# Cutting-Edge Trajectory Optimization through Quantum Annealing
## 1 Introduction
During QA, the system gradually transitions from an initial state to a low-energy state, similar to the cooling process in metallurgy, hence the term \"annealing\". QA promises more efficient optimization solutions, especially for large-scale and complex problems [24; 25]. In particular, current quantum annealers can handle computations with about an order of magnitude higher number of qubits than gate-based quantum machines. Proposed applications of QA include factorization of prime numbers [26], systems of polynomial equations [27], and differential equations [28; 29]. The research is also exploring the application of quality control to practical engineering challenges, despite the limitations of current quantum annealers. In aerospace engineering, some applications have been proposed, such as addressing some artificial intelligence problems framed as combinatorial tasks [30], optimizing aircraft trajectories to prevent conflicts [31] and allowing agile planning of the Earth observation maximization of image acquisition [32]. The goal of this study is to explore the cutting-edge capabilities of QA and its potential applications. Although QA has garnered considerable attention in the optimization community, its potential in the specific context of space trajectory optimization has yet to be fully explored. The paper presents a transcription procedure that converts trajectory optimization problems into equivalent binary optimization problems using the QUBO representation. This versatile procedure can be applied to a wide range of optimal control problems, encompassing dynamic, boundary, and path constraints. Specifically, the trajectory is discretized and approximated using polynomials for each component. By employing inverse dynamics, the required control to follow the approximated trajectory can be determined. Consequently, the optimization function becomes the system's energy, represented by the integral of the square of the control. As this function depends on the form of the polynomial, the focus lies on optimizing the polynomial itself. This novel approach demonstrates significant potential for efficiently exploring extensive solution spaces, making it particularly promising for addressing complex trajectory design challenges in fields like space exploration and satellite missions. This paper is organized as follows. With the purpose of providing the appropriate context and establishing the necessary nomenclature, Section 2 provides a brief description of the basic principle behind Quantum Annealig. Section 3 presents an overview of the transcription procedure for trajectory optimization, elucidating the mathematical formulation of the problem, including the cost function and various types of constraints. Moving on to Section 4, the lunar landing and rendezvous problems and their simplifications are reported. The results of the transcription procedure for this specific problem are discussed in Section 5, with an emphasis on specific considerations and limitations. Finally, in Section 6, conclusions and potential future research directions in the field of QA for trajectory optimization are presented.