*4.1. Experiment 1: Comparison with Existing Methods*

The first batch of tests compared the proposed model to comparable techniques [44] that used the GA, PSO, and CS to find the 3D Lorenz chaotic system characteristics solely using chaotic signals. Default swarm parameters were utilized. Table 3 shows that the suggested model is superior to the prior techniques. QFOA's calculated parameters matched the original parameters' real values. According to [49], the 3D Lorenz chaotic system's initial parameters are *θ*<sup>1</sup> = 10, *θ*<sup>2</sup> = 28, and *θ*<sup>3</sup> = 8/3, allowing complete synchronization between the master and slave chaotic systems. The estimated parameters matched the CS-based model, but the QFOA outperformed in terms of the optimal function's mean and standard deviation. Most data points were close to the mean with a low standard deviation (more reliable). QFOA was more effective and resilient than other chaotic system parameter estimation strategies. The model and system responses were synchronized. This gain was due to the proposed model's higher coverage and exploration of the searching space, which improved parameter estimate accuracy and led to the discovery of optimum chaotic parameter values compared with existing techniques.


**Table 3.** Comparison of statistical results for the Lorenz system, in case of only using chaotic signal.

#### *4.2. Experiment 2: Effect of QFOA Iteration*

The second set of experiments investigated the effect of the QFOA number of iterations on the proposed model to identify the correct parameters of the 3D Lorenz chaotic system using only chaotic signals and masking voice signals with the chaotic signals. QFOA was performed 30 times every iteration, with 50 iterations total and *W* = 30 for data sampling. Default swarms were utilized. After 20 iteration, the parameters *θ*1, *θ*2, and *θ*<sup>3</sup> converge to the actual values. QFOA reached stable values in 25 iterations. As the fitness function value

declines rapidly to zero, indicating that QFOA may converge quickly to the global optimum. These few iterations did not require complex calculations. By adjusting the location of the QFOA swarms by modifying the number of iterations, the algorithm could reach an ideal balance between exploitation and exploration. At the same time, elitism in population iteration may have sped up the convergence and assured continual optimization. This highlights the remarkable efficiency of QFOA in accomplishing global optimization.

#### *4.3. Experiment 3: Effect of Number of Swarms*

The third set of experiments was implemented to find a suitable number of QFOA swarms that helped to reduce computational effort without sacrificing estimation precision. For the three-dimensional Lorenz system, the proposed model was run by setting the QFOA swarm numbers as 10, 30, and 100, respectively. In general, tiny populations provide poor outcomes. As the population grows, outcomes improve, but more fitness tests are needed. Beyond a certain point, outcomes are not significantly influenced. When there are too few swarms, the solution space is not sufficiently searched, resulting in unsatisfactory outcomes. Considering search quality and computational effort, a population size between 30 and 60 is suggested. A larger population size is suggested for estimating additional parameters. Size 25 performed well. Considering processing costs and estimating accuracy, a large population size is unnecessary.

#### *4.4. Experiment 4: Influence of the Data Sampling W*

The fourth series of experiments tested how data sampling affected model accuracy. To reduce the amount of parameter setting combinations, the model changed one parameter *W* at a time, while leaving other parameters (number of swarms, number of iterations, etc.) at default values. The impact of modifying these variables was also considered. General factors for selecting *W* were minimum fitness mean and highest estimate accuracy. All scenarios were run 30 times for comparison. Table 4 lists the estimation results and the means of the best fitness values for different data sampling *W*. As shown, the estimation accuracy declined as *W* increased. Moving from 30 samples to 100 decreased the mean of fitness values by 36%, whereas moving from 100 samples to 200 decreased the mean of fitness values by 45%. These three groups of input data may have provided a satisfactory estimate, but the 30 samples of data had the least variation. Different inputs impacted the first iteration, but for all instances, it took roughly 25 iterations for the algorithm to converge to zero, indicating these three conditions could all acquire quite accurate anticipated outcomes. As expected, chaotic parameter estimate accuracy falls as *W* rises. The crucial sensitivity of the nonlinear system to starting circumstances and parameters made the fitness function more difficult as *W* increased. To decrease estimate bias in target nonlinear systems, it is vital to sample enough data.


