# A Hybrid Optimization Algorithm for the Synthesis of Sparse Array Pattern Diagrams
## 2 Quantum Hybrid Particle Swarm Optimization Algorithm Combined with Quantum Behavior
### Quantum-Behaved Particle Swarm Optimization Algorithm
The radiation pattern of the array is shown when \(\varphi=0^{\circ}\) in Figure 10b. Figure 9: (**a**) Normalized pattern of a sparse multi-ring concentric circular array; (**b**) iteration curve; (**c**) 3D pattern. Table 4 presents the comparison results between the QPSO algorithm and the iterative convex optimization (ICO) and the improved differential evolution algorithm (DEA):

In the two references for the DEA and ICO, the maximum radius of the array is approximate, and the PSLL values of \(-24.67\) dB and \(-24.10\) dB are quite close. In contrast, QPSO achieves a significantly lower PSLL of \(-25.53\) dB, with a number of array elements comparable to those of ICO. It is worth noting that References [33; 34] primarily focus on optimizing a single variable, whereas the current work optimizes multiple variables, including element spacing, the number of elements, and the aperture between concentric rings. This comprehensive optimization approach provides a meaningful basis for comparison. ## 4 Conclusions

This study proposes a novel hybrid optimization algorithm (QPSO). Through the synergistic effect of the quantum potential well model and the adaptive genetic variation mechanism, it effectively solves the multi-peak optimization problem found in the pattern synthesis of sparse array antennas. This algorithm shows significant advantages in the optimization of linear arrays (PSLL = \(-20.45\) dB), planar rectangular arrays (full sidelobes \(<-40\) dB), and multi-ring concentric circular arrays (PSLL = \(-25.53\) dB), and represents a considerable improvement compared with the traditional PSO, GWO, and WOA algorithms. Its innovation is reflected in: (1) the quantum tunneling effect, which enhances global exploration ability; (2) dynamic mutation probability, which maintains population diversity; (3) the elitist retention strategy, which ensures convergence stability. These characteristics provide new solutions for the optimization of large-scale array antennas. Conceptualization, Y.L. and L.H. ; methodology, Y.L., L.H. and X.X. ; validation, X.X. ; writing--original draft preparation, Y.L. and L.H. ; writing--review and editing, Y.L. and H.Y. All authors have read and agreed to the published version of the manuscript. This work was supported by the Hubei Provincial Natural Science Foundation of China, 2024 AFB966. Not applicable. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Work** & **Type** & \begin{tabular}{c} **Array** \\ **Aperture** \\ \end{tabular} & \begin{tabular}{c} **Aperture** \\ **Requirement** \\ \end{tabular} & \
\begin{tabular}{c} **Array** \\ **Elements** \\ \end{tabular} &
\begin{tabular}{c} **PSLL** \\ **(dB)** \\ \end{tabular} \\ \hline
[33] & DEA & \(4.98\lambda\) & \(d=0.5\lambda\) & / & \(-24.67\) \\
[34] & ICO & \(5\lambda\) & \(d\geq 0.5\lambda\) & \(224\) & \(-24.10\) \\ This work & QPSO & \(3.5\lambda\sim 5.6\lambda\) & \(0.8\lambda\geq d\geq 0.5\lambda\) & \(230\) & \(-25.53\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of several concentric circles. Figure 10: (**a**) Element layout of a sparse multi-ring concentric circular array; (**b**) array radiation pattern at \(\varphi=0^{\circ}\). **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author and the first author. **Acknowledgments:** We thank the editor and the anonymous reviewers for their constructive comments, which helped to improve our work. **Conflicts of Interest:** The authors declare no conflicts of interest. ## Abbreviations

The following abbreviations are used in this manuscript: