1 Introduction

Quantum algorithms are computational procedures that leverage the power of quantum mechanics’ superposition and entanglement phenomena to solve problems at accelerated rates compared to classical algorithms. Many quantum algorithms have been developed for a wide range of applications, including cryptography, optimization, machine learning, and simulation (Montanaro 2016). For instance, the quantum approximate optimization algorithm (QAOA) is a hybrid quantum-classical algorithm that demonstrates its efficacy in tackling optimization problems in the field of logistics, finance, and scheduling (Farhi et al. 2014; Zhu et al. 2022). Shor’s algorithm stands as one of the most esteemed quantum algorithms, designed to factor large numbers, which is considered infeasible for today’s most powerful classical computers (Shor 1994). The algorithm capitalizes on the fact that prime factors of a number can be represented as the period of a function, which can be efficiently computed on a quantum computer using the quantum Fourier transform subroutine. Additionally, the variational quantum eigensolver (VQE) algorithm is a hybrid quantum-classical algorithm for simulating the behavior of molecules, which is important for drug discovery and materials science (Peruzzo et al. 2014). The comprehensive overview of various quantum algorithms is presented in Fig. 1, providing a holistic perspective.

Fig. 1
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Holistic view of quantum algorithms

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An optimization problem is a computational challenge that involves finding the best possible solution from a set of feasible options, typically in accordance with certain defined criteria (Cheng et al. 2016). It has three elements - decision variables, objective function, and constraints, and the key idea is to determine the value of the decision variable subject to constraints such that the objective function is optimized. Mathematical optimization plays a crucial role in solving various science, engineering, economics, and business-related real-life optimization problems. Conventional, deterministic optimization techniques such as calculus-based, enumerative, and dynamic programming provide exact solutions for linear, continuous, differentiable, and convex optimization problems. However, most real-life problems are complex and exhibit characteristics such as multimodality, high-dimensionality and non-differentiability. The deterministic algorithmic approach does not provide a cost and time-efficient solution for non-linear, multi-objective, non-differentiable and non-convex optimization problems. Therefore, a random optimization problem-solving approach becomes significant, offering an approximate solution to NP-hard class optimization problems in polynomial time. Broadly, these are classified into heuristic and metaheuristic algorithms.

Heuristic algorithms work on “trial and error” and produce near-optimal solutions to intricate problems in a feasible time. However, the solutions provided by heuristic algorithms stuck to the local optima and do not guarantee optimum global performance (Desale et al. 2015). Metaheuristic algorithms, on the other hand, mean- “upper level” or “beyond”, which provide a good way to extend the search to the global scale. A metaheuristic search is an iterative process that guides a subordinate heuristic by intelligently combining various concepts to explore and exploit the search spaces using learning strategies to organize information effectively, aiming to discover near-optimal solutions efficiently (Osman and Kelly 1997). Metaheuristics can be classified as single-solution and population-based approaches, with the former being exploitative and the latter being explorative (Baghel et al. 2012). Over the past few decades, the hybridization of metaheuristic algorithms with deep learning, machine learning, and data mining algorithms experienced a surge to leverage the power of both domains (Akay et al. 2022; Hussain et al. 2019). Moreover, introducing the computing advantages of quantum mechanics to existing algorithms has become a profound interest and widespread concern. QiMs are the approximate algorithms that exploit quantum computing (QC) mechanisms to improve computational complexity and performance. These algorithms can explore search spaces more efficiently and effectively by executing multiple search trajectories simultaneously, potentially finding better solutions than classical optimization techniques. Although these algorithms are designed to run on classical computers, they ameliorate the efficiency considerably (Ganesan et al. 2021).

Furthermore, various criteria could be applied to categorize QiMs like a type of candidate solution, type of search, and source of inspiration (Osman 2003; Binitha and Sathya 2012; Gendreau and Potvin 2005; Dragoi and Dafinescu 2021). The inspiration source of major metaheuristics is nature, such as evolutionary algorithms (EA) and genetic algorithms (GA) mimic biological processes; algorithms like particle swarm optimization (PSO), ant colony optimization (ACO), and artificial colony optimization are inspired by animal behavior. Additionally, metaheuristics like gravitational search (GS) and simulated annealing are based on physics laws, while others find inspiration in human activities, chemistry, mathematics, and musical laws (Rao et al. 2011; Abualigah and Diabat 2021; Alia and Mandava 2011; Bechikh et al. 2014). Due to the potential benefits offered by QiMs over classical metaheuristic algorithms, there is an extensive literature in the field. Scientometric analysis paves the way for novel technological paths, fostering researchers’ interactions. In contrast, the systematic literature review (SLR) aims to offer comprehensive insights by employing theoretical synthesis in a specific field. Hence, the present study addresses the research gap by conducting a scientometric and systematic analysis of QiMs publications from inception to date based on four broad categories viz., quantum-inspired evolutionary algorithms (QiEA), quantum-inspired genetic algorithms (QiGA), quantum-inspired particle swarm optimization algorithms (QiPSO) and quantum-inspired algorithms based on physics and chemistry laws (QiPC). The research does not encompass QiMs that utilize game, human, mathematical, and musical techniques, primarily due to the limited availability of data and research on these specific approaches. The proposed research explores explicitly the four categories mentioned above by performing publication trends, co-citation, and co-occurrence network analysis. In addition, the current research extensively analyzes the reputed scholarly work by adopting the SLR approach to get a deeper understanding.

