# Quantum Computing Approaches for Mission Covering Optimization
## 6 Summary and Conclusions  
In this paper, we introduced Mission Covering Optimization (MCO), implemented three different constrained optimization techniques (QA, QAOA, and QAOAH) to find solutions to two scenarios of MCO, and discussed results after running implementations on the IBM machine, D-Wave Machine, and on a state vector simulator. Results were compared based on timing, relative cost, qubits used, and constraints violated. From the 200 tests performed on each scenario, QA achieved the quickest results while using the least number of qubits and violating the least number of constraints. We conjecture that QAOA and QAOAH approaches may have taken longer for the gradient descent algorithm to be convinced that an optimal solution was found because of the abundance of noise in current hardware. It was found that it is nontrivial to engineer multiple constraints by embedding in the mixing-Hamiltonian, especially when compared to the ease of using Lagrange multipliers, in simulation, where adding constraints entails simply adding terms together. The study conducted here suggests that the additional complexity in the QAOAH approach poses potential scalability challenges (due to the additional qubits required to ID the constraints) for problems similar to MCO with multiple types of unique constraints. The results of the work presented here illustrate the fact that at the current nascent stage of quantum computing, one is still coding to the quantum computing paradigm; here, Adiabatic Quantum Computing Model (AQCM) (as instantiated by the D-Wave machine), and the Gate-Based Model (GBM) (as instantiated by the IBMQ). Even at the ideal simulation stage of this study, there are substantial differences in the resources required for implementing constraints under the AQCM and the GBM, with the latter requiring many more ancilla qubits. In the AQCM, mixing occurs naturally through the adiabatic evolution, while in the GBM, mixing has to be explicitly implemented, which for the model we investigated, entailed the construction of mixers that would not violate the constraints. In simulation, this construction worked seamlessly; however, in practice, it was limited by the actual noise in the qubit-based implemented quantum circuits involved. This is a well-recognized limitation of today's current noisy intermediate scale quantum (NISQ) hardware. Lastly, while not directly explored in this study, the network connectivity plays an important role in contributing to the noise (resulting in constraint violation) under each paradigm. It is well-known that under the AQCM, the implementation of constraints requires the 'chaining' together of many individual qubits in order to act as an effective  
Figure 9: QPU and Simulated Results **(a)** Average relative cost versus the number of qubits used. The relative cost is the mission cost obtained minus the lowest possible mission cost that can be achieved (while keeping constraints) for the problem. The lowest possible cost is found by brute force. Negative relative costs must mean that the solution violated constraints. **(b)** Average number of constraints of each method versus the number of qubits used. The number of constraints counts up how many extra qubits are within a column in the matrix view representation plus the absolute difference of qubits within a row between resource sets (\(\mathbf{R}_{1}\) and \(\mathbf{R}_{2}\)). This count includes the number of columns that do not have any qubits on at all. single qubit. In practice, the length of chain is finite (typically 10-20 qubits) and 'breakage' of the chain, due to local environmental decoherence, acts as another important source of circuit noise and degradation. In the GBM, extra ancilla are required to implement constraints, increasing the circuit depth of the specific problem under study, which is then subject to local environmental decoherence effects.