Optimization is rapidly progressing, with new algorithmic and theoretical techniques emerging across various fields of science and engineering. A key trend is the increasing focus on the interdisciplinary nature of optimization. Traditional approaches, such as gradient-based and Hessian-based methods, are efficient but face challenges with complex, non-differentiable, or discrete problems. These methods can get trapped in local minima and are sensitive to numerical noise. In contrast, meta-heuristic optimization approaches efficiently tackle multimodal, high-dimensional, and nonlinear problems. They can handle any objective function and are noted for their simple design and self-adaptive capabilities, which can yield global optima when properly configured. Meta-heuristic algorithms are generally grouped as follows: (i). Evolutionary Computing (EC): Considering natural evolution as the fundamental principle and includes Genetic Algorithms (GA) [1], Differential Evolution (DE) [2][3], and Genetic Programming (GP)[4], etc. (ii) Swarm Intelligence (SI): Inspired by social behaviors, examples include Particle Swarm Optimization (PSO) [5] [6], Ant Colony Optimization (ACO) [7] and Adaptive Social Behavior Optimization (ASBO) [8], etc. (iii) Physical/Chemical Principle-Based Algorithms: Includes Simulated Annealing [9] and DNA Computing [10],etc. A significant issue for meta-heuristics is maintaining a balance between exploration (finding new solutions) and exploitation (improving known solutions), which affects convergence and global optimality [11] [12]. Strategies to improve this balance include tuning algorithm parameters and using self-adaptive mechanisms based on population diversity and fitness. Furthermore, surrogate models can simplify the evaluation of complex objective functions, reducing computational costs [13]. Researchers are also combining quantum mechanics with meta-heuristics to improve the performance, leading to quantum meta-heuristic algorithms. There are two main approaches for incorporation, as shown in Fig. 1: (i) Quantum-Inspired [14] [15]: In this approach, the meta-heuristic model is transformed into the quantum realm to explore solutions in Hilbert space. (ii) Quantum Behaved [16]: This approach enhances meta-heuristic outcomes by selecting a potential well to obtain the wave function from the time-independent Schrödinger Equation and determining its position through quantum measurement.

In a quantum-inspired meta-heuristic, a quantum solution is represented by a sequence of qubits. The algorithm applies an evolutionary structure to the qubits through quantum logic gates, which function like operators within Hilbert space. These quantum gates manipulate the quantum solution. By considering the principle of quantum superposition, a quantum solution can represent multiple solutions simultaneously, facilitating a faster exploration of the vast solution space. Quantum entanglement introduces correlations among qubits, helps maintain a high level of diversity, and prevents premature convergence to local optima. These characteristics make quantum meta-heuristics (QMHs) powerful tools for solving complex optimization problems that have extensive and multimodal search domains and require rapid convergence. Several algorithms fall under this category, including Quantum-Inspired Genetic Algorithm (QGA)[17][18], Quantum Differential Evolution (QDE)[19], Quantum Ant Colony Optimization (QACO)[20], and Quantum Artificial Bee Optimization (QABO)[21].

In quantum-behaved meta-heuristics, a feasible potential is expressed in the time-independent form of the Schrödinger equation to obtain the wave function. From this wave function, a probability density function is derived, and a quantum measurement is subsequently applied to effect changes. New solutions are generated by incorporating the quantum change discovered into the meta-algorithm's solutions in a probabilistic environment. When the need arises for lower computational costs and moderate accuracy, Particle Swarm Optimization (PSO) has proven to be a highly effective algorithm, with numerous applications in the past [22]. Various versions and hybridizations with other meta-heuristics have emerged [23] [24]. Among them, a PSO variant that incorporates quantum mechanics principles to guide the search process is known as Quantum-Behaved Particle Swarm Optimization (QPSO) [25]. A significant challenge faced by all meta-heuristics is the tuning of the numerous associated parameters, which can vary across different applications. Conventional PSO encounters this issue, requiring tuning of three parameters: inertia weight, social constant, and cognitive constant [26]. In contrast, QPSO only necessitates tuning a single parameter [27] and demonstrated improvements over conventional PSO in terms of accuracy, convergence speed, variation, and robustness [25] [42].

