# Cutting-Edge Trajectory Optimization through Quantum Annealing
## 1 Introduction  
Trajectory optimization plays a pivotal role across various domains, including space exploration and satellite missions, where finding the most efficient trajectories is essential for mission success, fuel conservation, and resource optimization. Recent advancements in optimization techniques have been propelled by emerging artificial intelligence technologies, notably Genetic Algorithms (GA) and Neural Networks (NN). In the field of satellite constellations, [PERSON] et al. employed GA in [1] to optimize trajectories, maximizing coverage while minimizing revisit time. [PERSON] et al. in [2] utilized GA to optimize orbital trajectories for spacecraft swarms, facilitating collaborative tasks in space and collision avoidance. Particle Swarm Optimization demonstrated its progress in [3], addressing real-time guidance for autonomous lunar landings and hazard avoidance, with promising experimental results. In [4], Particle Swarm Optimization was exploited to maximize asteroid surface coverage using hopping trajectories in low-gravity environments. Application to asteroids Itokawa and Bennu showed significant surface coverage improvement. A comprehensive overview of genetic algorithms was presented in [5], providing insights into established algorithms, their implementations, and associated pros and cons. Neural Network techniques have also made significant contributions. [PERSON] et al. applied reinforcement learning in [6] to autonomously guide a spacecraft during a mission to impact a binary asteroid, utilizing neural networks for optical observation processing and achieving a high success rate despite uncertainties. [PERSON] et al. in [7] leveraged reinforcement meta-learning, combining hazard detection and avoidance with image and radar data to ensure safe landing points. Addressingchallenges in asteroid exploration, [PERSON] et al. in [8] proposed the use of hopping rovers and explored deep reinforcement learning for 3D terrain path planning. [PERSON] et al. in [9] focused on optimizing an asteroid hovering controller through reinforcement meta-learning, enabling stable positioning even with less precise asteroid data. Combination of these methods can yield advantageous outcomes, as seen in [10; 11], where frameworks synergized particle swarm optimization for trajectory optimization and extreme learning machines for precise gravitational acceleration approximation from asteroids. The advancements in this field have unlocked the potential for more advanced and efficient trajectory optimization methods, emphasizing the role of computational efforts in realizing these breakthroughs. As a result, the importance of computer processing power has become paramount in the field of trajectory optimization.Especially the computational limitations have become a significant challenge in handling complex trajectory design problems. Traditional computational resources struggle to handle the massive calculations required for optimal trajectory design, especially for scenarios involving intricate navigation through varying gravitational fields and orbital constraints. Quantum Computing (QC) has demonstrated the ability to outperform classical and more commonly used optimizers in specific cases [12]. Some problems where QC provides an advantage over classical algorithms are factorization of prime numbers [13], unstructured database search [14],linear systems of equations [15], and simulation of quantum systems [16; 17; 18]. Research is active, with many algorithms proposed in many different areas, such as differential equations [19], artificial intelligence and machine learning [20; 21] and optimization [22; 23]. While the aforementioned literature focuses primarily on gate-based quantum computing, which operates by applying a series of gates to manipulate qubits, similar to classical computers, another notable quantum computing model is QA. QA exploits and guides the natural evolution of a quantum system to solve combinatorial optimization problems.