# Cutting-Edge Trajectory Optimization through Quantum Annealing
## 6 Conclusions  
This methodology leverages QA through a QUBO Formulation, translating trajectory optimization problems into the binary optimization domain. The proposed procedure presents a general approach that can be applied to optimization problems with linear dynamics. Its strength lies in its versatility, making it suitable for a wide range of scenarios within the realm of linear applications. Although it is designed for linear systems, its adaptability to different problems is a noteworthy advantage. Future research endeavors might focus on developing methods to tackle optimization challenges posed by nonlinear systems. Moreover, while the applications explored in this study may appear straightforward, a notable advantage in computational time is already discernible with the QA. This early advantage lays the groundwork for a deeper exploration of the potential harbored by this novel technology. The significance of the computational time disparity is indeed striking. The QA demonstrates a remarkable level of computational efficiency, evident through its computational time, which is orders of magnitude smaller than that of IP and PSO methods. This discrepancy in computation time may be a very important finding that could highlight an innate advantage of quantum computing in its ability to efficiently address optimization challenges. By exploiting the principles of quantum mechanics, QA is able to explore solution spaces in a parallel and probabilistic manner, enabling it to navigate through a vast number of potential solutions more quickly. In contrast, classical optimization methods often rely on iterative and sequential procedures that require greater computational efforts. Nevertheless, it is worth noting that due to the polynomial-based approximation of the system state, an optimally exact solution might not be achieved. This approach strikes a  
Figure 10: Comparison of Monte Carlo results for Rendezvous application between QA (blue), PSO (red), and IP methods (green) in terms of computation time and cost function. balance between optimizing for fuel consumption and computational efficiency. Opting for higher-degree polynomials could offer more flexibility for optimization, potentially yielding more accurate outcomes. However, this decision introduces a trade-off by potentially prolonging computational time, a factor undesirable in applications requiring swift computations. Hence, the selected approach reflects a compromise between result optimality and computational efficiency. Another limitation is that the flight time, \(\Delta t_{f}\), has to be fixed. This is, again, due to the need for reducing the optimization into a QUBO form. Indeed, if \(t_{f}\) were a variable, it would introduce a multiplication factor with \(\alpha\) in Equation (10). Consequently, non-linear terms would emerge, potentially reaching higher than second-degree terms within the cost function. This intricacy would render the transcription into a QUBO problem infeasible. Given these encouraging outcomes and the numerous achievements in recent times, quantum computation has undeniably demonstrated its potential. Future research could enhance the methodology by exploring advanced trajectory approximation techniques to achieve greater precision while maintaining computational efficiency, such as B-Spline or Neural Network. In addition, the development of methods to handle nonlinear systems and to include time as a problem variable could expand the applicability of the methodology. Lastly, it might also be of interest to investigate the use of hybrid optimizers that combine quantum and classical optimization techniques, which could provide better solutions. Conceptualization, A.C. and F.D.G. ; methodology, A.C. and F.D.G. ; software, A.C. and F.D.G. ; validation, A.C.; formal analysis, A.C.; investigation, A.C.; resources, D.S. ; data curation, A.C.; writing--original draft preparation, A.C. and F.D.G. ; writing--review and editing, A.C., F.D.G. and D.S. ; visualization, A.C.; supervision, D.S.