## 4 MAMZI with Phase Noise Stabilization
### Analysis of the Results in Phase Noise Stabilization  
Figure 4 presents the results of applying the \(P\&O\) algorithm to the system with multidimensional noise (for \(N=2\), \(4\), \(6\), and \(8\)), according to the scheme shown in Figure 2. Due to the precision in the simulation and to illustrate the system's operation, we show algorithm \(P\&O\) operating with a value of \(\delta=4\) in each configuration on the complete set of noise matrices described above. Figure 4: Feedback phase noise control operation. Vertically, a pair of graphs is shown, depicting the control phases obtained from \(P\&O\) and their respective optical intensity as a function of dimension \(N\), that is, (**a,b**) \(N=2\), (**c,d**) \(N=4\), (**e,f**) \(N=6\), (**g,h**) \(N=8\). In addition, Figure 4 shows the calculated control phases and the respective intensity responses. For example, the results for dimension \(N=2\) are presented in the vertical pair Figure 4a,b; for \(N=4\), they are shown in Figure 4c,d, and so on for higher dimensions. The system control operates on each output sequentially, remaining controlled for \(\tau_{control}=4000\), generating ten temporary operation cycles (\(\tau_{control}=l_{max}/10\)), effectively keeping the outputs maximized and operating the changes at high speed. However, the stead-state signals show some variability. Figure 5 shows the system's response of a MAMZI for \(N=8\) under feedback control, using the above-mentioned parameters. In this figure, the steady-state variability and the operating speed differences throughout each work cycle can be seen more clearly. In our analysis of the control operation, we used the following four indicators to evaluate the performance of the proposed control method:  
**Control velocity** evaluates the speed at which the algorithm reaches its maximum intensity, equivalent to stabilizing the phase noise. Therefore, the time taken by the stabilized signal intensity to go from 10% (\(I_{k}^{out}[t_{0}]=0.1\)) to 90% (\(I_{k}^{out}[t_{1}]=0.9\)) is measured, as we show in Figure 5. It is important to note that the sampling period is an arbitrary variable, so the control speed (\(cv\)) is given by  
\[cv=\frac{0.8}{t_{1}-t_{0}}. \tag{8}\]  
**Root mean square error** quantifies the steady-state control error on the stabilized signal. Since the stabilization cycle is \(\tau_{control}\), the signal is assumed to reach the steady state from time \(t_{2}=\tau_{control}/3\). The upper limit is set at time \(t_{3}=2\tau_{control}/3\) to obtain a robust statistic (see Figure 5). The \(RMSE\) calculation is performed considering this time interval, being estimated from  
\[RMSE=\sqrt{\frac{1}{N_{L}}\sum_{N=t_{2}}^{t_{3}}(I_{k}^{out}[n]-I_{ref}[n])^{2}}, \tag{9}\]  
where \(N_{L}\) is the total number of points sampled between \(t_{2}\) and \(t_{3}\). **Visibility** quantifies the optical interference characteristics of a communications channel. It is especially crucial in quantum systems, where the visibility of the system directly influences the fidelity of quantum state preparation and measurement. Visibility is defined by  
\[V=\frac{I_{k}^{out}max-I_{k}^{out}min}{I_{k}^{out}max+I_{k}^{out}min}, \tag{10}\]  
Figure 5: Time labels for calculating control parameters in a stabilized MAMZI for \(N=8\). where \(I_{k}^{out}max\) and \(I_{k}^{out}min\) correspond to the maximum and minimum values of \(I_{k}^{out}\), respectively. **Crosstalk** measures the ratio of unwanted information (\(I_{p}^{out}\)) from an external channel to the information that this channel can transmit (\(I_{k}^{out}\)) with \(p\
eq k\). Mathematically, it is expressed as  
\[Cr=10 log\big{(}\frac{I_{p}^{out}}{I_{p}^{out}}\big{)} \tag{11}\]  
Figure 6 shows the results obtained for the \(P\delta O\) algorithm using a fixed step considering \(1<\delta<20\). It is observed that the RMSE remains below 1% for systems smaller than N = 6. For a larger N, the error does not exceed 2% with \(\delta=3\).