improve the performance of beetle antennae search optimizer. IEEE/CAA J. Autom. Sin.
**2020**, 7, 461–471. [Google Scholar] [CrossRef] - Lin, M.; Li, Q.; Wang, F.; Chen, D. An improved beetle antennae search algorithm and its application on economic load distribution of power system. IEEE Access
**2020**, 8, 99624–99632. [Google Scholar] [CrossRef] - Zhou, T.J.; Qian, Q.; Fu, Y. An Improved Beetle Antennae Search Algorithm. Recent Dev. Mechatron. Intell. Robot. Proc. ICMIR
**2019**, 2019, 699–706. [Google Scholar] - Shao, X.; Fan, Y. An Improved Beetle Antennae Search Algorithm based on the Elite Selection Mechanism and the Nieghbor Mobility Strategy for Global Optimization Problems. IEEE Access
**2021**, 9, 137524–137542. [Google Scholar] [CrossRef] - Yu, X.W.; Huang, L.P.; Liu, Y.; Zhang, K.; Li, P.; Li, Y. WSN node location based on beetle antennae search to improve the gray wolf algorithm. Wirel. Netw.
**2022**, 28, 539–549. [Google Scholar] [CrossRef] - Lin, M.; Li, Q.; Wang, F.; Chen, D. An improved beetle antennae search algorithm with mutation crossover in TSP and engineering application. Appl. Res. Comput.
**2021**, 38, 3662–3666. [Google Scholar] - An, J.; Liu, X.; Song, H. Survey of Quantum Swarm Intelligence Optimization Algorithm. Comput. Eng. Appl.
**2022**, 7, 31–42. [Google Scholar] - Kundra, H.; Khan, W.; Malik, M. Quantum-inspired firefly algorithm integrated with cuckoo search for optimal path planning. Mod. Pyhsics C
**2022**, 33, 2250018. [Google Scholar] [CrossRef] - Zamani, H.; Nadimi-Shahraki, M.H.; Gandomi, A.H. QANA: Quantum-based avian navigation optimizer algorithm. Eng. Appl. Artif. Intell.
**2021**, 104, 104314. [Google Scholar] [CrossRef] - Nadimi-Shahraki, M.H.; Fatahi, A.; Zamani, H.; Mirjalili, S. Binary Approaches of Quantum-Based Avian Navigation Optimizer to Select Effective Features from High-Dimensional Medical Data. Mathematics
**2022**, 10, 2770. [Google Scholar] [CrossRef] - Zhou, N.-R.; Xia, S.-H.; Ma, Y.; Zhang, Y. Quantum particle swarm optimization algorithm with the truncated mean stabilization strategy. Quantum Inf. Process.
**2022**, 21, 21–42. [Google Scholar] [CrossRef] - Sun, J.; Feng, B.; Xu, W. Particle swarm optimization with particles having quantum behavior. In Proceedings of the 2004 Congress on Evolutionary Computation, Portland, OR, USA, 19–23 June 2004; pp. 325–331. [Google Scholar]
- Hao, T.; Huang, X.; Jia, C.; Peng, C. A quantum-inspired tensor network algorithm for constrained combinatorial optimization problems. Frontiers
**2022**, 10, 1–8. [Google Scholar] [CrossRef] - Amaro, D.; Modica, C.; Rosenkranz, M.; Fiorentini, M.; Benedetti, M.; Lubasch, M. Filtering variational quantum algorithms for combinatorial optimization. Quantum Sci. Technol.
**2022**, 7, 015021. [Google Scholar] [CrossRef] - Fallahi, S.; Taghadosi, M. Quantum-behaved particle swarm optimization based on solitons. Sci. Rep.
**2022**, 12, 13977. [Google Scholar] [CrossRef] [PubMed] - Soloviev, V.; Bielza, C.; Larrañaga, P. Quantum Approximate Optimization Algorithm for Bayesian network structure learning. Quantum Inf. Process.
**2022**, 22, 19. [Google Scholar] [CrossRef] - Li, M.W.; Wang, Y.T.; Geng, J.; Hong, W.C. Chaos cloud quantum bat hybrid optimization algorithm. Nonlinear Dyn.
**2021**, 103, 1167–1193. [Google Scholar] [CrossRef] - Wang, T.; Yang, L.; Liu, Q. Beetle Swarm Optimization Algorithm: Theory and Application. Filomat
**2020**, 34, 5121–5137. [Google Scholar] [CrossRef] - Mirjalili, S. The ant lion optimizer. Adv. Eng. Softw.
**2015**, 83, 80–98. [Google Scholar] [CrossRef]

**Figure 4.**Convergence of the optimal solution of the Griewank function at different iterations. (

**a**) Optimal solution in natural number unit; (

**b**) optimal solution in logarithmic unit.

**Figure 6.**The iteration curves when solving the multimodal benchmark functions with four algorithms.

**Figure 7.**Population diversity at different iterations when optimizing the Griewank function with the population size = 30.