This research makes three main contributions: (i) it clarifies how potential well characteristics influence the solution exploration phase; (ii) it proposes a new global best-based characteristic length strategy to increase the accuracy of Quantum Particle Swarm Optimization; and (iii) it introduces a more accurate and robust quality measurement index for algorithm performance. To address the first contribution, the study analyzes the quantum behavior's effect as a function of random numbers depending on potential well probability density functions. It investigates various potential wells—Delta, Harmonic Oscillator, Square Well, Lorentz Potential Field, Rosen-Morse Potential Field, and Coulomb-like Square Root Field—and presents their effects graphically. The findings show that both linear and non-linear changes exist for all potential wells, influencing solution dimensions differently based on random number strength. This enables diverse exploration to support finding the global optima. The second contribution focuses on characteristic length (CL): while a mean self-best CL can cause slow convergence and risk local optima, the proposed global best-based CL enables asymmetrical changes according to each member's distance from the global best, promoting both local and global exploration. This approach accelerates convergence and improves global search capability. The third contribution is the introduction of the Quality Measuring index QM_index, a distance-based measure to evaluate how close an algorithm's result is to the absolute best. Smaller QM_index values indicate better performance. The research relies on two sets of benchmark functions from CEC2005 and CEC2017, incorporating characteristics of unimodal, multimodal, shifted and rotated, hybrid, and composite with different dimensions, to test and detail algorithmic performance. Overall, the proposed global-based CL in QPSO demonstrated superior performance compared to other QPSO variations and the recent Farmer and Season Algorithm (FSA) [64].

In this paper, the work is structured into several sections. Section 2 presents a review of related work, while Section 3 discusses the identified research gap and provides a brief overview of the proposed solution. The basic principles of Particle Swarm Optimization are outlined in Section 4, and Section 5 describes the association of quantum behavior with PSO. Section 6 analyzes the effective contributions of different potential wells in Quantum Particle Swarm Optimization. In Section 7, the proposed solution for the global best-based characteristic length in QPSO is discussed. A quality measurement index and its formulation are provided in Section 8. Section 9 covers the details of the experimental outcomes and their analysis over benchmark functions, along with the impact of high dimensions, as well as observed limitations with the QPSO algorithm. The outcomes of considering the multiple potential wells simultaneously in the solution exploration has been discussed in Section 10. The paper concludes with a discussion of future work.

In the context of the current research on Quantum Behaved Particle Swarm Optimization, previous literature has been studied over the involved fundamental principles, local attractor positioning, variations in characteristics of the contraction-expansion (CE) coefficient, characteristic length formulation, the effect of potential wells, hybridization with other algorithms, and applications over different fields.