Table 1 Summary of review articles
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1.1 Literature review

QC manifests breakthroughs in quite a large number of domains, and optimization is one among them. In addition, considerable interest has been observed in QiMs development during recent decades, owing to the remarkable performance exhibited by the algorithms. Consequently, a spurt in the scientific literature has been observed, highlighting the importance of structuring the literature outcomes to set a route for future research. The current section delves into the pertinent scientific literature concerning QiMs to provide a concise summary of the findings and to address the research gap to be tackled in the current study. Li et al. (2020) presented an overview of QiEA, QiPSO and quantum immune clonal algorithm, focusing on improvements made to operators and population size. Furthermore, the authors have discussed popular quantum learning algorithms, including quantum neural networks and quantum clustering algorithms. Additionally, the rationality of combining learning algorithms with quantum optimization algorithms, compared to the traditional counterparts, is demonstrated. Ross (2019) presented the survey that reports the analysis of QiMs from the perspective of their implementation over circuit-based quantum computers. The analysis shows that a quantum-inspired Acromyrmex evolutionary algorithm can be implemented over NISQ quantum computers. Hakemi et al. (2022) explored the domain of QiMs over the past five years and illuminated the strength of fusing quantum mechanical notions with optimization techniques. In addition, the authors present the practical applications of QiMs in addressing real-world problems and embark on exploring novel metaheuristic algorithms that have yet to incorporate quantum elements. Gharehchopogh (2023) presents a comprehensive review of QiMs based on the advantages and disadvantages offered by the algorithms over conventional analogs. The analysis provides valuable insights into the particular applications of QiMs in the science and engineering field, thereby emphasizing the practical relevance and potential impact in these fields.

1.1.1 Research gap

An extensive forage of popular databases such as IEEE Explore, Web of Science (WoS) and Scopus shows that very few review analyses have been conducted over QiMs research domain, and no study presented the knowledge structure in the current domain to set an evolutionary pathway for the researchers. Moreover, the state-of-the-art surveys limit in presenting the intellectual synthesis and future research directions for the researchers. Conspicuously, with improvements in data processing and visualization techniques that enable the in-depth exploration of a domain, knowledge mapping using scientometric analysis has become more and more potent. Several scientometric studies have been performed that explore the macro-structure of a domain, such as re-manufacturing (Ozcan and Corum 2021), convergence (Klarin et al. 2021), smart disaster management (Neelam and Sood 2021), smart learning (Kaur and Bhatia 2021), airborne disease prevention (Sood et al. 2023), and quantum computing (Sood and Pooja 2023; Sood and Chauhan 2023). On the contrary, the systematic literature analysis focuses more on the content to synthesize and explore the research methods, differences and similarities, and research gaps in the scientific data. Consequently, the present study employs the blend of scientometric and SLR, comparable to the approach used by Kaur and Bhatia (2022), Gao et al. (2022), and Goyal and Kumar (2021) to counter the research gap. Table 1 shows the comparison of significant review articles existing in the QiMs and in various other domains based on seven parameters, namely methodology, analytical tool, research area, domain categorization, co-occurence network analysis (CoNA), co-citation network analysis (CcNA) and research prospects. The existing literature in QiMs domain reveals that no article presents the comprehensive knowledge structure in QiMs using the scientometric methodology. Moreover, the significant articles of various domains utilizing scientometric and systematic methodology do not fully employ all the parameters listed in Table 1. Therefore, the proposed study concentrates on delivering both the co-citation and co-occurrence analyses. The scientometric approach in the current research presents the publication growth analysis, prominent research areas, high-yield authors and most collaborating countries. Moreover, an extensive SLR analysis is performed to reveal key challenges and future research scope in this knowledge domain.

1.2 Research questions

The primary goal of the study is to summarize the status quo of QiMs research, with the following research questions defining its purview:

  • RQ1 What are the temporal publication patterns?

  • RQ2 What are the potential research topics emerging application areas of QiMs in each category?

  • RQ3 Which are the prominent authors in each category of QiMs?

  • RQ4 Which active countries are involved in cooperative research in QiMs domain?

  • RQ5 What are the key insights and research gaps in the current knowledge domain?

Paper organization

The research article is structured into six sections to analyze QiMs literature. Section 2 delineates the methodology employed for gathering data and selecting appropriate tools for data processing. Section 3 provides an extensive SLR to present key insights and identify future research opportunities. Section 4 presents the growth of publications, keyword co-occurrence analysis, author co-citation analysis and country collaboration analysis. Section 5 presents the discussion, key insights, research gaps and future research directions. Section 6 presents the conclusion of the study.