Name | $\mathbf{Formulation}\mathbf{f}\left(\mathit{y}\right)$ | $\mathit{f}{\left(\mathit{y}\right)}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathbf{y}\left(\mathit{t}\right)$ |
---|---|---|---|

${F}_{1}$ | $-200{e}^{-0.2\sqrt{{y}_{1}^{2}+{y}_{2}^{2}}}$ | −200 | {0,0} |

${F}_{2}$ | $\sum}_{i=1}^{n-1}{\left({y}_{i}^{2}\right)}^{\left({y}_{i+1}^{2}+1\right)}+{\left({y}_{i+1}^{2}\right)}^{\left({y}_{i}^{2}+1\right)$ | 0 | {$0,0,\cdots ,0$} |

${F}_{3}$ | $-\frac{1+cos\left(12\sqrt{{y}_{1}^{2}+{y}_{2}^{2}}\right)}{\left(0.5\left({y}_{1}^{2}+{y}_{2}^{2}\right)+2\right)}$ | −1 | {0,0} |

${F}_{4}$ | $\sum}_{i=1}^{n}{\left|{y}_{i}\right|}^{i+1$ | 0 | {$0,0,\cdots ,0$} |

Name | Algorithm | Best | Worst | Average | Variance | Time(s) |
---|---|---|---|---|---|---|

${F}_{1}$ | PIO | −199.7120 | −175.2841 | −195.4801 | 19.4308 | 0.029 |

SOA | −199.9893 | −45.0261 | −185.6132 | 658.3112 | 0.010 | |

GWO | −200 | −200 | −200 | 10^{−28} | 0.011 | |

QBSO | −200 | −199.9999 | −200 | 10^{−10} | 0.087 | |

BSO | −200 | −177.8722 | −197.8545 | 10.3658 | 0.009 | |

BAS | −199.9965 | −199.8666 | −199.9398 | 10^{−4} | 0.017 | |

${F}_{2}$ | PIO | 10^{−4} | 31.4922 | 6.2104 | 60.2032 | 0.027 |

SOA | 0.0020 | 10^{4} | 10^{3} | 10^{7} | 0.018 | |

GWO | 0.0017 | 0.0391 | 0.0134 | 10^{−5} | 0.026 | |

QBSO | 10^{−7} | 10^{−5} | 10^{−6} | 10^{−11} | 0.152 | |

BSO | 0.1055 | 1.5872 | 1.2652 | 2.8857 | 0.018 | |

BAS | 8.366 | 33.949 | 19.747 | 24.237 | 0.017 | |

${F}_{3}$ | PIO | −0.9998 | −0.9291 | −0.9509 | 10^{−4} | 0.029 |

SOA | −1 | −0.0352 | −0.7232 | 0.0831 | 0.010 | |

GWO | −1 | −0.9362 | −0.9754 | 10^{−4} | 0.011 | |

QBSO | −1 | −1 | −1 | 10^{−23} | 0.088 | |

BSO | −1 | −0.9362 | −0.9641 | 10^{−4} | 0.010 | |

BAS | −0.996 | −0.465 | −0.897 | 10^{−3} | 0.018 | |

${F}_{4}$ | PIO | 10^{−6} | 10^{7} | 10^{5} | 10^{12} | 0.066 |

SOA | 0.0227 | 10^{45} | 10^{43} | 10^{88} | 0.025 | |

GWO | 10^{−4} | 10^{3} | 27.9445 | 10^{4} | 0.037 | |

QBSO | 10^{−19} | 10^{−15} | 10^{−16} | 10^{−31} | 0.181 | |

BSO | 10^{−4} | 10^{4} | 10^{3} | 10^{7} | 0.025 | |

BAS | 19.887 | 10^{4} | 10^{3} | 10^{8} | 0.017 |

Name | Algorithm | Best | Worst | Average | Variance | Time(s) |
---|---|---|---|---|---|---|

