# Indoor-Outdoor Detection in Mobile Networks Using Quantum Machine Learning Approaches
## 4 Quantum Machine Learning Approaches
### Quantum Annealing-Based SVM
First, we investigate what the biggest problem is that can be embedded on each of the available machines, without using decomposition techniques. For the Chimera architecture, this means using 25 data points for learning and 8 for validation. For the Pegasus architecture, this means 70 data points for learning and 23 for validation. The QPU uses less than 10 ms of annealing time for solving this problem. The results for these problems can be found in Figures 13 and 14. For reference, the result from the  
Figure 12: Resulting assignments of the simulated annealing-based SVM, 225 datapoints. If we do want to use decomposition techniques, the D-Wave system employs built-in routines to decompose the problem into smaller sub-problems that are sent to the QPU, and in the end reconstructs the complete solution vector from all sub-sample solutions. Here, we use D-Wave's Hybrid Solver, which implements state-of-the-art classical algorithms with intelligent allocation of the QPU to parts of the problem where it benefits most, i.e., the sampler uses the energy impact as the criterion for the selection of variables. These solvers are designed to accommodate even very large problems. This means that most parameters of the embedding are set automatically. By default, samples are iterated over four parallel solvers. The top branch implements a classical Tabu Search that runs on the entire problem until interrupted by another branch completing. The other three branches use different decomposers to sample out parts of the current sample set and send it to different samplers. HQPU acts as a black box solver and the specific part of the problem which gets embedded on the QPU is unknown. The hybrid solver can solve the problem using 225 data points and 75 validation points, and is faster than simulated annealing, with approximately the same result (see Figure 17). We show the results using the performance indicators Accuracy, \(F1\)-score, Precision and Recall as our KPIs again. Lastly, we present the energy of every solution, which is the score generated by the objective function in Equation (15). To give some idea about the optimal values, we also use the commercially available solver LocalSolver (www.localsolver.com, accessed on 10 May 2021), to solve both the original problem expressed by Equations (11)-(14) and the QUBO problem expressed by Equation (15). For all problems, a solution is found quickly, however, after 60 s, optimality gaps of 40-99% still  
Figure 16: Resulting assignments of the simulated annealing-based SVM, 70 datapoints. Figure 17: Resulting assignments of the Hybrid Solver-based SVM, 225 data points. remain. To make the comparison somewhat fair, we also give the Hybrid Solver one minute of calculation time, Simulated Annealing 4000 reads (resulting in one minute of calculation time) and the QPU 4000 reads as well (around 80 ms of calculation time). The scores for the three cases, using 25, 70 and 225 data points for training, are depicted in Table 3. We see that for almost all cases, the performance is good, all validation data points are labelled correctly. Only in the \(N=75\) case is there one point that is not labelled correctly in the case of the simulated annealing and the direct use of the QPU. The Hybrid Solver does label this point correctly. The main difference is found in the energy level, meaning the objective value. The best solution for all cases is found by LocalSolver, solving the original problem. The Hybrid Solver is the best performing approach of all quantum (inspired) approaches in all cases. The direct implementation on the QPU performs worst, which can be seen in the heat maps, where the direct QPU solution has a large area where the score is equal to one. LocalSolver also has problems solving the QUBO problem, probably due to the lack of structure of the problem.