## 4 Exemplary Optimization Problem  
In this paper, we consider the tactical aircraft deconfriction problem; however, the following approach can also be used to deal with similar constraints in other problems, such as the job shop scheduling problem [16]. The tactical deconfriction problem involves predicting and resolving conflicts between aircraft in airspace from 5 to 30 min into the future. A conflict is defined as a violation of the safety cylinder of an aircraft by another aircraft. Most approaches to solving this problem involve mixed-integer linear/non-linear programming [17], but there are also some recent quantum approaches. We base our work on the approach described by [PERSON] et al. [18], who first formulated the quantum approach for this problem. For a detailed explanation of the approach, we refer the reader to the original paper. Following this approach, for each of \(n\) aircraft, we propose \(m\) maneuvers and define a set of \(n\times m\) binary variables as follows:  
\[X=\{x_{ij}:\ i=1,\ldots,n,\ j=1,\ldots,m,\ x_{ij}\in\{0,1\}\}, \tag{6}\]  
where the variable \(x_{ij}\) taking the value 1 means that aircraft \(i\) is assigned maneuver \(j\); it takes the value 0 otherwise. Each variable directly corresponds to a specific qubit in a quantum environment. From this set of variables, we form two types of constraints. The first constraint ensures there are no conflicts, while the second constraint ensures that an aircraft performs one and only one maneuver. The Hamiltonian representing the no-conflict constraint is originally written as:  
\[H_{1}=\sum_{i,j,i^{\prime},i^{\prime},j^{\prime}\text{CM}(i,j,i^{\prime},j^{ \prime})=1}H_{\text{and}}(x_{ij},x_{i^{\prime}j^{\prime}}). \tag{7}\]  
where the CM matrix is defined as in [18]: \(H_{\text{and}}(x_{ij},x_{j^{\prime}j^{\prime}})=\frac{1}{4}I-\frac{1}{4}(Z_{ ij}+Z_{i^{\prime}j^{\prime}}-Z_{ij}Z_{i^{\prime}j^{\prime}})\). The Hamiltonian for the constraint that ensures an aircraft can perform one and only one maneuver is originally formulated as:  
\[H_{2}=\sum_{i=1}^{n}I-\sum_{j=1}^{m}\left(H_{x}(x_{ij})\prod_{ \begin{subarray}{ c}j=1\\ j\
eq j\end{subarray} }^{m}H_{\text{not}}(x_{ij^{\prime}})\right), \tag{8}\]  
where \(H_{\text{not}}(x_{ij^{\prime}})=\frac{1}{2}(I+Z_{ij^{\prime}})\) and \(H_{x}(x_{ij})=\frac{1}{2}(I-Z_{ij})\). If these two types of constraints are satisfied, it means that aircraft are assigned conflict-free trajectories and each aircraft performs only one maneuver. The deconfiction problem that is formulated this way does not have an optimization function, so a solution satisfying these two types of constraints is a correct, feasible solution to the problem. In this work, we focus on improving the second constraint. Note that due to the products in Equation (8), the number of summands containing Pauli-Z terms is \(n2^{m}\), with most necessitating entanglements of degree higher than two. This, in turn, can make it challenging to find the ground state of the final Hamiltonian and can make the quantum circuit prone to errors in noisy quantum environments. Classical optimization of hard problems almost always benefits from reducing the search space. Setting up more constraints while simultaneously keeping the number of feasible solutions constant prevents classical algorithms from wasting their computing cycles on exploring unprofitable dead ends that do not yield valuable solutions. In quantum computing, however, qubits can be put into superposition, allowing operations on many states simultaneously. This enables virtually cost-free computation on unfeasible solutions without adding any additional overhead. Considering this, we note that the one-hot constraint for the tactical aircraft deconfiction problem is superfluous when using quantum optimization algorithms, and removing it can lead to significant performance improvements. Counterintuitively, let us replace the constraint that an aircraft can perform one and only one maneuver with the constraint that an aircraft must perform at least one maneuver.