Several past studies can be regarded as foundational developments for QPSO, particularly regarding the principles of operation and control strategies for associated parameters aimed at enhancing exploration and improving convergence characteristics. One of the most significant contributions was made by Jun Sun et al. [25], who introduced quantum characteristics to PSO particles to enhance exploration from local attractors. They investigated the Delta potential well, which demonstrated effective performance. The analysis of the upper-level fixation of the CE parameter was thoroughly discussed in [27], proposing various approaches for the control strategy of QPSO. The positioning of local attractors significantly influences the performance of particles hence a Gaussian probability distribution-based strategy for identifying better local attractors was proposed in [28]. In this strategy, the mean value represents the local attractor position derived from a standard approach, while the standard deviation reflects the difference between the global and self-best positions. This hybrid method, utilizing both standard and Gaussian-based local attractors in a probabilistic environment, showed a slight performance improvement compared to standard strategy. A comprehensive analysis was presented in [29], examining the behavior of a single particle using a probabilistic approach. This analysis yielded the upper bound value of the CE coefficient to ensure convergence. Additionally, to enhance the exploration capability, diversity control was explored in [30]. This study revealed that the distance to the average point diversity is a crucial factor in defining search performance. The integration of quantum principles with meta-heuristics has been applied and simulated within classical systems. The effects of these quantum mechanics principles in conjunction with meta-heuristics in real quantum environments were investigated in [31]. For this purpose, the quantum cellular genetic algorithm was implemented on quantum simulations and machines to assess how the real quantum world influences performance. It is well-known that the effectiveness of any meta-heuristic algorithm is problem-dependent, raising the question of which characteristics of the problem contribute to algorithmic challenges. In [32], the effectiveness of Quantum Annealing (QA) in problem-solving was analyzed using meta-learning models. It was found that problem coefficients related to bias and coupling terms were more critical for discovering effective solutions than merely considering the density of these coefficients. The optimal utilization and challenges of the quantum approach with PSO were discussed in [33], emphasizing that PSO relies on an implicit probabilistic distribution rather than an explicit one. The position of the local attractor in QPSO significantly impacts both the accuracy and convergence characteristics. Finally, a collaborative learning-based strategy was proposed in [34], which generates local attractors using orthogonal and comparison operators. The orthogonal operators aim to leverage the information available between personal and global best positions, while the comparison operators address the potential for local optima traps. Overall, the characteristic length plays a crucial role in exploration. In most research, the focus is on the absolute difference between an individual's performance and the best mean performance of the group. A variation in the characteristic length, based on a fitness-weighted mean best position rather than the mean best, was proposed in [35]. In [36], a modification to the characteristic length using the global best instead of the mean best was implemented, which showed improvements in adaptive filter-based system identification applications. The quantum behavior of QPSO has been defined through the concept of potential wells, and various types of potential wells have been explored in past studies. A detailed discussion about the formulation of different potential wells, such as Delta, Harmonic Oscillator, and Square potential wells, was presented in [37], with applications in designing various types of antennas in the field of electromagnetics. The comparative performances were analyzed in the context of three different potential fields: Lorentz, Rosen-Morse, and a Coulomb-like square root potential, as noted in [38]. The concept of soliton particles in the formation of Quantum PSO allows for adaptations in the external potential field, which helps stabilize them without getting trapped in the potential wells [39]. A discrete version was proposed in [40]. Additionally, controlling parameters were introduced in [41] to balance the influences of personal best and global best positions, as well as to strike a balance between diversification and intensification strategies. To escape local optima, some of the worst solutions in the population were re-initialized. Furthermore, Lévy flight and straight flight techniques were integrated with QPSO to enhance performance in high-dimensional problem scenarios [42].