2 Methodology

2.1 Data search strategy

The widely recognized and appropriate method for assessing scientific data, known as the preferred reporting items for systematic reviews and meta-analysis (PRISMA), proposed by Liberati et al. (2009), has been employed to gather the most pertinent scientific literature on QiMs. Scopus database consisting of extensive content coverage and high-impact scientific journals has been employed to extract the dataset corresponding to each category. Equation 1 specifies the format of the search query employed on (TITLE-ABS-KEY) search field of the Scopus database for retrieving the dataset in each category. To retrieve data for a particular category, the set of keywords associated with that category are combined using OR operators. The combined keyword set is then used as a search string in the Scopus database to extract relevant records. Furthermore, to minimize overlapping among categories, records of other categories are excluded by implementing the AND NOT operator with their set of keywords.

$$\begin{aligned} cat_{i} =\{K\_cat_{i}\} \bigcap \; \left\{ \sim \left[ \bigcup \limits _{j=1}^{n} K\_cat_{j}, i \ne j\right] \right\} \end{aligned}$$
(1)

where, K_\(cat_{i}\) depicts the set of keywords corresponding to each category, n=4, and

K_\(cat_{1}\) “quantum genetic algorithm OR qga”,

K_\(cat_{2}\) “quantum evolutionary OR quantum memetic”,

K_\(cat_{3}\) “quantum particle swarm optimization OR qpso OR quantum ant colony OR quantum artificial bee colony OR quantum krill herd OR quantum cuckoo search OR quantum flower pollination” and

K_\(cat_{4}\) “quantum gravitational search OR qgsa OR multiscale quantum harmonic oscillator algorithm OR mqhoa OR simulated annealing OR chemical reaction optimization”.

The search terms for each query were identified from the previously published literature on QiMs, including the synonyms and relevant algorithms in each domain (Hakemi et al. 2022; Gharehchopogh 2023; Dehghani et al. 2022; Abdel-Basset et al. 2018). Figure 2 represents the review methodology adopted in the current study. The first column illuminates the PRISMA approach followed to search and collect data using four steps: Identification, screening, eligibility, and included records. In the identification phase, 879, 549, 949, 761 were identified, spanning from 1991 to 2023 falling within the QiGA, QiEA, QiPSO and QiPC categories, respectively. These records underwent screening by the exclusion criteria mentioned in Fig. 2. Finally, we employed the CiteSpace tool to eliminate duplicate records and restricted our dataset to English-language articles, resulting in a final selection of 443, 420, 535, and 452 papers for QiGA, QiEA, QiPSO, and QiPC categories, respectively. Moreover, to perform systematic literature analysis, the titles and abstracts of full-text articles were meticulously examined to ascertain their eligibility. The records which did not align with the research objectives were excluded and ultimately, 43 records were identified for systematic analysis. The records pertaining to each category were downloaded in the research information system (RIS) file format to perform scientometric and systematic analysis. Furthermore, the second column showcases the analyses performed in the paper, which include publication growth analysis, keyword co-occurrence network analysis, author co-citation analysis, country collaboration analysis and systematic literature review. The subsequent column in the framework diagram highlights the results obtained in the study.

Fig. 2
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QiMs literature analysis framework

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Fig. 3
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Major analytical steps supported by CiteSpace

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2.2 Tools

There is a range of data visualization tools at researchers’ disposal for conducting scientometric analysis including CiteSpace (Chen et al. 2012), Gephi (Jacomy et al. 2014), VOSviewer (Van Eck and Waltman 2010), SciMAT (Cobo et al. 2012), and UCInet (Johnson 1987). However, the authors have employed CiteSpace- a java-based visualization and analysis software (Chen 2016). CiteSpace is a professional scientometric and visualization tool developed by Dr. C. Chen under the aegis of Drexel University, USA. Figure 3 illustrates the procedural flowchart detailing the key steps followed to perform the visual analytics. It enables analysis of scientific contributions in plain text/Web of Science (WoS) format based on authors, institutions, references, keywords and countries categories. The network can be tailored according to specific requirements using parameters such as the look-back year, g-index, network pruning, and link retention factors. CiteSpace offers various structural and temporal metrics, including citation burstness, silhouette score, network density, modularity, and betweenness centrality (BC), which can be utilized to validate the results obtained through the visualization network. The parameters undergo iterative adjustments and testing procedures until the desired outcomes are achieved.

In addition, the current article utilizes the MATLAB tool to visually represent the publication patterns. MATLAB is a widely used programming language and integrated development environment with an extensive range of applications in fields such as engineering, mathematics, and scientific research. It offers various powerful and flexible built-in functions and toolboxes for complex numerical computations, data analysis, visualization, and algorithm development (Matlab 2012).

3 Systematic literature review

In recent studies, the researchers have explored different aspects pertinent to QiMs. To gain a comprehensive understanding of the state-of-the-art in a field, a thorough investigation of various works is required, under one unified perspective. The current survey identifies and addresses several challenges in QiMs domain by performing a SLR.

