# Cutting-Edge Trajectory Optimization through Quantum Annealing
## 5 Results
### Lunar Landing Results  
In Table 2, the lunar landing parameters are provided. Specifically, the initial position is selected to uphold the validity of the approximation applied to the dynamics equations, which assumes the lunar surface to be flat. As a result, the spacecraft is situated at an altitude of 3 km. This initial configuration is based on a previously published works from 2022 [36; 40], ensuring a consistent and valid reference point for our analysis.  
The QPU results are shown in Figure 3, in terms of position, velocity, and acceleration. The figures readily demonstrate that boundary conditions are fully met; notably, the final position is reached with zero velocity and acceleration. This implies that the QA successfully yields an optimal solution for the problem at hand.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Position (m)** & **Velocity (m/s)** & **Acceleration (m2/m)** \\ \hline Initial Condition & [0, 0, 3000] & [0, 50, \(-\)10] & Unconstrained \\ Final Condition & [3000, 2000, 0] & [0, 0, 0] & [0, 0, 0] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters to set the Lunar Landing problem.  
Figure 2: Lunar Landing and Rendezvous Cost Functions as a function of Number of Quibit.  
The trajectory profiles and the control law are visually presented in Figures 4 and 5, featuring a comparative analysis with trajectories and control generated using MATLAB's PSO and IP methods, respectively. In both figures, the substantial overlap between trajectories offers compelling evidence of the quality of QA's solutions. Additionally, on the right side of both figures, the alignment of QA's control trajectory with those of the IP and PSO methods is illustrated, further affirming their striking similarity. This robustly demonstrates QA's effectiveness in tackling these specific problems.  
Furthermore, a Monte Carlo simulation is executed employing 10,000 samples to comprehensively evaluate QA's performance in contrast to IP and PSO methods. The Monte Carlo comparison results are visually represented in Figure 6, en  
Figure 4: Comparison of trajectory and control law for Lunar Landing application between Quantum Annealing (blue line) and Particle Swarm Optimization (red line).  
Figure 5: Comparison of trajectory and control law for Lunar Landing application between Quantum Annealing (blue line) and Interior Point (green line).  
Figure 3: Solution of Lunar Landing by QPU.  
time and cost function optimization. In particular, the upper row reports the comparison between QA and PSO, while the bottom row reports the comparison between QA and IP methods. An insightful observation can be made when analyzing the results presented in Figure 6. Although the average value of the cost function is similar to those of QA (approximately \(-362.19\)), the PSO method (approximately \(-362.13\)), and the IP method (approximately \(-362.11\)), the QA method seems to exhibit a slightly better performance. Moreover, there is less data dispersion in QA's results. This indicates a more deterministic outcome compared to those of the other two methods. A significant difference is evident in terms of computational efficiency. Notably, the computational time required for QA (0.015 s) is an order of magnitude smaller than that of PSO (approximately 0.30 s) and IP methods (approximately 0.62 s).