This section describes other metaheuristic algorithms and their modifications for antenna array optimization problems. In [39], Panduro et al. used a genetic algorithm for the design of non-uniform circular arrays and obtained a large reduction of SLL. In [40], Khodier et al. used the particle swarm algorithm to synthesize the minimum side lobe level and compared it with other algorithms to verify the effectiveness of the particle swarm algorithm. In [41], Guney et al. used the honeybee algorithm to adjust the phase and amplitude of the array elements, and the effectiveness of the method was verified on a Chebyshev antenna model. In [42], Singh et al. used the Biogeography-based optimization algorithm to adjust the optimal excitation amplitude of the antenna array elements and effectively synthesized the antenna radiation map. In [43], Pappula et al. used a cat swarm optimization algorithm for the optimal array element location to suppress the SLL, which has applications in electromagnetic optimization. In [44], Singh et al. used the cuckoo optimization algorithm (COA) to optimize the linear and circular arrays, and the experimental results showed that COA has superior performance. In [45], Dib et al. used a symbiotic bio-search algorithm to synthesize the radiation map of symmetric antenna arrays and effectively obtained a radiation map with low interference. In [46], Saxena et al. introduced the gray wolf optimization algorithm into the field of electromagnetic and antenna optimization to optimize the excitation amplitude and array element position to achieve the minimum SLL optimization. In [47], Salgotra et al. proposed an extended version of the gray wolf optimizer (GWO-E) that utilizes multiple swarm partitioning and adversarial learning strategies to improve the performance of the algorithm. Experiments show that GWO-E obtains the largest SLL reduction in a non-uniform antenna array. In [48], Wu et al. used a spider monkey optimization algorithm to optimize the synthesized sparse line array in order to reduce the side level of the whole array. In [49], Sun et al. used the weed invasion algorithm for beam synthesis of LAA with circular antenna array (CAA) and obtained better synthesis characteristics in a real environment. In [50], Durmus et al. applied the equilibrium optimizer algorithm to symmetric LAA with non-uniform CAA for radiation map synthesis to achieve narrow half-power beam widths. In [51], Sharaqa et al. used a bio-geographic optimization algorithm (BBO) to optimize the maximum SLL reduction and validated the effectiveness of BBO in comparison with other well-known optimization methods. In [52], Das et al. determination of excitation amplitude and antenna array spacing to reduce SLL values using the moth flame optimization method. In [53], Subhashini et al. used the runner-root algorithm (RRA) for linear antenna array optimization and obtained good results. In [54], Almagboul et al. proposed to optimize the beamformer weights using an atomic search optimization algorithm to minimize the SLL value. In [55], Singh et al. proposed a novel algorithm by fusing a differential evolutionary algorithm with harmonic search to synthesize the loop antenna array to minimize the SLL. In [56], Vegesna et al. improved flower pollination algorithms for beam forming techniques with highly minimized SLL. In [57], Liang et al. introduced migration strategy and elite learning mechanism in Biogeography-based optimization algorithm and proposed a new method with an adaptive variation strategy. The proposed method is then tested on a classical function and used for the antenna array beam optimization problem. In summary, the antenna array optimization problem has gradually become a hot area of research. The linear antenna array is the most basic antenna geometric arrangement and the most practical geometric arrangement, but it still suffers from the defect of insufficient maximum SLL reduction. Finding a more effective method to solve the linear antenna array optimization problem is still a hot research topic.