3.1 QiGA

GAs are optimization techniques that draw inspiration from the mechanism of natural selection and genetics. These algorithms are employed to solve intricate optimization problems by mimicking the processes of selection, crossover, and mutation. QiGAs are a variant of classical genetic algorithms incorporating quantum-inspired techniques to enhance their capabilities. In the domain of QiGA, an array of algorithms has been formulated and advanced, including quantum-inspired genetic programming algorithm (Pereira et al. 2020), quantum-inspired memetic algorithm (Zhang et al. 2022), and qauntum-inspired cultural algorithm (Guo et al. 2018). Table 2 represents the basic quantum-inspired algorithms corresponding to each category in QiMs. QiGA finds applications in optimization problems, machine learning, image processing, bioinformatics, finance, energy optimization, and quantum computing. In this context, Li and Wang (2007) presented a hybrid QiGA for addressing the multi-objective flow shop scheduling problem. The proposed approach incorporates the updating operator of a quantum gate. The simulation results and performance metric comparisons demonstrated effective and robust performance in multi-objective optimization, achieving solutions with reasonable proximity and diversity. Xiao et al. (2010) proposed a QiGA for k-means clustering (KMQGA) on a Qubit-based representation. The proposed algorithm uses a quantum gate rotation operator with traditional genetic algorithm operations such as selection, crossover and mutation. By operating within the discrete 0–1 hyperspace, KMQGA effectively combines exploration and exploitation, resulting in improved clustering performance and enhanced exploitation capabilities. Konar et al. (2017) presented hybrid QiGA, an efficient real-time task scheduling method in a multiprocessor environment. The algorithm utilized rotation gates to explore variable chromosomes represented by qubits in Hilbert hyperspace to improve convergence in the scheduling process. Simulation results demonstrate its superiority over classical GAs and hybrid PSOs in terms of fitness values and scheduling time. Furthermore, QiGAs have diverse applications, including fuzzy network training (Zhao et al. 2009), antenna positioning (Dahi et al. 2016), multi-document text summarization (Mojrian and Mirroshandel 2021), and workflow scheduling optimization (Hussain et al. 2022).

3.2 QiEA

QiEA is an optimization technique that mimics natural evolution processes to solve complex problems. The pioneering work on QiEA was introduced by Han and Kim (2002), demonstrating the applicability of QiEA to combinatorial optimization problems. In the domain of QiEA, several algorithms have emerged, including quantum-inspired evolutionary programming (Yasin et al. 2010), quantum-inspired differential evolution algorithm (Deng et al. 2020), and quantum-inspired Acromyrmex evolutionary algorithm (Montiel et al. 2019). QiEA can be applied to improve the efficiency of various optimization problems, quantum algorithms, and quantum circuit design (Wright and Jordanov 2017; Zhang 2011). It involves tuning the parameters of quantum gates by designing quantum-inspired evolutionary operators, mitigating the effects of decoherence, and optimizing control pulses (Arpaia et al. 2011). The quantum rotation gate is the popularly used operator for carrying quantum gate updates, impacting the performance of QiEA remarkably (Xiong et al. 2018). Tayarani and Akbarzadeh (2014) made significant advancements in enhancing the performance of QiEA by introducing novel operators, including a diversity-preserving operator and a reinitialization operator. These operators play a crucial role in maintaining population diversity, mitigating premature convergence, and ultimately facilitating the algorithm in discovering better solutions. Furthermore, hybrid approaches are popular in QiEA that combine evolutionary algorithms with other optimization techniques, such as metaheuristics, differential evolution, and machine learning, to leverage the strengths of different algorithms (Wang and Li 2010; Kolahdoozi et al. 2019; Bharill et al. 2019). A hybrid quantum evolutionary algorithm is proposed by Cia et al. (2021), which leverages the techniques of cooperative co-evolution, random rotation direction and Hamming adaptive rotation angle to optimize the airport gate allocation problem. QiEA exhibits an enhanced performance in terms of reduced population size, preventing premature convergence rate and balance between exploration and exploitation of the search space, compared to the classical evolutionary algorithms (Zhang 2011).

3.3 QiPSO

QiPSO is an optimization algorithm that merges the fundamental principles of PSO, originally introduced by Kennedy and Eberhart (1995), with innovative quantum-inspired techniques. It incorporates quantum-inspired operators and encoding schemes to enhance exploration and convergence, making it suitable for complex optimization problems. The QiPSO was brought forth by Yang et al. (2004) that harness the quantum mechanics properties to nurture the capabilities of classical PSO. In the realm of quantum-inspired optimization algorithms, various versions of PSO have been developed, including ant-colony optimization (Li et al. 2019), Krill Herd Optimization (Li et al. 2019), Firefly algorithm (Wong et al. 2014), grey wolf optimizer (Srikanth et al. 2018), bat-inspired algorithm (Sharma and Sharma 2022) and whale optimization algorithm (Agrawal et al. 2020). Furthermore, the hybridization of QiPSO with other algorithms like genetic algorithms, simulated annealing, and machine learning algorithms aim to leverage the strengths of each algorithm and overcome limitations the algorithm possesses (Ali Ghorbani et al. 2018; Zouache et al. 2016; Fahad et al. 2022; Mirsadeghi and Khodayifar 2021). In this context, Singh and Mahapatra (2016) proposed a QiPSO algorithm hybridized with a genetic algorithm to prevent immature convergence of the population and mitigate the completion time to a significant extent. In an article by Jeong et al. (2010), the authors proposed a novel quantum-inspired binary particle swarm optimization (QiBPSO) algorithm specifically tailored for solving the unit commitment (UC) problems in power systems. The authors also proposed a new rotation gate to update the Qubit individual. The application of QiBPSO to UC demonstrated promising results, showcasing a remarkable optimizer for bulky UC problems. Although the scope of QiPSO and QiEA is indistinguishable, however, based on a comparison study by Cheng et al. (2023), QiPSO demonstrated superior efficiency in terms of fidelity and robustness when compared to QiEA. The study found that QiPSO outperformed QiEA in terms of maintaining solution quality and handling variations and uncertainties in optimization problems.

