# Mdpli Mitigation of Routing Congestion on Data Networks: A Quantum Game Theory Approach
## 4 Results
### Pareto Optimality and Nash Equilibrium
#### 4.1.2 Pure Quantum Strategies  
As mentioned earlier, the quantum player's strategy consists of a sequence of \(1\) qubit quantum gates. To demonstrate the potential that quantum strategies have, we are going to start by studying a particular case: the system with rotations in the \(X\), \(Y\) and \(Z\) axes. The one-qubit quantum gate matrices for rotations are:  
\[R_{X}(\varphi)= \begin{pmatrix}\cos(\frac{\varphi}{2})&-i\sin(\frac{\varphi}{ 2})\\ -i\sin(\frac{\varphi}{2})&\cos(\frac{\varphi}{2})\end{pmatrix}  \tag{2}\]  
\[R_{Y}(\varphi)= \begin{pmatrix}\cos(\frac{\varphi}{2})&-\sin(\frac{\varphi}{ 2})\\ \sin(\frac{\varphi}{2})&\cos(\frac{\varphi}{2})\end{pmatrix}  \tag{3}\]  
\[R_{Z}(\varphi)= \begin{pmatrix}e^{-i\frac{\varphi}{2}}&0\\ 0&e^{i\frac{\varphi}{2}}\end{pmatrix}  \tag{4}\]  
Therefore, every player must choose \(3\) angles that we will call \(\varphi_{X}\), \(\varphi_{Y}\) and \(\varphi_{Z}\). In our design, we propose the strategy \(\varphi_{X}=\frac{\pi}{2}\), \(\varphi_{Y}=\frac{\pi}{4}\) and \(\varphi_{Z}=0\) for every player as shown in Figure 5.  
Figure 4: Trade-off between traveling and routing time for different \(p\) values between \(0\) and \(0.9\). Values of \(p\) closer to \(0\) give a high traveling time and low routing time. Values of \(p\) closer to \(1\) give a low traveling time and high routing time. Figure 5: Game model for \(2\) players. In Figure 6a,b the quantum protocol performance when all players are using \(S_{1}=(\frac{7}{2},\frac{7}{4},0)\) is shown against the classical performance. The quantum game shows the lowest traveling time and a medium routing time compared to the classical one. With this in mind we can recalculate Figure 4 and add the quantum case. By doing this we obtain Figure 7 where we can observe how the classical traveling-routing time trade-off barrier is crossed by this quantum protocol. Thus, we will obtain a performance that enhances any classical performance by simultaneously reaching less routing time and less traveling time. Figure 6: Graphs for different probabilities \(p\) and the quantum case: (**a**) Traveling time as a function of the number of packets. (**b**) Routing time as a function of the number of packets. Figure 7: Trade-off barrier broken by quantum protocol. In red the quantum strategy, in blue different mixed classical strategies with \(p\) values between 0 and 0.9. Another way of visualizing the advantage of the quantum protocol is by measuring the total_time = routing_time + traveling_time. In Figure 8, it is clear that when the total time is measured, the performance of the quantum protocol surpass the performance of the classical protocol as the number of packets increases and the network becomes increasingly congested. This advantage may be understood considering the quantum state of the circuit for two players (Figure 5) just before measuring, is: \(|\psi_{out}\rangle=J^{\dagger}(R_{z}(0)\otimes R_{z}(0))(R_{y}(\frac{\pi}{4}) \otimes R_{y}(\frac{\pi}{4}))(R_{x}(\frac{\pi}{2})\otimes R_{x}(\frac{\pi}{2})) J|00\rangle=-j\frac{|01\rangle+|10\rangle}{\sqrt{2}}\). That is, state \(|01\rangle\) will be measured with a 50% probability and state \(|10\rangle\) with a 50% probability. This strategy turns out to be Pareto optimal since no player can improve their performance without worsening someone else's. This \(|\psi_{out}\rangle=-j\frac{|01\rangle+|10\rangle}{\sqrt{2}}\) means that one of the two players will always take the channel and the other will not. By avoiding state \(|11\rangle\) we are avoiding the case where the two players take the channel, therefore, avoiding network congestion. By taking advantage of the entanglement generated at the beginning of the circuit in the quantum game, an unattainable behavior in the classical case is obtained: having a medium routing time with minimal traveling time. However, \(S_{1}=(\varphi_{X},\varphi_{Y},\varphi_{Z})=(\frac{\pi}{2},\frac{\pi}{4},0)\) is just one of the possible pure strategies. In Figure 9, the performance of different quantum strategies is shown.