# Cutting-Edge Trajectory Optimization through Quantum Annealing
## 5 Results
### Rendezvous Results  
Table 3 provides the parameters for the rendezvous scenario. Notably, the final conditions are uniformly set to zero, given the utilization of equations that pertain to relative motion. It is worth noting that this scenario was taken from the study detailed in [41], where the same polynomial approximation method is used to address a rendezvous problem. This ensures consistency and aligns with the methodology established in the scientific literature. In particular, the radius of the target spacecraft orbit is set to 773 km.  
The outcomes in terms of position, velocity, and acceleration are showcased in Figure 7. Similar to the previous application, the rendezvous scenario successfully achieves the targeted boundary conditions, with all values equating to zero.  
Figures 8 and 9 provide a comparative analysis with trajectories and control laws generated using MATLAB's IP and PSO methods, respectively. Once again, it is evident that the results obtained from QA are highly comparable to the outcomes achieved through IP and PSO methods.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline  & **Position (m)** & **Velocity (m/s)** & **Acceleration (m\({}^{2}\)/m)** \\ \hline Initial Condition & [\(-10\), \(-45\), \(3\)] & [\(0\), \(0\), \(0\)] & Unconstrained \\ Final Condition & [\(0\), \(0\), \(0\)] & [\(0\), \(0\), \(0\)] & [\(0\), \(0\), \(0\)] \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters to set the Rendezvous problem.  
Figure 6: Comparison of Monte Carlo results for Lunar Landing application between QA (blue), PSO (red), and IP methods (green) in terms of computation time and cost function.  
Additionally, a Monte Carlo simulation is conducted using 10,000 samples to thoroughly assess QA's performance in comparison to that of IP and PSO methods. The Monte Carlo comparison results, presented in Figure 10, encompass computational time and cost function optimization. Specifically, the upper row presents the comparison between QA and PSO, while the bottom row provides the comparison between QA and the IP method. Once again, the average values of the cost function are remarkably consistent across all three methods, with QA yielding approximately \(-0.370\), IP yielding \(-0.361\), and PSO yielding \(-0.361\).  
Figure 8: Comparison of trajectory and control law for Rendezvous application between Quantum Annealing (blue line) and Particle Swarm Optimization (red line).  
Figure 7: Solution of Rendezvous by QPU.  
Figure 9: Comparison of trajectory and control law for Rendezvous application between Quantum Annealing (blue line) and Interior Point (green line).  
Furthermore, there is notably less data dispersion in the results obtained using the QA method, indicating a higher degree of determinism compared to the other two approaches. Moreover, QA demonstrates an impressively low computational time of 0.023 s, even when applied in the context of the Rendezvous scenario. This computational efficiency is notably superior, being an order of magnitude lesser than that of the IP and PSO methods, which require approximately 0.141 and 0.226 s, respectively.  
Upon analyzing the results presented across different scenarios, a consistent trend emerges. While the average cost function values for QA and fmincon demonstrate comparability, a remarkable supremacy of QA in terms of computational efficiency emerges.