3.4 QiPC

In the realm of quantum-inspired optimization algorithms based on physics and chemistry laws, various algorithms have been developed, including gravitational search algorithm (Soleimanpour-Moghadam et al. 2014), multiscale quantum harmonic oscillator algorithm (Wang et al. 2018), simulated annealing (Rutenbar 1989), colliding bodies optimization (Kaveh et al. 2020), charged system search (Talatahari et al. 2022). Furthermore, QiPC often uses techniques like adiabatic computing or variational approaches to explore solution spaces, optimize objective functions, or simulate quantum systems. In this context, Bandyopadhyay et al. (2008) proposed an adaptive multi-objective simulated annealing algorithm (AMOSA) for multi-objective optimization problems that incorporate simulated annealing and adaptive techniques to efficiently explore the solution space and find diverse sets of Pareto optimal solutions. Experimental findings presented that AMOSA performs efficiently in the context of convergence, diversity, and solution quality compared to other traditional multi-objective optimization algorithms. Li et al. (2019) presented a competitive modified multiscale quantum harmonic oscillator algorithm to tackle multi-modal optimization problems while incorporating strict meta-stability constraints. The experimental results show enhanced robustness and exploitation ability in local search space as compared to other metaheuristic algorithms. The authors proposed a quantum-inspired gravitational search algorithm that demonstrated efficient performance in binary encoded optimization problems (Nezamabadi-Pour 2015). Additionally, various QiPC algorithms have been developed for feature selection (Barani et al. 2017), classification (Han et al. 2013), and facial expression recognition (Kumar et al. 2020) applications. Consequently, QiPC algorithms offered valuable computational enhancements and insights in various optimization, machine learning, and computational optimization problems.

Table 2 Framework of quantum-inspired algorithms in each category
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4 Analyses

4.1 Publication trends

The following section analyzes publication trends in the field of QiMs from inception to 2023. Figure 4 visualizes the overall publication’s growth within QiMs literature. The horizontal axis shows the years from 1991 to 2023, while the vertical axis represents the number of publications received during this time span. The blue trend line in the graph depicts the year-wise publications count of QiMs literature, whereas the red line shows the percentage share of QiMs publications corresponding to the quantum algorithms (QAs) research domain. The QA constitutes algorithmic development in various areas, such as error mitigation, pattern recognition, classification, machine learning, quantum simulation, and secure post-quantum communication. The analysis infers that QiMs literature solely contributes with 26.66% average publications share corresponding to the QA domain, depicting a significant contribution and the current research interest. Additionally, research progression has incurred gradually from 2008 to 2020, and a spurt is observed in 2021 and 2022. Figure 5 depicts the publication counts in each category of QiMs literature. The analysis depicts a steady growth of publications in QiGA, QiEA and QiPSO genres till 2006 and a significant rise in publication count afterward. Remarkably, 2022 shows the highest frequency of published papers in each category. Moreover, the research interests in QiGA and QiPSO categories have increased significantly in the last 5 years. However, the QiEA shows a decline in the publication count in the 2018 (18), 2021 (26) and 2022 (23) years. The decline of trending lines in the graphs in 2023 can be attributed to the inclusion of records only until March.

Fig. 4
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Cumulative publication trends every year

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Fig. 5
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Publication trends in each domain

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4.2 Keyword co-occurrence

Keyword co-occurrence network (KCN) analysis provides significant insight into the emerging research topics and core themes of publications in a succinct manner. KCN analysis for each category of QiMs is performed using the CiteSpace tool. Figures 6, 7, 8, 9 illustrate the KCN of QiMs corresponding to each category. In visualizations, each node represents a keyword, and the node size corresponds to the frequency of co-occurrence of that keyword. A larger node size indicates a higher frequency and, thus, a research trend. In KCN, the degree denotes the cardinality of connections or links established by a keyword with other keywords within the analyzed dataset. The link color indicates the year in which the linked keywords initially co-occurred. The width of the purple ring surrounding a node indicates the BC score in a network. This metric gauges the significance and potential impact of a keyword within the research domain. The nodes with the yellow color represent the emerging research area. Moreover, the networks omit some irrelevant keywords, such as “atoms” and “non-human,” thereby enhancing the clarity and visual impact of the network diagrams by reducing unnecessary keywords and links clutter. Table 3 displays the keywords that occur most frequently, along with the accuracy, degree and BC values. The accuracy value assesses the percentage of relevancy of the keyword within its category as compared to others. It is computed by a percentage-based calculation method, which entails the ratio of the keyword’s frequency within its designated category to the total frequency across the remaining categories. Moreover, the average keywords such as “quantum computer”, “quantum inspired genetic algorithm”, and “quantum evolutionary algorithm” are omitted; and overlapping keywords are categorized under the most relevant category based on our current understanding.