The concept of “quantum” first appeared in the last century in the field of physics, and it is in quantum computers that quantum properties are fully exploited [58]. This concept has attracted the attention of many scholars and introduced quantum theory into meta-heuristic algorithms. In [59], Yu et al. introduced the idea of quantum rotating gate to encode the population for the defect that the basic dragonfly optimization algorithm easily falls into local optimum. In [60], Zhang et al. proposed a quantum-inspired satin bowerbird optimizer algorithm to encode populations based on the quantum bits of Bloch spheres. In [61], Deng et al. proposed an efficient DE algorithm for solving large-scale optimization problems by introducing quantum computing features and co-evolutionary ideas to improve DE for the shortcomings of differential evolutionary algorithms. In [62], Zhou et al. proposed a quantum wind-driven optimization algorithm for unmanned aerial vehicle path planning problems with quantum non-gates for variation. In [63], Coelho et al. proposed a mutation-based quantum particle swarm algorithm (QPSO) inspired by quantum mechanics to prevent premature convergence of the algorithm. In [64], Zou et al. introduced dynamic population optimization strategies in TLBO and used quantum learning strategies to maintain population diversity. In [65], Yu et al. introduced the quantum rotating gate strategy and the water cycle mechanism into the basic smile mould algorithm to compensate for the early convergence. In [66], Cui et al. proposed a quantum-inspired moth flame optimizer for clustering on the UCI benchmark dataset to test efficiency. In [67], Li et al. proposed a new quantum coded pathfinder optimization algorithm for parameter optimization problem in proton exchange membrane fuel cells (PEMFC). Some scholars combine quantum computing ideas with meta-heuristic algorithms in the field of antenna optimization. In [68], Ho et al. proposed an improved quantum particle swarm algorithm that escapes from local optima by introducing diversification and intensification phases, and applied it to the optimization problem of a 28-element LAA. In [69], Patidar et al. used the quantum particle swarm method to optimize a broadband frequency-invariant pattern. In [70], Gao et al. combined the TLBO algorithm with quantum theory, made the population evolve in a good direction through the method of quantum revolving gate, and finally applied it to linear array pattern synthesis. In [71], Liu et al. combined quantum computing ideas with genetic algorithms to optimize the maximum signal output power. In [72], Mikki et al. used quantum particle swarm optimization for linear array antenna synthesis problems. The above articles have successfully integrated quantum ideas and heuristic algorithms and applied to the field of antennas, but there are still shortcomings in maximizing the effect of reducing SLL. While reducing the SLL problem is not novel, more efficient methods are absolutely needed. Therefore, this paper combines the physics-based equilibrium optimizer algorithm with quantum ideas for the first time, and proposes an efficient quantum equilibrium optimizer for reducing the SLL value, and fully compares it with other methods to show the advantages of QEO.

In a linear antenna array, all the elements are placed in a straight line. For the linear antenna, the array factor can be expressed as the combination of array excitation amplitude and position in array space. The linear antenna array element has the characteristics of uniform radiation pattern, but the array element excitation is concentrated in the center of the array when radiating together as an antenna array. The array factor can be expressed as follows [73]:

(1)AF(φ)=2∑n=1MIncos⁡(kdcos⁡(φ)+ψn)where k=2π/λ is the wave number, In is excitation amplitude of elements, d represents the position of each element in the line array, ψn indicates the phase of each element, φ is the azimuth angle.

In this paper, the excitation amplitude of the array elements in the linear antenna array is adjusted to make the antenna highly radioactivity in a specific direction. The optimization objective is to get the maximum SLL reduction and the fitness function can be expressed as:

(2)fitness=max{20log10⁡|AF(φSL)||AF(φML)|}

(3)s.t. φML=arg⁡max|AF(φ)|,φ∈[0,π]

(4)φSL∈[0,φFN1]∪[0,φFN2]

(5)0<Ii<1,∀i∈Mwhere φSL and φML are the interval between the values of the side lobe and the main lobe; φFN1 and φFN2 are the first nulls of the pattern; λ is wave length and array spacing fixed d=0.5λ; In order to ensure the maximum width of the first nulls of the pattern in the optimization process, φFN1 and φFN2 select the appropriate value.

The equilibrium optimizer’s inspiration comes from the dynamic mass balance equation in the container for a control volume that describes the conservation of mass entering. The equation is defined as follows:

(6)VdCdt=QCeq−QC+Gwhere C is the concentration of particles, V is volume, Vdcdt is the rate of mass change, Q is the flow of volume,Ceq represents the equilibrium condition of particles, and G is the mass generation rate.

After obtaining the mass balance equation, the turnover rate (λ=QV) is introduced to rearrange the equation, and the solution time of the rearranged equation is solved as an integral of the function to obtain as follows:

(7)C=Ceq+(C0−Ceq)F+GλV(1−F)where F is expressed as follows:

(8)F=exp⁡[−λ(t−t0)]where t0 and C0 is initiate time and start concentration.

As with most optimization methods, the searching flow path starts with initializing the container particle.

(9)Cinitial=Cmin+rand(Cmax−Cmin)where Cinitial is the initial concentration of the particle, Cmin and Cmax are the maximum and minimum values of particle concentration, rand is a random number.

After that, the particle concentration is evaluated and four better fitness particles are chosen to constitute the equilibrium pool. The first four candidate solutions in the equilibrium pool are the particles with the best fitness values identified. However, and the last one of the equilibrium pool is the arithmetic mean of the four previous candidates as given in Eq. (10). Each iteration selects equilibrium candidate solutions with equal probability from the equilibrium pool.