${F}_{1}$ | PIO | −200 | −156.4101 | −186.9236 | 126.6304 | 0.025 |

SOA | −199.9457 | −8.5804 | −119.6113 | 10^{3} | 0.009 | |

GWO | −200 | −199.9999 | −200 | 10^{−11} | 0.010 | |

QBSO | −200 | −199.9996 | −200 | 10^{−9} | 0.075 | |

BSO | −199.9958 | −10^{−4} | −177.3426 | 10^{3} | 0.004 | |

${F}_{2}$ | PIO | 0.0082 | 137.1152 | 26.5353 | 771.5456 | 0.020 |

SOA | 73.2925 | 10^{4} | 10^{4} | 10^{8} | 0.015 | |

GWO | 3.2853 | 57.7149 | 16.7590 | 111.2829 | 0.015 | |

QBSO | 10^{−8} | 10^{−6} | 10^{−6} | 10^{−12} | 0.095 | |

BSO | 3.4566 | 67.7782 | 21.5658 | 155.9654 | 0.010 | |

${F}_{3}$ | PIO | −1 | −0.6185 | −0.8981 | 0.0072 | 0.025 |

SOA | −0.9635 | −0.0045 | −0.2380 | 0.0722 | 0.009 | |

GWO | −1 | −0.9362 | −0.9478 | 10^{−4} | 0.010 | |

QBSO | −1 | −1 | −1 | 10^{−21} | 0.076 | |

BSO | −1 | −0.4877 | −0.9238 | 0.0079 | 0.003 | |

${F}_{4}$ | PIO | 10^{−4} | 10^{13} | 10^{11} | 10^{24} | 0.031 |

SOA | 10^{10} | 10^{48} | 10^{47} | 10^{95} | 0.013 | |

GWO | 10^{5} | 10^{19} | 10^{17} | 10^{36} | 0.018 | |

QBSO | 10^{−20} | 10^{−14} | 10^{−15} | 10^{−30} | 0.115 | |

BSO | 10^{17} | 10^{49} | 10^{47} | 10^{97} | 0.048 |

Name | $\mathbf{Formulation}\mathit{f}\left(\mathit{y}\right)$ | $\mathit{f}{\left(\mathit{y}\right)}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathit{y}\left(\mathit{t}\right)$ |
---|---|---|---|

Ackley | $-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}}\right)-\mathrm{exp}\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}\mathrm{cos}(2\pi {x}_{i})\right)+20+\mathrm{exp}\left(1\right)$ | 0 | {$0,0,\cdots ,0$} |

Griewank | $\sum _{i=1}^{n}}\frac{{x}_{i}^{2}}{4000}-{\displaystyle \prod _{i=1}^{n}}cos\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | 0 | {$0,0,\cdots ,0$} |

Rastrigin | $10\mathrm{n}+{\displaystyle \sum _{i=1}^{n}}\left({y}_{i}^{2}-10\mathrm{cos}(2\pi {y}_{i})\right)$ | 0 | {$0,0,\cdots ,0$} |

Quarrtic | $\sum _{i=1}^{n}}i{y}_{i}^{4}+random\left[0,1\right)$ | $0+\mathrm{rand}$ | {$\sqrt{i},\sqrt{i},\cdots ,\sqrt{i}$} |

Name | Algorithm | Best | Worst | Average | Variance | Time(s) |
---|---|---|---|---|---|---|

