*Discussion*

Once the different models had been implemented, we achieved the following results. Through the results of the Figures 4–7 and Tables 1–5, the good performance of our formulation compared to the general TSP [41] can be observed. An almost identical operation is seen with the generic TSP, except that we are improving at least the number of qubits for the same cases in our proposal. Although the time difference is not significant again, the difference between path lengths is. Let us remember that the advantage of the formulation in which we have worked is based on improving the number of qubits used. We then have that the larger the problems we are working on, the better this difference will be appreciated in the number of variables.

The MTZ model does not offer positive results. This is since *Annealing* presents many difficulties to find minimum expressions in which the representation of integers appears in their binary format. This is because although the numbers 2*k* − 1 and 2*k* are close, they are not close in their binary form since they differ in *k* variables, so the *annealing* tends to present bad results. Apart from that adjusting, the Lagrange coefficient of this type of constraint is also a complicated task.

Native TSP and GPS modelling show better results. While it is true that general modelling gives slightly better results, it requires the use of a higher number of qubits. This may be since the function to be optimised for this model has a smaller number of local minima where the *Annealing* can ge<sup>t</sup> stuck or there can be a bad of the Lagrange coefficients.

The problem on which the simulations are carried out consists in finding the optimal path when the points are placed on the vertices of the regular polygons that have the same number of vertices as nodes in our problem.

**Figure 4.** Path length comparison for *N* = 9. In this graph, we see how the length of the solution paths for the case of 9 Cities is very similar so that both models give good results.

**Figure 5.** Time comparison for *N* = 9. This graph shows the time taken to carry out the executions in the case of 9 cities. Although it seems that there is a lot of difference, it only represents 10% of the total time, which, as we have seen in other experiences, is not significant.

**Figure 6.** Path length comparison for *N* = 11. For the example of 11 cities, we can observe that the outcomes are quite similar. Although the time difference is not significant again, the difference between path lengths is. Let us remember that the advantage of the modelling we have worked is based on improving the number of qubits used. We then have that the larger the problems we are working on, the better that difference will be appreciated in the number of variables.

**Figure 7.** Time comparison for *N* = 11. For the example of 11 cities, we can observe that the outcomes are quite similar because although there is a mean difference of about 20 s between the results of both simulations, the experience with this problem and other similar ones is that this very small difference does not affect the results on the length of the solution path.

One of the behaviours and results that we believe is important to mention is the following. We realized that it is even more important to consider the number of edges that our model generates. The vertex/connections in a quantum computer are limited and define our quantum computer's typology and quality for error mitigation. Thus, a model that produces many edges (direct links) may request more from a computer than another which generates fewer. The Figure 8 offers us a comparative study between our GPS model and the native TSP. This figure shows the exponential behaviour and the number of interconnections that each model offers. Our model improves the number of qubits and gives us a grea<sup>t</sup> result reducing the number of connections a lot. The native TSP behaves as 0.8(*N* + 2)<sup>5</sup> while the GPS as 2(*N* + 2)3.

One aspect of GPS worth commenting on here is to generalize it also to be used for the Cutting-plane method. We must change the current constraint (13) since this methodology only works with linear constraints. The way to do this is as follows. For each *i*, *j*, *k*:


In these equations, the variables *<sup>w</sup>pi*,*j*,*<sup>k</sup>* are auxiliaries. The purpose of these variables is to satisfy the said constrains. These two restrictions are satisfied by all cases of (*xj*,*i*,2, *xk*,*j*,2, *xk*,*i*,<sup>2</sup>) except for (0, 0, 1) (because it doesn't satisfy the second constraint) and (1, 1, 0) (because it doesn't satisfy the first constraint).

**Figure 8.** In this figure we can appreciate the exponential behaviour and the number of interconnections that each model offers. Our model (GPS) improves the number of qubits and gives us a grea<sup>t</sup> result reducing the number of connections a lot. The Native\_TSP behaves as 0.8(*N* + 2)<sup>5</sup> while the GPS as 2(*N* + 2)3.

#### **9. Conclusions and Further Work**

The importance of finding a good formulation in the QUBO model that minimises the number of variables to be used is crucial for the computing era we are in, as we have commented throughout this work. It is true that, although the technology of annealingbased quantum computers allows us to have much more qubits than gate-based computers, it remains a limitation and, therefore, a challenge to try to solve. Hence highlighting the importance of our research.

With this work, we offer a new formulation for the TSP called GPS and apply it to find an optimal formulation for the VRP that minimises the time the vehicles make their journey. We have also seen that the results of the D-Wave simulator solver are consistent with the expected solution. However, we consider it unnecessary to test it in gate-based quantum computers, given their limitations today in the number of qubits. Still, we emphasise that our current formulation is valid for such computers. The improvement in our models represents a fairly significant order of magnitude because we went from *N*<sup>3</sup> variables to 3*N*2. The Figures 9 and 10 summarises the major contribution of this article.

Our GPS formulation and the VRP proposal can help in optimisation problems when we want to reduce the number of variables and therefore reduce the number of qubits quite a bit. In addition, it is interesting in situations, such as the one raised in the future line of the article [14], by modelling some biological activities on selected sets of organic compounds as can be seen in [51], or resource optimization problems such as gasoline and aircraft travel. Another interesting application could be to compare GPS with the approach offered by this reference [52] using deep reinforcement learning to address combinatorial optimisation problems with feasibility constraints. This leads us to project on how to make this comparison in quantum computing using the proposal made in this reference [53].

**Figure 9.** Comparison of the different models based on the number of qubits. This graph shows the behaviour and evolution of the numbers of qubits for each model. We see the best performance of our GPS model compared to the other models.

**Figure 10.** Benchmark between MTZ and GPS model based on the number of qubits. We can appreciate that for 30 cities, GPS model needs 2700 qubits while the MTZ 4458.

The results obtained from our VRP formulation and all the experiments carried out maintain the number of variables *QN*<sup>2</sup> and allow us to offer the community new formulations that minimise the time it takes for vehicles to travel.

Future work will apply the ideas developed in the QUBO model of these problems to similar ones. In particular, we will look for other variants of the TSP to use the modelling of this that we have carried out.

**Author Contributions:** Conceptualization, P.A.-A.; Methodology, P.A.-A. and S.G.-B. and G.A.- L.; Software, S.G.-B. and P.A.-A. and G.A.-L.; Validation, P.A.-A. and S.G.-B. and G.A.-L.; Formal Analysis, S.G.-B. and P.A.-A. and G.A.-L.; Investigation, S.G.-B. and P.A.-A.; Resources, P.A.-A.; Data Curation, P.A.-A.; Writing—Original Draft Preparation, S.G.-B. and P.A.-A.; Writing—Review and Editing, P.A.-A. and G.A.-L. and S.G.-B.; Visualization, P.A.-A. and G.A.-L. and S.G.-B.; Supervision, P.A.-A.; Project Administration, P.A.-A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** This article does not contain any studies with human or animal subjects.

**Informed Consent Statement:** Informed consent was obtained from all individual participants included in the study.

**Data Availability Statement:** Data sharing not applicable. No new dataset were created or analyzed in this study. Data sharing is not applicable to this article.

**Conflicts of Interest:** The authors declare no conflict of interest.
