# Quantum-Based Analytical Techniques on the Tackling of Well Placement Optimization
## 1 Introduction
Following this methodology, many researchers combined different algorithm's best features and developed many hybrid strategies ([29; 30]) that showed superior performance in well placement optimization problems. However, this type of technique often lacks theoretical studies to support the proposed combination. Again, when the comparison aspect is considered DE is better than particle swarm optimization (PSO) and CMA-ES [31]. According to Reference [32], gravitational search algorithm (GSA) is superior than genetic algorithm (GA). Again, Hybrid differential evolution and particle swarm optimization (HPSDE) is compared with DE and PSO and found to outperform DE and PSO [33]. In addition, PSO provided more net present value than GA in another study [34]. Again, ACO-GA-PSO have shown superior performance than PSO, GA, S-PSO, and RS [25]. Furthermore, BA has provided better net present value than GA and PSO [24]. The performance of ICA is compared with SPSO and SCGA and the tuned ICA has outperformed other algorithms [26]. The above-stated comparison between algorithms does not establish a single algorithm as a superior algorithm. Moreover, the compared algorithms are not consistent, and a wide set of existing algorithms is not considered in the previous study. Again, because of the reservoir heterogeneities [35; 36], the nature of the cost functions is discontinuous, high dimensional, and nonconvex, which contains multiple local optima. In addition to that, due to the excessive amount of computational expenses, parameter tuning requires a substantial amount of reservoir simulations. Hence, the trial and error approach to find the best configuration is impractical. This can be attributed to the optimization algorithms erratic performance. Again, because of the lack of study in exploration and exploitation analysis, the selection of proper strategy is difficult Furthermore, in most cases, only one reservoir is utilized to evaluate the performance of the algorithms. Such an approach does not establish the superiority of the algorithm in this field since reservoir heterogeneity changes the surface of the search field. Moreover, the result of the techniques can differ with respect to the nonlinearity, dimensionality, and surface of the search field. Previous research shows that only a few algorithms are used for comparison purposes [37]. Hence, a wide range of algorithms should be evaluated in a common synthetic reservoir to claim the superiority of a particular algorithm. Furthermore, metaheuristic algorithms are stochastic. So, statistical analysis and certain criteria should be considered to establish a technique in this field. In addition, the large search space having high dimensionality is rarely reliable for stochastic algorithms. Therefore, perceiving an efficient algorithm with an inherent ability to execute this complex optimization problem is profoundly significant. To the best of our knowledge, there have been no researches on the application of quantum computation to deal with well placement optimization problem. Therefore, in this study, to resolve the problems mentioned before, quantum-behaved bat algorithm (QBA) and quantum particle swarm optimization (QPSO) are implemented and compared with a wide range of established algorithms. Moreover, collecting the end result in terms of cost function value does not help to fully understand the internal behavior of the algorithms. Hence, this study attempts to explain the internal behavior of the algorithms on well placement optimization with graphical illustrations. This paper is ordered in the following manner: Section 2 is dedicated to discuss the problem statement of well placement optimization. In Section 3, we discuss the proposed techniques in detail. The comparison of different optimization methods and their analysis are provided in Section 4. Section 4 focused on the statistical and convergence analysis of different metaheuristic algorithm's results as well as critical findings with discussion. This section also focuses on dimension-wise diversity analysis.