Ackley | PIO | 0.0210 | 5.6406 | 2.4490 | 2.5567 | 0.032 |

SOA | 0.0620 | 21.3100 | 19.4798 | 19.0403 | 0.021 | |

GWO | 20.6624 | 21.1627 | 20.9935 | 0.0081 | 0.030 | |

QBSO | 10^{−4} | 0.0030 | 0.0011 | 10^{−7} | 0.140 | |

BSO | 10^{−5} | 6.1147 | 1.9238 | 2.2163 | 0.023 | |

BAS | 3.753 | 5.506 | 4.399 | 0.1266 | 0.017 | |

Griewank | PIO | 10^{−4} | 0.1750 | 0.0390 | 0.0020 | 0.034 |

SOA | 10^{−5} | 4.4718 | 1.3344 | 0.8192 | 0.022 | |

GWO | 10^{−4} | 0.1750 | 0.0390 | 0.0020 | 0.029 | |

QBSO | 10^{−9} | 10^{−7} | 10^{−8} | 10^{−15} | 0.149 | |

BSO | 1.0792 | 5.4872 | 1.6651 | 0.4517 | 0.134 | |

BAS | 0.371 | 0.949 | 0.655 | 0.0156 | 0.017 | |

Rastrigin | PIO | 5.7557 | 247.5033 | 138.0620 | 10^{3} | 0.037 |

SOA | 0.4156 | 10^{4} | 10^{3} | 10^{7} | 0.017 | |

GWO | 27.9906 | 143.6525 | 55.9303 | 359.9034 | 0.029 | |

QBSO | 10^{−6} | 10^{−4} | 10^{−5} | 10^{−9} | 0.143 | |

BSO | 5.57 | 10^{4} | 10^{3} | 10^{6} | 0.022 | |

BAS | 82.358 | 10^{2} | 10^{2} | 10^{2} | 0.017 | |

Quarrtic | PIO | 0.0042 | 10^{3} | 10^{5} | 156.2653 | 0.056 |

SOA | 0.0084 | 10^{8} | 10^{7} | 10^{16} | 0.030 | |

GWO | 0.1165 | 1.1138 | 0.3810 | 0.0330 | 0.038 | |

QBSO | 10^{−6} | 0.0027 | 10^{−4} | 10^{−7} | 0.175 | |

BSO | 0.6687 | 10^{8} | 10^{8} | 10^{16} | 0.046 | |

BAS | 10^{2} | 10^{2} | 10^{2} | 10^{4} | 0.018 |

Name | Algorithm | Best | Worst | Average | Variance | Time(s) |
---|---|---|---|---|---|---|

Ackley | PIO | 0.0694 | 8.2592 | 4.3592 | 4.2878 | 0.023 |

SOA | 11.3070 | 21.3684 | 21.0656 | 1.1158 | 0.016 | |

GWO | 20.7051 | 21.1816 | 21.0613 | 0.0065 | 0.015 | |

QBSO | 10^{−4} | 0.0015 | 10^{−4} | 10^{−8} | 0.097 | |

BSO | 10^{−5} | 20 | 3.5073 | 25.7410 | 0.006 | |

Griewank | PIO | 0.0011 | 1.0355 | 0.6206 | 0.1467 | 0.029 |

SOA | 1.0423 | 13.6958 | 5.2995 | 10.9620 | 0.017 | |

GWO | 0.1494 | 0.9737 | 0.5470 | 0.0272 | 0.016 | |

QBSO | 10^{−10} | 10^{−8} | 10^{−8} | 10^{−16} | 0.095 | |

BSO | 1.2868 | 10.8748 | 3.9758 | 2.7785 | 0.007 | |

Rastrigin | PIO | 13.1031 | 312.7558 | 131.6508 | 10^{3} | 0.028 |

SOA | 159.2776 | 10^{4} | 10^{4} | 10^{8} | 0.011 | |

GWO | 85.6812 | 329.1141 | 193.3601 | 10^{3} | 0.016 | |

QBSO | 10^{−6} | 10^{−4} | 10^{−5} | 10^{−9} | 0.093 | |

BSO | 229.76 | 10^{4} | 10^{4} | 10^{7} | 0.006 | |

Quarrtic | PIO | 0.0047 | 10^{4} | 10^{3} | 10^{7} | 0.026 |

SOA | 11.1423 | 10^{9} | 10^{8} | 10^{17} | 0.018 | |

GWO | 62.0423 | 10^{4} | 10^{3} | 10^{7} | 0.018 | |

QBSO | 10^{−5} | 0.0109 | 0.0023 | 10^{−6} | 0.109 | |

BSO | 10.51 | 10^{9} | 10^{8} | 10^{18} | 0.013 |

D | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
---|---|---|---|---|---|---|---|---|---|---|

PIO | 17.5 | 48.1 | 68.7 | 90.1 | 10^{2} | 10^{2} | 10^{2} | 10^{2} | 10^{2} | 10^{2} |

SOA | 758 | 10^{3} | 10^{3} | 10^{4} | 10^{4} | 10^{4} | 10^{4} | 10^{4} | 10^{4} | 10^{4} |

GWO | 4.63 | 19.9 | 40.7 | 72.4 | 10^{2} | 10^{2} | 10^{2} | 10^{2} | 10^{2} | 10^{2} |

BSO | 15.99 | 10^{2} | 10^{2} | 10^{2} | 10^{3} | 10^{3} | 10^{3} | 10^{3} | 10^{3} | 10^{3} |

QBSO | 10^{−7} | 10^{−5} | 10^{−5} | 10^{−4} | 10^{−4} | 10^{−4} | 10^{−4} | 10^{−4} | 10^{−3} | 10^{−3} |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yu, L.; Ren, J.; Zhang, J.
A Quantum-Based Beetle Swarm Optimization Algorithm for Numerical Optimization. *Appl. Sci.* **2023**, *13*, 3179.
https://doi.org/10.3390/app13053179

**AMA Style**

Yu L, Ren J, Zhang J.
A Quantum-Based Beetle Swarm Optimization Algorithm for Numerical Optimization. *Applied Sciences*. 2023; 13(5):3179.
https://doi.org/10.3390/app13053179

**Chicago/Turabian Style**

Yu, Lin, Jieqi Ren, and Jie Zhang.
2023. "A Quantum-Based Beetle Swarm Optimization Algorithm for Numerical Optimization" *Applied Sciences* 13, no. 5: 3179.
https://doi.org/10.3390/app13053179