Various possibilities have been explored to integrate other domain algorithms with Quantum Particle Swarm Optimization to enhance performance. A quantum-behaved PSO algorithm utilizing a hyper-chaotic discrete system was presented [43], where all particles are confined within an identical particle system, and the theory of two-dimensional hyper chaotic sequences is applied. Initially, seasonal fluctuation inference is eliminated by employing identical particles, followed by a chaotic search conducted through the hyper-chaotic sequence. Additionally, a mimetic algorithm combined with a memory mechanism was proposed in QPSO [44], enabling particles to gain experience through a local search mechanism before evolving. The memory mechanism is applied subsequently to improve search capabilities. To enhance convergence, the same random number is applied to each dimension of a particle. A Gaussian distribution-based random number generator was utilized that had a linearly decreasing variance, allowing a global search advantage in the initial phase and a local search in later phases [45]. A weighted mean of the best position was also employed to balance the search between global and local approaches. A chi-square-based mutation strategy integrated with QPSO in [46] for the CEED problem has demonstrated better performance compared to other mutation forms based on Gaussian and Cauchy distributions. A local search strategy in QPSO [47] was developed to enhance performance by creating a "super particle," formed by randomly selected particles, with contributions to this formation being fitness-oriented. A hybrid QPSO was proposed in [48] and applied to the Energy Demand Predictions (EDPs) problem, using Solis and Wets methods to boost local search performance. This algorithm aims to identify optimal solutions when constraints are met, employing two evolutionary operators to balance local and global searches. The principle of quantum entanglement from quantum mechanics has been leveraged in [49] to develop meta-heuristics. For a given population of proto-born particles, properties of entanglement and superposition are utilized to create twin-born and combination-born particles, respectively. An elite-born particle is generated via a localized searching approach. To address high-dimensional optimization, a quantum PSO based on diversity migration was proposed [50], where migrating individuals are selected based on their fitness and physical positioning. Furthermore, the superposition characteristic of quantum mechanics was proposed [51] to update the velocity parameters of particles in PSO, with enhancements to local searches assisted by the Kangaroo Algorithm. Quasi-opposite-based learning was introduced in the initialization stage to accelerate convergence, and a double evolutionary mechanism was applied to update individual positions [52]. This approach combines the advantages of particle mining and population guidance during iterations to improve QPSO searches. Additionally, a combination of cuckoo search and QPSO has been integrated for solving differential equations [53].

Numerous studies have explored real-world applications of QPSO. These applications include resource allocation in cloud computing [54], optimizing truss structures using quantum angle encoding and a perturbation operator [55], and minimizing glycemic loads for specific foods in the context of nutrition [56]. Additionally, QPSO has been integrated with deep Q-networks to enable path definition and obstacle avoidance for autonomous underwater vehicles [57]. A QPSO-based clustering method has also been employed to conserve energy in sensor nodes within wireless sensor networks (WSN) [58]. Other applications include Code Division Multiple Access (CDMA) [59], the Internet of Things (IoT) [60], and multi-task allocation [ 61], among others.

The analysis of previous research primarily focused on the re-localization of local attractor positions, variations in contraction coefficients, the nature of applied potential wells, population initialization, and the integration of Quantum Particle Swarm Optimization with other meta-heuristics to enhance performance. Numerous application-oriented studies have been observed, but several critical areas have received minimal attention: (i) how the characteristic length affects exploration and performance, and its appropriate formulation. (ii) What should be a more appropriate formulation of characteristic length? (iii) Although probability density functions (PDFs) of potential wells were observable, after measurement, there was no clear information available about the characteristics of the obtained function of random numbers and their behavior in the exploration. (iv) the need for a unified quality measuring parameter that encompasses both the accuracy and convergence characteristics of algorithms throughout the convergence process.

To address these gaps, this work introduces a new strategy for characteristic length, defined through the global best instead of the mean of the self-best. The results demonstrate that the global best characteristic length enables simultaneous multi-level and multi-scale exploration, facilitating both global and local level exploration in parallel. This significantly increases the chances of escaping local optima traps and establishes a faster global convergence. The performance of the global best characteristic length in high-dimensional problems was found to be less affected compared to the self-best mean approach. An innovative metric, the Quality Measuring Index, was proposed to simultaneously account for accuracy and convergence characteristics, providing a means for relative comparison of algorithms. The proposed approach was first tested on CEC2005 benchmarks that included both unimodal and multimodal numeric optimization problems. The performance of the global best characteristic length was compared against the mean self-best approach across different types of potential wells to ensure that the observed improvements were substantial and consistent. A graphical approach was developed to illustrate the final effect of potential wells on solution exploration, clearly showing how QPSO with the global best characteristic length can explore more effectively and rapidly. In the later stage of work, the proposed approach was applied over the CEC2017 single objective function real parameters function that carried the shifted, rotated, hybrid, and composite structure of functions to ensure the capability of solution exploration even in the most difficult form of landscape.