Table 3 Top five keywords by frequency, BC and degree in each category
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Figure 6 depicts the KCN of QiGA, consisting of 471 nodes and 2556 links, indicating a high network density. The analysis reflects that the research trends in the QiGA domain are combinatorial optimization (28), support vector machine (26), regression analysis (24), fault detection (23) and cloud computing (14). Moreover, the network reveals the potential influence of combinatorial optimization, with the highest degree (49) and BC (0.08) values. The KCN analysis also identifies emerging application areas in the QiGA category as job shop scheduling, aerodynamics, and compressed sensing.

Fig. 7 presents the KCN of QiEA, consisting of 485 nodes and 2195 links, indicating a high network density. The results reveal that the primary research topics in the QiEA domain are quantum theory (47), artificial intelligence (14), learning algorithms (24), integer programming (12), and knapsack problem (11). Additionally, the network reveals the potential influence of quantum theory, with the highest degree (65) and BC (0.09) values. Furthermore, the KCN analysis identifies emerging research areas in the QiEA category as 6 g, agricultural monitoring, quantum cryptography, vehicle routing, quantum simulators.

Fig. 6
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KCN analysis of QiGA

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Fig. 7
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KCN analysis of QiEA

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Fig. 8
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KCN analysis of QiPSO

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Fig. 9
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KCN analysis of QiPC

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Figure 8 represents the KCN of QiPSO, consisting of 395 nodes and 2028 links, indicating a high network density. The results reveal that the main research topics in the QiPSO domain are ant-colony optimization (38), quantum chemistry (34), clustering algorithms (13), internet of things (11), and molecular structure (10). The network also reveals the potential influence of clustering algorithms, with the highest degree (23) and ant-colony optimization with BC (0.11) values. The KCN analysis also identifies emerging research areas in the QiPSO category as power management (telecommunication) and spectrometry, wireless sensor network.

Fig. 9 shows the KCN of QiPC, consisting of 457 nodes and 2175 links, indicating a high network density. The results reveal that the primary research topics in the QiPC domain are quantum annealing (40), monte carlo methods (26), ground state (23), density function theory (19), and ising model (14). The network also reveals the potential influence of the ground state, with the highest degree (46) and BC (0.12) values. The KCN analysis also identifies emerging research areas in the QiPC category as quantum entanglement, adiabatic quantum computation.

Moreover, examining the co-occurrence patterns of keywords over time in the QiMs domain reveals that current research predominantly focuses on developing quantum adaptations of machine learning algorithms. Additionally, emerging research areas within the QiMs domain encompass 6 g, agricultural monitoring, quantum cryptography, routing algorithms, power management, and wireless sensor network. Notably, quantum-inspired algorithms such as integer programming, ant-colony optimization, monte carlo, and quantum annealing have observed the rise as the most popular methods for solving optimization problems.

4.3 Author co-citation analysis

By mapping networks of scientific literature, the direct citation analysis straightforwardly reflects real-world connections, whereas co-citation analysis tracks pairs of papers often mentioned in the same source article to analyze development patterns and relationships between authors, documents, and journals. Henceforth, the author co-citation analysis identifies authors based on frequently co-cited publications, presenting an intellectual association of authors. This approach outlines the emerging authors in a particular study field that are actively contributing to the development of their research area. The author co-citation analysis for each category of QiMs is performed using CiteSpace and the visualizations for each category are displayed in Figs. 10, 11, 12, 13. In the network diagrams, nodes depict the authors and links between nodes represent the co-citation strength. The network density indicates the extent of the overall cooperation among researchers (Ma et al. 2022). Node labels’ size in the diagrams is analogous to the citation count, where the larger the font size of the node label, the greater the co-citation frequency. Table 4 highlights the top 10 influential authors indexed by F1 score, frequency, BC, and burst value. The BC value gauges the significance of nodes in the network and the high burst value shows that citations to the authors’ work have increased suddenly. The F1 score gauges the overall performance of an analysis in terms of precision and recall and is calculated by using Eq. 2 given below:

$$\begin{aligned} F1-score= 2*(precision*recall)/(precision+recall) \end{aligned}$$
(2)

where, precision is the measure of accuracy and recall is the measure of relevancy of an author among all categories in author co-citation network. The author co-citation network diagram of QiGA Fig. 10 has 430 nodes and 2263 links with an overall network density value of 0.0245. The analysis shows that HAN KH (209), NARAYANAN A (188), and WANG L (99) have received the highest citations; GROVER LK (8.98), SHOR PW (7.54) and ZHANG Y (7.33) are most significant authors; NARAYANAN A (0.21), HAN KH (0.17), and WANG L (0.11) have attained abrupt citations growth.

The author co-citation network diagram of QiEA Fig. 11 has 471 nodes and 1971 links with an overall network density value of 0.0178. The network analysis depicts that HAN KH (134), NARAYANAN A (63) and ZHANG G (56) have received the highest citations; HAN KH (6.47), ZHANG G (5.72) and GROVER LK (5.42) are most significant authors; HAN KH, NARAYANAN A and WANG Y have gained sudden growth in citations count with 0.6 burst value in QiEA category.

