# Quantum Computing Approaches for Mission Covering Optimization
## 5 Analyses of Results
### Scenario 1  
Figure 6 shows the timing averages for QA, QAOA, and QAOAH, respectively, versus the problem qubit size.  
Because the timing differed by different levels of magnitude, three y-axes with different scales are displayed. QA run time outperforms the other methods by having an overall constant run-time of 0.012 s, while QAOAH can use the QPU for 5 h across all jobs in certain worse-case instances (this excludes queue-time and creation time on QPU for IBMQ devices). For all methods, timing was calculated based on qpu time (not wall-clock time). For the QA method, 'qpu_sampling_time' was used to calculate the total qpu time, while 'running time' is used for QAOA and QAOAH.  
Figure 7a shows the average costs for all MCO configurations versus the number of qubits it took to encode each scenario 1 run. These costs do not include the cost accumulated from the embedded constraint functions Equations (9) and (24) as these are used to make the constraints hold. Furthermore, all the costs plotted are actually the cost for that particular run minus the best possible cost it can receive. This best possible cost is computed by using  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Results**} & \multicolumn{2}{c}{**Execution Environment**} & \multicolumn{2}{c}{**Average Violated Constraints**} \\
**Machine** & **D-Wave** & \multicolumn{2}{c}{**IBMQ**} & \multicolumn{2}{c}{**D-Wave**} & \multicolumn{2}{c}{**IBMQ**} \\ \hline
**Algorithm** & **QA** & **QAOA** & **QAOAH** & **QA** & **QAOA** & **QAOAH** \\ \hline S1 Hardware & ✓ & ✓ & ✓ & 0.05 & 0.97 & 3.89 \\ S1 Simulation & - & - & - & - & - & - \\ S2 Hardware & ✓ & ✓ & - & 0.0 & 0.96 & - \\ S2 Simulation & - & - & ✓ & - & - & 0.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: S1 and S2 stands for Scenario 1 and 2, respectively. MCO implementations in hardware and simulation execution environment. And number of constraints violated in the various cases.  
Figure 6: QPU Results QPU Times for Scenario 1 Runs. QA timing includes the anneal time of each 50 samples, 20 microseconds per sample. QAOA and QAOAH varied in the number of jobs that ran, each of which ran the parameterized circuit 1000 times.  
brute force search methods in simulation. This difference is referred to as the relative cost. Therefore, the best possible relative cost a run could have is zero. For most runs, it can be seen that QA and QAOA runs have a relative cost of around zero, but as the number of qubits increase, QAOA becomes less optimal compared to QA. The QAOAH approach, when using a large amount of qubits, actually had a negative relative cost indicating that it must have violated some constraints in order to achieve a cost below the solution found with brute-force.  
In Figure 7b, the average number of constraints violated is plotted against qubit size. The number of constraints is calculated by counting the number of resources that were assigned to more than one mission. For example, if resource 1 was assigned to two extra missions, and resource 2 was assigned to three extra missions, then the number of constraints violated is five. QA mostly had no constraints violated at any sized qubit problem, while QAOA and QAOAH suffered constraint violations when the problem size increased.