Natural swarms exhibit complex emergent behaviors resulting from local interactions, decentralized control, and self-organization. The mathematical abstraction of these behavioral properties, such as survival and reproduction, has led to the development of a branch of artificial intelligence known as Swarm Intelligence (SI). Various algorithmic models have been formulated to correspond to different types of swarms, including Particle Swarm Optimization (PSO), Ant Colony Optimization, and Bee Optimization. These algorithms are iterative meta-heuristics. PSO, in particular, has demonstrated significant success across diverse application domains and employs a computational approach inspired by leadership and self-motivation. During each iteration, three populations are maintained: the position population, which encodes candidate solutions; the velocity population, which determines changes in the solution population; and the self-best population, which stores the best solutions achieved by individuals. The updates to velocity (V) and position (S) for each member in each dimension for the iteration are defined by Eq. 1 and Eq. 2 [26].

The inertia weight ), along with two other two positive constant parameters cognitive ( and social( constant decide the nature of convergence characteristics. The inertia weight determines how much previous velocities influence the current one, and a better choice helps to balance between global and local search. Higher inertia weights help the algorithm to search more broadly, while lower weights make it focus on a local region. The cognitive and social parameters are less crucial, and proper values help the algorithm to converge faster and to avoid getting stuck in local optima. Uniformly generated random values of parameters , in range of 0 to 1 help to maintain the population diversity through different sampling of influencing factors. The performance of PSO depends a lot on these three parameters’ value settings, but there is no standard method that exists to find their best values. There are three possible approaches to assume values for the inertia parameters in algorithms: (i) static approach: where a constant value less than 1 is assumed throughout all iterations, (ii) dynamic approach: where a decreasing function of value is applied over iterations, and (iii) self-adaptive approach: where a function adjusts the inertia weight based on the current state of population diversity or fitness diversity. Regardless of the approach applied [5] [62] —whether in algorithm structure, parameter variations, or both—PSO faces challenges in global exploration, especially with large and multimodal problems, and requires enhancements to improve its performance.

Classical PSO was developed on the concept of inspirational factors of social culture, in which the influences of the leader and self-motivation were considered to provide changes in the individual. The Convergence characteristics of PSO have been demonstrated through the trajectory analysis in [5], and was established that convergence of each particle to its local attractors ensured the algorithm’s convergence. Considering the population size of M for a D- dimensional problem, the local attractor position ( ) of a particle in the iteration can be estimated as a linear combination of its personal–best and the global-best position as defined by Eq. (3)

Where are the cognition and social constants while are the vector of random numbers, generated through uniform distribution in the range of [0 1]. In compact form, Eq. 3 can be written as Eq. 4.

Where

In the classical approach of the PSO, the attractive field pulls all the particles towards their local attractors, and many don’t have enough energy to escape eventually. With time, there was no proper exploration of regions beyond the local attractor’s field, causing of the solution to get trapped in the local minima.

Assuming a quantum system corresponding to PSO where particles described their quantum behavior and followed the quantum mechanics rules. The local attractors for a particle can be considered as the random average of personal-best and global-best positions, using the proper potential well, which provides the attraction towards their local attractors along with a good possibility of exploring the regions far beyond the potential well center. This provides the opportunity to increase the chance of obtaining the global optima. The exploration displacements from the local attractors were achieved through the quantum behavior formalism as given below. To use the Quantum behavior-based phenomenon in PSO, there are two sections under which the whole scenario has to be considered: (i) formalism of the Quantum environment and (ii) exploitation of quantum behavior with PSO in search of a better solution.

