# Quantum Computing Approaches for Mission Covering Optimization
## 5 Analyses of Results
### Scenario 2  
For scenario 2, the QAOAH method was calculated via IBM's state-vector numerical simulation tool due to the long running-time of runs in scenario 1. Because of this shift from running on actual hardware (scenario 1), to simulation (scenario 2), the QAOAH method is plotted using its wall-clock times against QA's and QAOA's QPU time in Figure 8. Even with this change, the magnitudes of running-time are very diverse, so a third y-axis is added, as before. Furthermore, for the QAOAH approach, the mixing operator in Equation (33) is used instead of Equation (27) because removing the column-swap terms reduces the total gate count of the overall computation, making simulation times faster.  
In scenario 2, QA times are quite faster than QAOA methods. It can be seen that the QAOAH method has far less flexibility in terms of qubit-range. This is because in our MCO algorithm implementation, extra qubits are required to represent each resource's ID. To keep consistency with the QA and QAOA methods, the x-axis in each plot in this section represents the number of qubits used minus the amount used to represent the IDs.  
Figure 7: QPU Results (**a**) Average relative cost versus the number of qubits used. The relative cost is the mission plus the precedence cost obtained minus the lowest possible mission plus precedence cost that can be achieved (while keeping constraints) for the problem; the lowest possible cost is found by brute force: Relative cost = algorithm’s solution cost–brute force solution’s cost. Negative relative costs imply that the solution violated constraints. (**b**) Average number of constraints violated versus the number of qubits used. The number of constraints counts up how many extras qubit are on within a column in the matrix view representation. This count includes the number of columns that do not have any qubits on it at all.  
As in scenario 1, the relative costs for each method are plotted in Figure 9a. Both QAOAH and QA methods have zero relative costs. However, QAOA by itself did not exhibit a positive relative cost. For both cases in this subplot, the data are insufficient to deduce whether or not these violated constraints.  
In Figure 9b, the average number of constraints violated is plotted against the number of qubits the problem encodes. The number of constraints violated is calculated similarly to scenario 1, but now including all violations of the second constraint. For example, if four resources of type-1 and 2 of type-2 were allocated to a mission, then the number of constraints violated is \(4-2=2\). Figure 9b shows that QA and QAOAH did not violate constraints at any qubit size, while QAOA did on average. QAOA violated constraints mostly likely because it found a solution where the mission cost exceeded the cost incurred by the constraint function. QAOAH in simulation, however, did not violate constraints because the mixing operator that was used transforms solutions without leaving the constrained space. These results contrast with the QPU runs for QAOAH in Scenario 1 where it did violate constraints. Since the simulation ran without any noise profiles, it is expected that QAOAH should not violate constraints in theory, but this is not the case when running on the actual quantum machine. The source of noise on actual hardware is most likely due to the gate noise and noise from measurement.  
Figure 8: QPU and Simulated Results QA and QAOA QPU times and simulated QAOAH wall-clock times for Scenario 2. QA timing includes the anneal time of each 50 samples, each 20 microseconds per sample. QAOA ran with 1000 shots. QAOAH ran an undetermined amount of jobs using the statevector-simulator to obtain results. Unlike Scenario 1, the times recorded are total wall-clock times for QAOAH.