(10)C⟶avg=(C⟶eq(1)+C⟶eq(2)+C⟶eq(3)+C⟶eq(4))/4

(11)C⟶eq,pool={C⟶eq(1),C⟶eq(2),C⟶eq(3),C⟶eq(4),C⟶avg}where C⟶avg is the arithmetic mean, C⟶eq(1),C⟶eq(2),C⟶eq(3),C⟶eq(4) are the first four candidate solutions. C⟶eq,pool is an equilibrium pool.

Exponential terms help enhance the exploration capabilities of the EO, and it is described as follows:

(12)F⟶=a1sign(r⟶−0.5)[e−λ→t−1]where a1 is a constant value, sign(r⟶−0.5) is a symbolic function in Eq. (12) has an effect on the searching direction, and time t is a nonlinear parameter. r⟶ and λ⟶ are a random vector between 0 and 1. It is calculated as follows:

(13)t=(1−IterMax_iter)(a2IterMax_iter)where a2 is a constant value that controls exploitation capabilities, experiments show that when a1=2 and a2=1, EO has better optimization capability.

The mass generation rate is helpful in improving the exploitation ability of the algorithm, and it is described as follows:

(14)G⟶=G⟶0e−k⟶(t−t0)

(15)G⟶=G0⟶F⟶

(16)G0⟶=GCP→(Ceq⟶−λ⟶C⟶)

(17)GCP→={0.5r1,r2≥GP0,r2<GPwhere r1 and r2 are the random number, GCP is the generation rate control parameter.

The updating rule of EO is given as:

(18)C⟶=C⟶eq+(C0⟶−Ceq⟶)F⟶+G⟶λ⟶V(1−F⟶)

The first term in Eq. (18) selects a balanced candidate from the equilibrium pool. The existence of the equilibrium pool strategy enhances the exploration capability of the algorithm. The second is responsible for improving the global exploration capability of the algorithm, and the symbolic function controls different exploration directions. The third improves the exploitation capability of algorithms to control small changes in particle concentrations. The pseudocode of the basic EO is illustrated in Table 1.

This strategy is similar to the individual optimal location preservation strategy of PSO, which records the best location searched by individuals.

The smallest unit in quantum computing is the quantum bit, which can be expressed as “0” and “1” for two states of quantum existence. A quantum bit can be expressed as follows:

(19)|ψ⟩=α|0⟩+β|1⟩where α and β are the probability symbols of the quantum bits being 0 and 1, respectively; Due to the variable nature of the probability amplitude, it satisfies the equation |α|2+|β|2=1. The encoding of the individual quantum equilibrium optimizer inspired by the above quantum computing ideas is obtained as follows:

(20)Xi=[α1α2…αDβ1β2⋯βD]=[cos⁡θ(i,1)cos⁡θ(i,2)…cos⁡θ(i,D)sin⁡θ(i,1)sin⁡θ(i,2)⋯sin⁡θ(i,D)]where θ=2π×rand, rand is a random number between 0 and 1, i is the population size, D is the problem dimension.

Thus, quantum chromosomes correspond to two positions in quantum space, as follows:

(21)Pic=(cos⁡θ(i,1),cos⁡θ(i,2),…,cos⁡θ(i,D))

(22)Pis=(sin⁡θ(i,1),sin⁡θ(i,2),…,sin⁡θ(i,D))

In QEO, obtaining the particle positions corresponding to the problem requires mapping the quantum space of the quantum chromosome to the problem space σQEO=[a,b]. The quantum chromosome solution space transformation equation is as follows:

(23)Xic=12[b(1+αi)+a(1−αi)]

(24)Xis=12[b(1+βi)+a(1−βi)]where Xic is the quantum state |0⟩ calculated by the probability amplitude αi, Xis is the quantum state |1⟩ calculated by the probability amplitude βi.

In order to improve the defect that the basic equilibrium optimizer algorithm tends to converge prematurely in the late iteration, this paper introduces a quantum rotation gate strategy with a quantum equilibrium pool strategy inspired by quantum computing. The angle variation of the quantum probability amplitude is adjusted by the quantum rotation gate, and the variation of the position is realized by the quantum equilibrium pool strategy.

In order to show the QEO algorithm more clearly, Table 2 and Fig. 2 respectively show the pseudocode and flow chart of the quantum equilibrium optimizer.

The performance of different comparison algorithms in the maximum SLL optimization problem was evaluated using the following measures:

Max and Min: Maximum values and minimum values in the calculation results of thirty independent runs.

