## 3 Results and Discussions
To intuitively observe the ability to predict the chaotic dynamics of the GS-PC\({}_{x}\), ES-PC\({}_{x}\), GS-PC\({}_{y}\) and ES-PC\({}_{y}\) in our system, Figure 4 presents their predictive results. In this figure, \(T=10\) ps, \(\theta=40\) fs, and \(N=250\). The samples of the delayed GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\) from the driving QD Spin-VCSEL output are denoted as \(C_{Gx}(n-L_{Gx})\), \(C_{Gy}(n-L_{Gy})\), \(C_{Ex}(n-L_{Ex})\) and \(C_{Ey}(n-L_{Ey})\), respectively. The samples of the trained GS-PC\({}^{\prime}_{x}\), GS-PC\({}^{\prime}_{y}\), ES-PC\({}^{\prime}_{x}\) and ES-PC\({}^{\prime}_{y}\) from the reservoir QD spin-VCSEL output are denoted as \(y^{{}^{\prime}}_{Gx}(n)\), \(y^{{}^{\prime}}_{Gy}(n)\), \(y^{{}^{\prime}}_{Ex}(n)\) and \(y^{{}^{\prime}}_{Ey}(n)\), respectively. As observed from Figure 4, the chaotic trajectories of the \(C_{Gx}(n-L_{Gx})\), \(C_{Gy}(n-L_{Gy})\), \(C_{Ex}(n-L_{Ex})\) and \(C_{Ey}(n-L_{Ey})\) are almost identical to those of the \(y^{{}^{\prime}}_{Gx}(n)\), \(y^{{}^{\prime}}_{Gy}(n)\), \(y^{{}^{\prime}}_{Ex}(n)\) and \(y^{{}^{\prime}}_{Ey}(n)\), respectively. In Figure 5a, when \(T=10\) ps, \(\theta=40\) fs, and \(N=250\), the prediction errors (\(NMSE_{Gx}\) and \(NMSE_{Gy}\)) of the GS-PC\({}_{x}\) and GS-PC\({}_{y}\) are 0.0359 and 0.0375, respectively. The \(NMSE_{Ex}\) and \(NMSE_{Ey}\) for the ES-PC\({}_{x}\) and ES-PC\({}_{y}\) are 0.0995 and 0.0865, respectively. These indicate that the four parallel reservoirs based on the reservoir QD spin-VCSEL can accurately predict the chaotic dynamics of the GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\), respectively. To comprehensively observe the prediction abilities of the four parallel reservoirs on the chaotic dynamics of the delayed GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\), Figure 5a illustrates the relationship between the prediction errors (\(NMSE_{Gx}\), \(NMSE_{Gy}\), \(NMSE_{Ex}\) and \(NMSE_{Ey}\)) and the sampling period \(T\) when \(\theta\) is 40 fs. As shown in Figure 5a, \(NMSE_{Gx}\) and \(NMSE_{Gy}\) exhibit an almost linear decrease from 0.0362 to 0.0350 and from 0.0376 to 0.0366, respectively, as T increases from 2 ps to 128 ps. Similarly, the \(NMSE_{Ex}\) and \(NMSE_{Ey}\) also reveal a linear decrease from 0.0998 to 0.0961 and from 0.0867 to 0.0836, respectively. The reason why a longer sampling period \(T\) leads to reduced training error might be explained as follows. In this work, \(\theta=T/N\) is fixed at 40 fs, and a smaller \(N\) is associated with a smaller \(T\), resulting in a lower-dimensional state space. This situation can make the training of the four parallel reservoirs based on the reservoir QD spin-VCSEL become unstable and more difficult, consequently leading to a larger \(NMSE\). Additionally, when \(T\) is fixed at a certain value, the \(NMSE_{Ex}\) and \(NMSE_{Ey}\) are significantly larger than \(NMSE_{Gx}\) and \(NMSE_{Gy}\). This may be explained by the fact that ES-PC\({}_{x}\) and ES-PC\({}_{y}\) have more complex chaotic dynamics than GS-PC\({}_{x}\) and GS-PC\({}_{y}\), respectively, making the predictions of ES-PC\({}_{x}\) and ES-PC\({}_{y}\) more challenging compared to those of GS-PC\({}_{x}\) and GS-PC\({}_{y}\). Figure 5b shows the relationship between the prediction errors (\(NMSE_{Gx}\), \(NMSE_{Gy}\), \(NMSE_{Ex}\) and \(NMSE_{Ey}\)) and the virtual node interval \(\theta\) when \(T\) is fixed at 10 ps. From the observations in Figure 5, it can be seen that as \(\theta\) increases from 1 fs to 320 fs, the \(NMSE_{Ex}\) and \(NMSE_{Ey}\) slowly increase from 0.0979 to 0.0998 and from 0.0853 to 0.0868, respectively. Then, they gradually stabilize at 0.0998 and 0.0865. On the other hand, the \(NMSE_{Gx}\) and \(NMSE_{Gy}\) remain nearly constant at 0.0363 and 0.0376, respectively. The results indicate that when \(T=10\) ps, the choice of the virtual node interval \(\theta\) has a slight impact on the prediction accuracy for the GS-PC\({}_{x}\) and GS-PC\({}_{y}\).