The performance of Quantum Particle Swarm Optimization is largely influenced by the characteristics of the chosen potential well. Various types of potential wells have been examined in previous studies. Table 1 lists the equations associated with commonly used potential wells, including Delta, Harmonic Oscillator, Square Well, Lorentz Potential Field, Rosen-Morse Potential Field, and Coulomb-like Square Root Field. These potential fields exhibit significant differences in their functional formulas and probability density functions [37] [38]. The primary focus here is to understand how these different potential fields impact the exploration of new solutions when utilized within QPSO. The final solution update equations for QPSO corresponding to each potential well are also included in Table 1. For each potential well, the contribution function indicating the final effect is presented in the last column of the table, where is a random number generated from a uniform distribution between 0 and 1. The values associated with represent changes in each dimension that facilitate exploration. To comprehend the varying effects of different potential wells on the exploration,100 randomly selected values of, the corresponding, values have been plotted as shown in Fig. 3. The variability in the performance contributions can be observed. From Fig. 3, it is evident that all contribution graphs can be divided into two regions: (i) the nonlinear region, applicable when, and (ii) the linear region, applicable when. The first region is characterized by non-linearity for all potential wells, except for the Square Well. The degree of non-linearity is greatest with the Lorentz Potential Field, particularly for lower values of. Harmonic, Coulomb, and Rosen-Morse fields exhibit a similar pattern with a lower degree of non-linearity. The Delta well shows a moderate level of variation compared to the two extremes. In the second region, all potential wells display linear characteristics, with the steepest slope observed in the Delta well, resulting in greater changes as it moves from 0.5 to 1.

The effects of the other potential wells show nearly similar variations. This nonlinear variation in the region has a significant impact on the direction and distance of the next exploration attempt due to the asymmetrical changes in different dimensions. These changes facilitate global exploration. In contrast, the linear region provides comparatively smaller changes across associated dimensions, supporting local solution exploration. In summary, there exists a mechanism that balances local and global exploration by providing equal opportunity across all dimensions. Approximately 50% of the dimensional changes are directed towards global exploration through the nonlinear regions, while the remaining 50% support local exploration through smaller changes in the linear regions. The overall effect of a specific dimension is illustrated in Fig. 4.The large differences in nonlinearity patterns among various potential wells in the nonlinear region, particularly for the Lorentz potential well, indicate that using the same value of the CE coefficient for all types of potential wells may not be advisable. The Lorentz potential well requires a smaller range of CE variation compared to the Delta potential well. Otherwise, the search process may move abruptly from one distant location to another, which can significantly decrease the efficiency of the algorithm. Similarly, potential wells that exhibit smaller nonlinear changes may need to operate with a larger range of CE values.

The effects and benefits of integrating quantum mechanics with Particle Swarm Optimization can be understood by analyzing the parameters that come into play after the inclusion of quantum behavior. A more generalized solution upgrade equation for Quantum Particle Swarm Optimization can be represented by Eq. 15 and Eq. 16, where a local attractor is defined as a point on the line between the self-best explored solution and the global best-achieved solution. The exploration of new solutions from the local attractor is further defined by three parameters: (i) a constant known as the contraction-expansion coefficient, (ii) the characteristic length, (iii) the potential well contribution function and (iv) the probabilistic selection of additive or subtractive arithmetic. The inclusion of quantum behavior in PSO enhances the stochastic approach during the search process by providing different levels of scaling across various dimensions. This leads to a comprehensive range of search possibilities concerning the local attractor in conjunction with probabilistic additive or subtractive arithmetic. The first parameter, η, serves as a scaling factor that helps control convergence over time and must be less than 1. The other parameter and function contribute to the magnitude variation in the search process by scaling the dimensions of available solutions. The provided scaling reflects the absolute differential of the current solution relative to a reference point (usually the mean of the explored self-best solutions). The function that governs this behavior is influenced by random variables, and its characteristics depend on the type of potential well considered. Additionally, this function provides scaling across different dimensions at varying levels, encompassing both global and local exploration. These parameter variations are inherently positive, imparting directional changes through additive and subtractive operations based on the local attractor position within a probabilistic environment. The entire process is compactly illustrated in Fig. 5.