The author co-citation network diagram of QiPSO Fig. 12 has 382 nodes and 2774 links with an overall network density value of 0.0381. The network shows that SUN J (96), BRANCHINI BR (66) and WANG Y (54) are the most cited authors; MIRJALILI S (8.44), FRISCH MJ (7.32) and KENNEDY J (4.84) are the most significant authors; BRANCHINI BR (0.19), WANG Y(0.15) and ANDO Y (0.14) have seen a dramatic increase in the citations count in QiPSO category.

The author co-citation network diagram of QiPC Fig. 13 has 487 nodes and 1773 links with an overall network density value of 0.015. In the QiPC category KIRKPATRICK S (135), KADOWAKI T (70) and FARHI E (63) have the highest citations count; LUCAS A (7.6), BOIXO S (5.7) and DAS A (4.02) are most significant authors; KIRKPATRICK S (0.61), FARHI E (0.1) and ALBASH T (0.06) have the highest burst score.

In addition to identifying the most influential authors, the network density of co-cited author networks in the field of QiMs depicts substantial interconnections and collaboration among authors. It indicates a highly cohesive research community recognizing and acknowledging mutual contributions.

Fig. 10
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Author co-citation network analysis of QiGA

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Fig. 11
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Author co-citation network analysis of QiEA

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Fig. 12
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Author co-citation network analysis of QiPSO

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Fig. 13
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Author co-citation network analysis of QiPC

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Table 4 Top 10 authors by frequency and BC in each category
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4.4 Country collaboration

Country collaboration network analysis states the cooperative relationship and the arrangement of influential countries in a research domain. The analysis provides a deep insight to the scholars about the active countries involved in cooperative research; moreover, the outcomes serve as motivation for scholars to seek cooperation opportunities internationally in this knowledge domain. Figure 14 visualizes the country cooperation network that comprises 81 nodes and 344 links. Each node depicts the publications count resulting from collaborative efforts between different countries, with larger nodes indicating higher publication counts. The number of links that a particular country has with other countries in the network quantifies the level of collaboration between countries; more links count conveys close inter-country cooperation. The layers in the node of the country collaboration network represent the number of citations, with a broader layer indicating more citations. The color of the layer corresponds to a specific time period in which it has received the citations. This helps the researchers to identify the temporal evolution of citations of a node. The red centers in the nodes represent the burstness, which refers to the publications that have experienced a sudden significant increase in citations within a specific time period. The purple ring around a node representing BC value serves as the visual indicator of its level of influence within the collaboration structure. The analysis shows that developed countries such as China, the United States, Japan, the United Kingdom, Germany, France, Italy, and Canada are leading collaborative countries in the QiMs research domain. Furthermore, the node of China in the collaborative network is the largest, indicating that China (757) acquired a leading cooperation position in QiMs, followed by the United States (253) and India (157). Developed countries tend to have more established networks of resources and researchers, leading to accelerating and fostering collaborative research. Furthermore, it is evident from the intricate, entangled linkages that QiMs have become a research topic with global interest. Collaboration between nations in this knowledge domain is increasing, showing that countries are dedicated to conducting research together. Table 5 lists the top 10 most collaborative countries as per publications count, BC and link count (number of neighboring nodes). According to the results, the United States shows the highest degree of cooperation with a maximum link count (41), followed by China (39) and India (33), indicating global participation in QiMs research. In the context of BC, China (0.32), the United States (0.29), and the United Kingdom (0.20) produced the most insightful publications in QiMs. Besides, it can be seen that instead of a remarkable publications count, BC of Japan is low, depicting the requirement of researchers’ efforts on quality enhancement.

Fig. 14
figure 14

Country collaboration network diagram

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Table 5 Top 10 most collaborating countries of QiMs research
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5 Discussion

The current research is carried out to answer the research questions mentioned earlier by scientometric and systematic literature analysis approach of the nature-inspired QiMs of the Scopus dataset since its inception. The analysis is performed over four categories based on the source of inspiration: quantum-inspired evolutionary algorithms, quantum-inspired genetic algorithms, quantum-inspired particle swarm optimization algorithms, and quantum-inspired algorithms based on physics and chemistry laws.

  1. 1.

    To address the first research question regarding publication trends, the authors have conducted a publications pattern analysis of QiMs literature. The analysis shows that the contribution of QiMs algorithms, amounting to 26.66% of overall quantum algorithms, indicates its status as a current research hotspot. Furthermore, the distribution of literature corresponding to each category reveals a significant growth rate of 42.59% in research over the last decade, depicting the knowledge domain holds immense research scope. A review of the publication trends corresponding to each category advocates that QiPSO (526) pertains to the maximum publications count. The analysis emphasizes that the potential of QiMs remains largely untapped, highlighting the opportunity for radical alteration through the assimilation and deployment of diverse QiMs in optimization. By exploring and harnessing the capabilities of various QiMs, significant advancements and transformative changes can be achieved in the field of optimization.

  2. 2.

    To address the second research question regarding popular research topics, the authors have conducted KCN using the CiteSpace visualization tool. The analysis depicted that combinatorial optimization, support vector machine, job shop scheduling, and aerodynamics in QiGA; artificial intelligence, learning algorithm, 6 g, and agricultural monitoring in QiEA; ant-colony optimization, quantum chemistry, power management, and wireless sensor network in QiPSO; quantum annealing, monte carlo methods, quantum entanglement, and adiabatic quantum computation in QiPC, are the potential research topics and emerging application areas corresponding to each category.

