# Cutting-Edge Trajectory Optimization through Quantum Annealing
## 5 Results  
In this section, the empirical results obtained using the D-Wave Advantage 4.1 QPU through the D-Wave cloud service \"Leap\" are presented. This quantum annealer adheres to a topology named Pegasus, featuring a pool of approximately 5000 qubits for computational tasks. Within this topology, each qubit establishes connections with fifteen others, resulting in a total of around 35,000 interconnections. The main objective is to assess the QA's performance in terms of computational time and optimized cost function. This evaluation is carried out through a comparative analysis with established optimization techniques. Specifically, the Quantum Annealer's performance is compared with the Interior-Point (IP) optimization method provided by the \"fmincon\" function in MATLAB R2022b. Additionally, a comparison is made with the genetic algorithm, Particle Swarm Optimization (PSO), which is implemented using the \"particleswarm\" function on MATLAB [39]. Both the IP and the PSO method are implemented on a workstation equipped with 32 GB of RAM, an Intel Core i7-12700H 2.70 GHz CPU from the 12 th Generation, and an NVIDIA GeForce RTX 3080 Ti GPU with 16 GB of dedicated RAM (manufactured by Intel and NVIDIA Corporation, Santa Clara, CA, USA).  
The parameters used to set the simulations are provided in Table 1.  
We note that the variables within the problem being examined are considerably fewer in number than what the physical constraints demand. Indeed, by choosing to use 32 bits to convert the three real components into binary, the number of variables of the problem are kept very low: 3 \(\times\) 32, which amounts to 96 variables, while the non-zero components of the matrix \(Q\) count 1488. Regarding the embedding procedure, it is carried out automatically by the built-in algorithms of D-Wave. The resulting embeddings are easily found and have chain lengths of eight components at maximum. As a result, the  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **Final Time (\(t_{f}\))** & **Step** & **Parameters (\(\mathbf{G}\times\mathbf{n_{b}}\))** & **Upper Limit (\(\mathbf{\alpha_{max}}\))** & **Lower Limit (\(\mathbf{\alpha_{min}}\))** \\ \hline LL & 100 s & 0.1 s & 96 qubit & [10, 10, 10] & \(-\)[10, 10, 10] \\ RV & 320 s & 0.1 s & 96 qubit & [10, 10, 10] & \(-\)[10, 10, 10] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters to set the QUBO problem.  
question of embedding does not arise as a concern and is resolved indirectly through the application of a substantially smaller-sized problem. The decision to limit the qubit count to 32 is not arbitrary but based on a systematic analysis of the cost functions in the context of both Lunar Landing and Rendezvous applications. Indeed, the number of qubits directly impacts the precision of conversion from real to binary numbers, as indicated in Equation (17). The outcomes of this analysis are showcased in Figure 2, where the cost functions are plotted against the base-2 logarithm of the number of qubits (\(log_{2}(n_{b})\)). This visualization simplifies the presentation of results and underscores that a plateau is reached for both applications when \(log_{2}(n_{b})\) equals five, corresponding to \(n_{b}\) = 32. As such, pursuing a greater number of variables proves to be superfluous.  
In addition, a Monte Carlo simulation is carried out for each test case, utilizing 10,000 samples to provide a comprehensive evaluation of QA in comparison to IP and PSO, with a focus on assessing the achieved cost function and computational time. This extensive analysis allows for a thorough understanding of the relative performance of the methods in terms of optimization quality and efficiency.