The rotational exploration of new solutions is depicted in Fig. 6, where the search behaviors around the local attractor can be easily observed. It is assumed that the local attractor (LA) is situated in the first quadrant. The rotational behavior during the search for new solutions concerning the local attractor is influenced by two factors: (i) the amplitude of the quantum contribution (assuming a delta potential well and (ii) the probabilistic selection of addition and subtraction as arithmetic operators. The potential search areas in different quadrants are defined under varying conditions, as illustrated in Table 2. It can be concluded that new solutions are explored throughout the entire spatial region surrounding the local attractor.

Considering the usefulness of the potential well in exploring regions far from the local attractor is a key advantage of Quantum Particle Swarm Optimization over classical Particle Swarm Optimization. Analyzing the position update equation provided in Eq. 16 reveals several interesting aspects. Any quantum version of PSO driven by a potential well incorporates three multiplicative factors: - A constant factor that controls the convergence rate and the scaling of change. - A characteristic length that defines the absolute radial distance from the corresponding local attractor for determining positional changes. - Random number-based functions that determine both the scaling of change and the direction of that change. The directional change arises from different dimensions being scaled with varying values. It is fascinating to observe that this can be seen as an exploration of new positions surrounding the local attractors.

Benchmark functions for numeric optimization are groups of functions that embody optimization problems, displaying various characteristics such as being constrained or unconstrained, having continuous or discrete variables, and being either unimodal or multimodal. For any new algorithm, testing and validation using benchmark functions is crucial for evaluating its performance compared to other algorithms. This benchmarking is also essential for gaining a better understanding of the algorithm's advantages and disadvantages.

Previous analyses of individual potential wells aimed to enhance exploration around local attractors. Section 6 details the characteristics and effects of each potential well on solution evolution. This raises the question of whether utilizing a set of potential wells, rather than a single type, throughout the evolutionary process affects performance. To investigate this, two arrangement formats were considered. In the first format, each population member has been given the chance to evolve using a different potential well. In the second format, each dimension of a population member evolves under a distinct potential well. In the first approach, during each iteration, a population member selects a potential well from a predefined set through uniform random selection, and all dimensions of that member evolve with the same potential well. Consequently, the population evolves under multiple potential wells. In the second approach, each dimension of a member evolves with a potential well selected randomly from the set, aiming to increase diversity in the exploration process. The considered set of potential wells carries the Delta, Square Well, Lorentz Potential Field, Rosen-Morse Potential Field, and Coulomb-like Square Root Field.

The performance impact of incorporating multiple potential wells in QPSO was evaluated using 28 functions of dimension 10 (F₁ to F₂₉, excluding F₂) from the CEC2017 benchmark, with a global best strategy for characteristic length. Fifty-one independent trials were conducted. Performance was compared to a single potential well-based strategy implemented by Q^gDPSO, with all other experimental parameters consistent with those in Section 9.2. Results showed close competition among all strategies as can observed in Fig. 27. To assess both convergence behavior and accuracy, the QMindex was calculated. The mean values of relative area under convergence and accuracy across all functions are presented in the Table 13. Accuracy values were nearly identical and close to 1 for all strategies, and the convergence characteristic area was also close to 1 in all cases. QMindex values for all strategies were close to 0. These results suggest that the inclusion of multiple potential wells does not confer a significant advantage in the solution exploration process of QPSO in compare to single potential well.