  3. 3.

    Regarding contribution to the third research question, the high-yield authors are Han Kh, Sun J and Kirkpatrick S. At the same time, Grover Lk, Han Kh, Mirjalili S and Lucas A are the most significant authors.

  4. 4.

    In response to the fourth research question regarding most collaborating countries, it is evident that China, the United States, and India emerge as the dominant contributors in research on QiMs literature conducted in collaboration with other countries. Moreover, the study highlights that China produces the highest quality literature and the United States is the most collaborative country.

In conclusion, the QiMs have received more scholarly attention in the last decade; among them, QiPSO has been very mature. Thus, the QiGA, QiEA and QiPC categories imply a scope of research. Moreover, the dense author co-citation network diagram indicates a good collaboration among authors in each category, which signifies a highly cohesive research community. Finally, the research is currently focused on QiMs in emerging fiscal realms such as China, the United States, and India.

5.1 Key insights and research challenges

The current study presents a systematic review of scholarly literature aiming to develop state-of-the-art of QiMs. The analysis implies the potential that the efficiency of metaheuristic algorithms can be boosted by integrating operators such as quantum bit encoding, quantum gate and quantum rotation gate. Moreover, QiMs exhibit enhanced performance in terms of reduced population size, preventing premature convergence, a balance between exploitation and exploration, reasonable proximity and diversity, and reduced execution iterations compared to classical metaheuristic algorithms. Consequently, the scientometric and SLR uproots following research gaps:

  1. 1.

    Research in the QiMs realm should focus on driving a thorough comparative analysis of algorithms, in terms of performance and computational complexity, with the traditional counterparts.

  2. 2.

    Combining QiMs with other metaheuristic, simulated annealing and machine learning algorithms is an emerging area; therefore, software practitioners should focus on experimental and theoretical analysis for the optimum utilization of optimization methodologies.

  3. 3.

    Research in combinatorial and multi-objective optimization effectively leverages quantum computing technologies to improve performance. However, the research for quantum-inspired constraint-handling optimization problems is still in the preliminary stage. Therefore, exploring real-world problems relevant to constraint handling can be considered to advance this field of research.

  4. 4.

    Most published practical implementations primarily focused on comparing QiMs with their classical counterparts. However, exploring the strengths and weaknesses of different QiMs with respect to each other is still an open area of research.

  5. 5.

    Exploration of the performance of hybrid and pure quantum optimization algorithms based on computational complexity matrices circuit width, size, and length can be carried out to incentivize the research.

5.2 Future Research Directions

The present research elucidates an in-depth and intent examination of QiMs literature to propel scientific advances and to provide innovative significance to software engineers.

  1. 1.

    The presented research can be further extended by incorporating document co-citation cluster analysis and co-authoring institution analysis in the current study.

  2. 2.

    The article meticulously employs the SLR technique to gain valuable insights into the research domain. Throughout the study, the authors rely on the comprehensive and widely utilized Scopus database for extracting relevant research articles. Although, the results presented in the current study are clearly illustrated to depict the scientific evolution and key insight into the literature. However, the records from other databases, such as Web of Science, IEEE Explore, and Google Scholars; other document types, such as books, book chapters, editorial notes, and websites; and languages other than English, can be considered to broaden the scope of the study.

6 Conclusion

The article serves as a comprehensive representation of the current research progress in optimization using QiMs. The scientometric implications of the current research offer a detailed exploration of the publication trends, keyword co-occurrence, author co-citation and country collaboration and prominent literature corresponding to each opted category of QiMs. Through analysis, it is discerned that QiMs exclusively include 26.66% of publication share in quantum computing and 42.59% growth rate in the past decade, indicating the emergence of QiMs as a prominent research hotspot. The study identifies job shop scheduling, aerodynamics, 6 g, agricultural monitoring, quantum cryptography, vehicle routing, power management, and wireless sensor network are the emerging research areas in QiMs domain. Significantly, among QiMs, integer programming, ant-colony optimization, monte carlo simulation, and quantum annealing have garnered prominence as the most popular algorithms for addressing optimization problems. Moreover, a meticulous examination of most prominent literature reveals that hybridization of QiMs with various machine learning algorithms is the prominent research field. The author co-citation network analysis identifies Han Kh, Sun J, Mirjalili S and Lucas A as influential authors that assist the researchers in keeping abreast with the latest developments and to commence joint initiatives. Furthermore, the proposed country collaboration analysis identifies China, the United States, and India that allows policymakers and funding organizations to draw conclusions about the value of fostering collaborations, allocating resources, and encouraging international partnerships in this domain. Acting as a valuable guide for academicians and researchers interested in this field, it also provides an impetus for solving complex constraint-based real-world optimization problems. Moreover, more accurate and optimized solutions can assist decision-makers in exploring the best possible alternatives and making informed choices, leading to risk mitigation and improved resource allocation. In the future, the authors intend to implement the QiMs by leveraging the tools and technologies provided by Amazon Braket quantum computing services.