## 3 Results and Discussions
Moreover, the prediction performance is bolstered by implementing chaotic mask signals derived from two coupled semiconductor lasers, detailed in [24]. These mask signals are normalized with standard deviations set to 1 and mean values calibrated to 0. Each reservoir's virtual node interval, denoted by \(\theta\), is fixed at 40 fs. Here, all rates for the QPSK, 4 QAM, 8 QAM and 16 QAM are 100 Gb/s. The input data sampling period \(T\) is maintained at 10 ps, resulting in a data processing rate of 100 Gb/s. We establish the number of virtual nodes, \(N\), at 250, where \(N=\tau/\theta\) and \(\tau=T\). We maintain the scale factor \(\gamma\), at a value of 1. To assess the predictions for the GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\) made by these four parallel reservoirs using the reservoir QD spin-VCSEL, we introduce the normalized mean square error (\(NMSE\)) as a metric to compare the delayed predictive targets against their associated reservoir outputs, which is given as follows: \begin{table}
\begin{tabular}{c c c c} \hline
**The Parameter and Symbol** & **Value** & **The Parameter and Symbol** & **Value** \\ \hline The field decay rate \(\kappa\) & 300 ns\({}^{-1}\) & Central frequency detuning \(\Delta\omega_{z}\) & \(-20\times 10^{9}\) rad/s \\ Line-width enhancement factor \(\alpha\) & 3 & The capture rate \(\gamma_{0}\) & 600 ns\({}^{-1}\) \\ Total pump intensity \(\eta\) & 4 & Intradot relaxation rate \(\gamma_{21}\) & 40 ns\({}^{-1}\) \\ Dichroism \(\gamma_{a}\) & 0.1 ns\({}^{-1}\) & Spin relaxation rate \(\gamma_{s}\) & 20 ns\({}^{-1}\) \\ Birefringence \(\gamma_{p}\) & 20 ns\({}^{-1}\) & Carrier recombination rate \(\gamma_{n}\) & 1 ns\({}^{-1}\) \\ Center frequency \(\omega_{c}\) & \(2\times 10^{14}\) rad/s & Injection strength \(k_{inj}\) & 35 ns\({}^{-1}\) \\ Center frequency \(\omega_{z}\) & \(10^{14}\) rad/s & Feedback strength \(k_{f}\) & 30 ns\({}^{-1}\) \\ Central frequency detuning \(\Delta\omega_{c}\) & 0 rad/s & The gain coefficient \(h_{1}\) & 1.1665 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter values of the reservoir QD Spin-VCSEL. Figure 3: Power spectral density (PSD) distributions of the four polarization components GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\) from the driving QD Spin-VCSEL output. Here, (**a**) the PSD of GS-PC\({}_{x}\) (PSD\({}_{Gx}\)); (**b**) the PSD of the GS-PC\({}_{y}\) (PSD\({}_{Gy}\)); (**c**) the PSD of ES-PC\({}_{x}\) (PSD\({}_{Ex}\)); and (**d**) the PSD of the ES-PC\({}_{y}\) (PSD\({}_{Ey}\)). \[ \begin{split} NMSE_{jx}&=\frac{1}{L}\frac{\sum\limits_{n=1}^{L }\left(y^{{}^{\prime}}_{jx}(n)-C_{jx}(n-L_{jx})\right)^{2}}{\text{var}(y^{{}^{ \prime}}_{jx}(n))},(j=G,E)\\ NMSE_{jy}&=\frac{1}{L}\frac{\sum\limits_{n=1}^{L }\left(y^{{}^{\prime}}_{jy}(n)-C_{jy}(n-L_{jy})\right)^{2}}{\text{var}(y^{{}^{ \prime}}_{jy}(n))},(j=G,E)\end{split}  \tag{23}\]  
where the subscripts \(Gx\), \(Ex\), \(Gy\) and \(Ey\) represent GS-PC\({}_{x}\), ES-PC\({}_{x}\), GS-PC\({}_{y}\) and ES-PC\({}_{y}\), respectively. \(L_{Gx}\), \(L_{Gy}\), \(L_{Ex}\) and \(L_{Ey}\) are the defined lengths of the testing data set for each variable. \(L\) represents the total number of data points in the testing data set. The term \"var\" denotes the variance of the data. When \(NMSE_{jx}\) and \(NMSE_{jy}\) are both 0, it means that the outputs of the reservoirs (GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\)) perfectly match with their corresponding predicted targets (\(C_{Gx}(n-L_{Gx})\), \(C_{Gy}(n-L_{Gy})\), \(C_{Ex}(n-L_{Ex})\) and \(C_{Ey}(n-L_{Ey})\), respectively). On the other hand, if \(NMSE_{jx}\) and \(NMSE_{jy}\) both are 1, it means that the reservoir outputs are completely different from the predicted targets. When \(NMSE_{jx}\) and \(NMSE_{jy}\) are both less than 0.1, it indicates that each reservoir is able to accurately infer the chaotic dynamics of its corresponding predicted target, which is the PC of the driving QD Spin-VCSEL output.