This work thoroughly discusses and analyzes the purpose and methodology of integrating quantum principles with standard Particle Swarm Optimization (PSO). It addresses existing gaps in prior research to enhance exploration, accelerate convergence, and provide a deeper understanding of how Quantum PSO (QPSO) can improve exploration outcomes. The integration of quantum principles in the search process introduces a characteristic length and a random variable function, which generates new solutions from local attractors by applying varying degrees of change in each dimension. These changes are implemented through probabilistic selection, allowing for both additive and subtractive modifications. Consequently, the new solutions can reach far areas while maintaining directional variation around local attractors. It was observed that the characteristic length plays a more critical role in generating new solutions than the types of potential wells used. The characteristic length based on the global best particle supports parallel exploration of both global and local search, resulting in significantly better exploration and convergence characteristics than the best mean or weighted best mean approaches. Particles that are farther from the global best position engage in global exploration, while those closer to the global best focus on refining their immediate surroundings. This dual strategy enables particles that previously lagged—due to constraints with the best mean characteristic length—to overcome limitations. In contrast, particles using the global best characteristic length can explore more distant regions in search of new solutions. This also enhances the management of complex dimensions due to the capabilities of multilevel and multistage exploration inherent in the global best-based characteristic length. Furthermore, the performance of the global best characteristic length consistently outperforms that of the best mean characteristic length across various types of potential wells, with the best results achieved using the Delta potential well. Additional accuracy improvements were realized by hybridizing different types of characteristic lengths in a probabilistic environment, allowing for random selection among population members to utilize either the global best or the best mean approach. A new metric for estimating algorithm quality was proposed to ensure that both accuracy and convergence dynamics are assessed together. This index is particularly useful for comparing algorithm quality, especially when accuracy is closely aligned and convergence characteristics intersect. Several numeric benchmarks were tested with various types of characteristic lengths and potential wells. The results indicated that the combination of characteristic length with the global best, especially when associated with the Delta potential well, yielded the best performance. Specifically, over 19 functions of the CEC2005 benchmark with the Delta potential well, a 6.8% improvement in normalized accuracy and a 66.62% improvement in normalized convergence characteristics were observed compared to the best mean-based approach. For the Soliton and Harmonic Oscillator potential wells, the improvements in normalized accuracy were 31.09% and 6.52%, respectively, while the normalized convergence characteristics improved by 66.54% and 42.19%. The experiments over 28 functions from the CEC2017 benchmark that carry the hybrid and composite nature of functions, the performances of the proposed algorithm structure with Delta potential well were compared to the newly introduced metaheuristic Farmer and Seasons Algorithm, and a betterment of 14.28% was observed in the performance features. There was an improvement of 2.41% in normalized accuracy and 16.86% in normalized convergence area also in compared to the mean self-best based characteristics length version of QPSO. Availability of different existing potential wells has been used to check the benefits of using multiple potential wells at different levels. Considering multiple potential wells in the solution upgradation at dimension-wise and member-wise does not generate any useful extra benefit over a single potential well. It was observed that under a very diverse environment of landscape, the Delta potential well carried the capability to explore the solution space efficiently.

When considering further research possibilities in this area, several opportunities and challenges emerge simultaneously. The following key areas exist where further research can improve the algorithm's performance:

(iv)Addressing High-Dimensional Challenges: The use of cooperative coevolution approaches, previously applied in evolutionary algorithms, could effectively manage high-dimensional issues.

(v) Hybridization with Other Meta-Heuristics: Currently, the trend in quantum meta-heuristics involves hybridizing with other algorithms. However, it is essential to thoroughly explore the core algorithm before engaging in hybridization; otherwise, it may merely increase computational costs and hinder research progress.

(vi)Optimizing the Exploitation of Quantum Principles: It is crucial to integrate quantum principles into algorithms in a way that enables effective utilization of quantum hardware. The potential for computation presented by quantum principles is vast, and now is the time to optimally and innovatively exploit these opportunities to transform the field of optimization and its applications

(vii) Increasing the reliability strength and reduction in computational burden: even though very satisfactory performances were achieved with Q^gDPSO, for some functions, high values of standard deviation were observed; hence, some mechanism is needed to improve the reliability. The possibility of using a hierarchical structure in the algorithm needs to be explored, where the first stage involves a multi-population that may have evolved under a heterogeneous environment. Later, in the second stage, the better-evolved members from the first stage can be considered as a population to evolve further. This approach can be observed in the present human society. Many practical problems involve a complex objective function and demand a high computational cost. Integrating the involvement of a surrogate model can reduce the computational demand significantly.