**8. Results**

To test the correct VRP model developed in QUBO, which minimises the maximum distance that all the vehicles travel, we will present some comparisons of the results obtained through the simulator of the different models that have been discussed in this paper.

The code has been implemented on the Ocean library [49] from D-Wave in python. The reader can find the code at [50].

Figure 2 offers a sample of our GPS formulation's results when using the D-Wave solver in different scenarios. We highlight some important cases that help us see the good functioning of the algorithm.

Figure 3 offers a sample of our VRP formulation's results based on the GPS when using the D-Wave solver in different scenarios. We highlight some important cases that help us see the good functioning of the algorithm. It is important to note that our algorithm minimizes the maximum distance travelled by all the vehicles (this is equivalent to reducing the time spanned by all cars). It is worth mentioning that the number of the qubits needed in the case *N* = 8 and *Q* = 3 is 1778. Where *N* is the number of cities and *Q*, the vehicles. In the discussion section, we will analyze this point and its impact on the topology of the QPU architecture and in this case of the D-Wave.

Let us observe in Tables 1–5 the comparison of the number of qubits, time during which the D-Wave Quantum Annealing simulator has been executed, and the length of the path found. The sign "-" represents that the algorithm did not find a possible way during the elapsed time (in minutes). In this examples, the cities which form the TSP to solve are the vertex of the regular polygon with these number of vertex.

**Figure 2.** In these graphs, we can observe the algorithm's results in different scenarios of the GPS formulation. We can follow the correct scalability of the algorithm. We provide the code [50] to check its proper functioning and to allow others to simulate lower values or values higher than *N* = 16.

**Figure 3.** In these graphs, we can observe the algorithm's results in different scenarios of the VRP formulation. We can follow the correct scalability of the algorithm. We provide the code [50] to verify the proper functioning of the formulation. Vehicle number 1 is red, and the next is light-steel-blue. While the depot is the 0 node in pale-green colour, and the rest are represented in light-steel-blue. In this case, we have variables cities from 4 to 12 and using up to 2 vehicles. It is important to highlight that this VRP minimises the time travelled by the cars. The number of qubits used is 2418 to test the last case.

**Table 1.** A regular polygon layout has been taken where the cities occupy the positions of the nodes [50] for the elaboration of all tables. In this scenario of 4 cities, we set comparison with the 3 models, MTZ, native TSP and GPS. The comparison is based on the number of times to find the solution, the distance travelled, and the number of qubits. We can appreciate the good performance of our GPS model, and above all the savings it offers us in the number of qubits.


**Table 2.** A regular polygon layout has been taken where the cities occupy the positions of the nodes [50] for the elaboration of all tables. In this scenario of 6 cities, we set comparison with the 3 models, MTZ, native TSP and GPS. The comparison is based on the number of times to find the solution, the distance travelled, and the number of qubits. We can appreciate the good performance of our GPS model, and above all the savings it offers us in the number of qubits.


**Table 3.** A regular polygon layout has been taken where the cities occupy the positions of the nodes [50] for the elaboration of all tables. In this scenario of 8 cities, we set comparison with the 3 models, MTZ, native TSP and GPS. The comparison is based on the number of times to find the solution, the distance travelled, and the number of qubits. We can appreciate the good performance of our GPS model, and above all the savings it offers us in the number of qubits.


**Table 4.** A regular polygon layout has been taken where the cities occupy the positions of the nodes [50] for the elaboration of all tables. In this scenario of 10 cities, we set comparison with the 3 models, MTZ, native TSP and GPS. The comparison is based on the number of times to find the solution, the distance travelled, and the number of qubits. We can appreciate the good performance of our GPS model, and above all the savings it offers us in the number of qubits.


**Table 5.** A regular polygon layout has been taken where the cities occupy the positions of the nodes [50] for the elaboration of all tables. In this scenario of 12 cities, we set comparison with the 3 models, MTZ, native TSP and GPS. The comparison is based on the number of times to find the solution, the distance travelled, and the number of qubits.


These results have been obtained using a simulator because we would require access to a quantum computer for a time similar to that needed to perform the simulations (in some cases more than an hour). However, it is the benefits of modelling with few qubits (such as GPS modelling) that will be much more notable when these problems are implemented on real quantum computers. Other studies that did not require many hours of the quantum computer were carried out on the D-Wave\_2000Q\_6. In the discussion section, we detail some interesting cases.