The experiment compared QEO with other novel algorithms, such as EO [20], SMA [18], PSO [11], SCA [74], WOA [15], GWO [17]. N is the population size and T is the maximum iterations. It is worth mentioning that the 16, 24 and 32 elements LAA, the population size is set to 30 and maximum iterations are set to 100. The parameter settings of the comparative algorithms in the experiment are shown in Table 3. The experimental parameter settings of the comparison algorithm are from its references. The experimental results come from all the algorithms running independently for 30 times under the same conditions, and the results were statistically analyzed.

In this section, we investigated the LAA optimization problem, namely verifying the effectiveness of the quantum equilibrium optimizer in reducing the SLL of LAA. QEO is tested with six other novel meta-heuristic algorithms in a linear array with different arrays of elements and four performance evaluation metrics are used to evaluate the optimization capability of the algorithms. The optimization results of different algorithms in the 16-element LAA with the excitation amplitude can be seen in Table 4. QEO obtains the optimal SLL value is −29.6939 dB, while the conventional method obtains the SLL is −13.1529 dB, and QEO improves 16.5410 dB compared to the conventional method. And the maximum SLL obtained by EO, SMA, PSO, SCA, WOA, and GWO is −28.4299, −28.3060, −23.7392, −25.6562, −28.7441, and −28.5342 dB. Beam patterns after optimization of the excitation amplitude by different algorithms are shown in Fig. 3. It can be seen that the QEO SLL is the smallest and the main beam has good radiation capability. The optimization results of different algorithms in 24-element LAA are shown in Table 5. Compared to the conventional values without optimization, the maximum SLL of QEO, EO, SMA, PSO, SCA, WOA, and GWO is reduced by 16.49, 14.75, 13.56, 8.12, 9.37, 12.59, and 13.71 dB, respectively. Compared to the conventional method, QEO showed the largest reduction. The QEO-optimized beam patterns partials beam is lower and the main beam has the best radiation intensity compared to other algorithms as seen in Fig. 4. As seen in Table 6 in the 32-element LAA, the maximum SLL obtained by QEO is significantly better than the other algorithms. As seen in Fig. 5, the optimized beam patterns of the proposed method are significantly better than other algorithms waveform levels are more distinct. As the number of LAA elements increases, QEO is able to adapt well to the computational difficulties associated with the increased problem size and obtains a larger SLL reduction compared to other algorithms. Figs. 6–8 show the three-dimensional antenna patterns of different array elements obtained by different optimization algorithms. From the figure, it can be observed that the QEO obtains the smallest side lobe level.

In this section, we analyze the convergence curves of different algorithms for LAA optimization of SLL with the different number of array elements. The convergence curves give a good indication of the optimization ability of the quantum equilibrium optimizer as well as other methods. The algorithm's search ability is related to its exploration ability and exploration ability. A stronger exploration ability makes the algorithm converge easily and early, while a stronger exploration ability decreases the convergence speed of the algorithm. As seen in Figs. 9–11, the best convergence curves are obtained for QEO in all three different systems, and the convergence speed and the accuracy of the searchare the best. Compared with the basic equilibrium optimizer, the quantum coding-based strategy improves the population diversity avoiding the algorithm from falling into premature convergence and improves the optimization accuracy of the equilibrium optimizer. The particles in the quantum equilibrium pool undergo quantum mutation to produce phase angle increments and update the probability operators through a quantum rotation gate. Based on this method, QEO can effectively escape from local optima in the late iterative stage and avoid overcrowding of the quantum population.

The intelligent optimization algorithm is stochastic, so that the computational results are different each time. In this section, we analyze the computational stability and the optimization performance of the algorithms. In order to observe the fluctuation of the data more intuitively, the optimization results of different optimization algorithms in three different LAA for thirty times are plotted, as shown in Figs. 12–14. It can be clearly observed from the figure that the mean value of QEO is the lowest located at the bottom of the picture, indicating that the proposed method has higher solution accuracy and robustness in optimizing the maximum SLL reduction problem. In order to quantitatively analyze the computational stability of the algorithm, the mean and standard deviation of different algorithms in the three test systems are shown in Table 7, and the method of Friedman’s average ranking is used for statistical analysis of the experiment result. FAR represents the score of Friedman’s average ranking, and Rank represents the ranking of statistical analysis of different algorithms. In order to verify whether the proposed method is significantly different from other optimization methods, the statistical analysis method of Wilcoxon p-value test is used in this section, and the results show that QEO significantly outperforms other methods in Table 8. The algorithm significantly outperforms the EO, SMA, PSO, SCA, WOA, and GWO algorithms in solving the linear antenna array optimization problem, which proves that the proposed quantum equilibrium optimizer has strong optimization capability.