## 5 Computational Experiment  
Having introduced the NOT XOR Hamiltonian, we compare it with standard one-hot encoding with many entanglements of high degree and with quadratic encoding, which uses only second-degree entanglements. We benchmark these three approaches against nine artificially generated instances of the same size: \(n=3\) aircraft with \(m=5\) alternative maneuvers. The difference between these instances lies in the number of potential conflicts \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Number of Maneuvers** & **0** & **1** & **2** & **3** & **4** & **5** \\ \hline \multirow{6}{*}{Bitstring} & & 00001 & 00011 & 00111 & \multirow{6}{*}{0111} & \multirow{6}{*}{0111} \\  & & 00010 & 00101 & 01011 & & \\  & & 00010 & 0110 & 01101 & & \\  & & 00000 & 0100 & 01100 & & \\  & & 01100 & 10101 & & & \\  & & 01000 & 10001 & 10110 & & \\  & & 10000 & 10100 & 11001 & & \\  & & 10000 & 10100 & 11010 & & \\  & & 10000 & 11000 & 11100 & & \\  & & 10000 & 11000 & 11100 & & \\ \hline NOT XOR & 1 & 0 & 1 & 0 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of possible maneuver assignments for one aircraft and the corresponding evaluation of the NOT XOR function. Figure 1: Quantum circuit representations resulting from Hamiltonian evolution for three different encodings of an aircraft (\(n=1\)) with \(m=3\) possible maneuvers. The symbol \(R_{Z}\) represents a parameterized quantum gate rotation around the Z-axis. between maneuvers, resulting in a varying number of feasible solutions to the problem. We choose to test instances with difficulties of 1, 3, 5, 10, 20, 30, 50, 100, and 124 solutions. The reason for choosing such a set of instances is that the first constraint for conflict avoidance (see Equation (7)) already contains the quadratic Pauli-Z terms. Thus, with an increasing number of potential conflicts, the entanglement complexity of the constraint determining the number of maneuvers performed by the aircraft may lose its significance. The test instances are generated using an iterative graph-based algorithm that detects cycles between nodes. Each node in the graph represents an aircraft performing a maneuver, while edges represent possible coexistence between these maneuvers. By removing an edge between two nodes, we introduce a conflict, reducing the number of solutions. Starting from a complete graph with no conflicts, the number of solutions removed corresponds to the number of cycles that include removing an edge. We tested the instances in a noisy simulator environment using the noise model derived from the _ibm_torino_ quantum device. The \(3\times 5\) size of instances was the largest we could compute within a reasonable time frame and required several days of HPC computations across multiple nodes. For the optimization of QAOA parameters, we used the Constrained Optimization BY Linear Approximation (COBYLA) [19] algorithm with the default 1000 iteration threshold. We fixed 10 random initial points, optimized each instance starting from each of these random points, and averaged the results. The results are presented in Figure 2.  
The figure answers two questions: Firstly, is the NOT XOR encoding better in terms of the number of feasible solutions we can find, i.e., does it make the optimization landscape simpler? Secondly, does the NOT XOR encoding introduce less noise to the circuit? One could argue that it would be unfair and biased to report that the NOT XOR solution gives better results when we treat all solutions that have no conflicts and where aircraft perform at least one maneuver as feasible. This is because optimizers with the one-hot and QUBO  
Figure 2: Probabilities of measuring a feasible solution as a function of the number (denoted by \(\#\)) of feasible solutions in the instance and the chosen encoding. The probabilities are evaluated in two ways. First, we sum the probabilities for all correct one-hot-measured bitstrings (blue bars).