**Table 4.** Statistical results for the extended Lorenz chaotic system with varied data sampling.

#### *4.5. Experiment 5: Comparison with another Quantum Metaheuristic Algorithm*

The fifth series of tests compared the proposed model with a comparable strategy that used the quantum firefly (QFA) algorithm to determine the ideal chaotic parameters of the 3D Lorenz chaotic system exclusively using chaotic signal and masking speech sounds with chaotic signal. Both techniques were performed 30 times to compare fitness means and standard deviations. Default swarms were utilized. Table 5 shows that the estimated chaotic parameters while masking speech signals with chaotic signals are similar to the QFA-based model. The mean fitness values and standard deviations of QFOA were 37 and 66% lower than in QFA.


**Table 5.** Statistical results for the Lorenz system.

In general, the quantum-inspired firefly algorithm (QFA) ensured the diversification of firefly-based generated solution sets, using the superstitions quantum states of the quantum computing concept. However, it suffered from premature convergence and stagnation; this was mainly dependent on the ability of the employed potential field to handle movement uncertainty. The suggested QFOA algorithm, inspired by the delta potential field, presented the most balanced computational performance in terms of exploitation (accuracy and precision) and exploration (convergence speed, and acceleration). The advantage of such models, on the one hand, is that they are "exactly solvable", e.g., the spectrum and eigenvectors are explicitly known; on the other hand, many interesting physical features are retained, despite the simplification involved in approximating short-range with zero-range. Thus, QFOA was more effective and resilient than QFA in estimating chaotic parameters.

#### *4.6. Experiment 6: Estimation Accuracy with Different Chaotic Systems*

The sixth group of experiments was conducted to determine the efficiency of the proposed model regarding the different chaotic systems, including the 3D Chen and 3D Rossler chaotic systems in cases of only using the chaotic signal. The algorithm was run 30 times and the default parameters of QFOA were used. Table 6 shows that the estimated parameters derived by QFOA were close to the original parameters for chaotic systems. As stated in [44], the original parameters of 3D Chen chaotic system were *θ*<sup>1</sup> = 35, *θ*<sup>2</sup> = 3, and *θ*<sup>3</sup> = 28; whereas, as stated in [73], the original parameters of 3D Rossler chaotic system were *θ*<sup>1</sup> = 0.2, *θ*<sup>2</sup> = 0.4, and *θ*<sup>3</sup> = 5.7, through which perfect synchronization could be obtained between the master and slave chaotic systems. In the search process, fruit flies modified their places based on individual and swarm experiences. This expanded the solution search space and prevented premature convergence. This also improved the algorithm's convergence speed. Generalized synchronization was possible with certain parameters [74].


**Table 6.** Estimation accuracy for different chaotic system using default QFOA parameters.

Computer simulations of the three 3D chaotic systems and comparisons with other metaheuristic approaches proved the suggested method's efficiency. The impact of data sampling, iterations, and swarms on estimating accuracy was also studied. Theoretical study and computer simulation led to the following conclusions: (1) A shorter data sample length improves estimate accuracy because a longer sample length complicates the objective function. (2) The highest number of iterations improves estimating accuracy by moving the swarms. Thus, exploitation and exploration balance each other. (3) Many swarms will investigate enough space for study, improving estimate accuracy. These swarms are computationally intensive. To decrease estimate bias in chaotic systems, use the right data sampling, iterations, and swarms.

For our simulations, we used some of the most famous chaotic systems as examples. The number of parameters for these chaotic systems was not large, and the system was not complex. At present, the most studied chaotic neural network systems have many parameters, and the weight of these systems affects the complexity of the network. However, the suggested simpler model may be adapted to deal with chaotic neural network systems and other complicated chaotic systems. In our case, instead of searching for only three chaotic parameters, which represented the final solution picked from the search space based on a quantum-inspired particle's movement, more parameters could be correctly estimated by increasing the number of fruit flies. Therefore, there is a trade-off between computational cost and required best fitness evaluation function that must be balanced.
