## 5 Numerical Results  
In this section, we conduct extensive simulation experiments to evaluate the communication performance of of our proposed algorithm in an IRS-assisted photon-counting MU downlink communication system. Specifically, we use the widely adopted Monte Carlo method to randomly generate transmission data for all users and evaluate the MSE and BER performance of the communication system. To reduce the error, we take the average of 100 independent tests for each data point in the figure. In the subsequent simulation experiments, we set the wavelength to \(\lambda=1550\) nm and the PD efficiency to \(\eta=0.5\). According to [19], we set the scintillation parameters \(\alpha=1\) and \(\beta=1\), which indicate the situation of strong turbulence fading. In fact, the scintillation index is defined as \(\text{S.I. }=1/\alpha+1/\beta+1/(\alpha\beta)\)[43], and its value represents the strength of the channel turbulence fading. In addition, we set the symbol duration to \(\tau=1\) us and the background radiation energy per bit is \(-170\) dBJ. Our subsequent research on the robustness of the solution can be conducted by setting different scintillation indexes and background radiation values. Table 1 shows the detailed parameter settings of the simulation system. To evaluate the performance improvement offered via the proposed AO algorithm, we selected two comparison schemes. (1) ZF precoding: This scheme employs typical ZF precoding under the AWGN model as a reference. The specific calculation method is expressed as \(\mathbf{W}_{ZF}=\mathbf{G}^{T}\Big{(}\mathbf{G}\mathbf{G}^{T}\Big{)}^{-1}\) and needs to be scaled by multiplying the coefficient to satisfy constraints (36) and (37). (2) Random IRS scheme: in this scheme, the binary matrix \(\mathbf{V}\) in \(\mathcal{P}3-b\) is generated randomly, needing only to satisfy constraints (30) and (31). To begin with, we present the convergence performance curve of the proposed algorithm, illustrated in Figure 3. The simulation parameters are configured for a \(12\times 4\) communication system assisted by 32 optical IRS units. As shown in Figure 3, the proposed AO algorithm achieves convergence after 10 iterations, while the only precoding \(\mathbf{W}\) scheme reaches convergence after 15 iterations. The convergence speed in both cases is very fast, indicating the effectiveness of the proposed AO algorithm. However, the AO algorithm has fewer iterations, indicating that it is more efficient. In addition, the simulation results indicate that compared with the only precoding scheme, the convergence value of the AO algorithm is lower and can reach close to \(10^{-2}\). This shows the necessity of adding IRS optimization, which can significantly reduce the MSE of the system and improve the system communication performance. Next, we explore the comparison between the proposed scheme and other baseline schemes, as shown in Figure 4. Regarding the coding methods, Figure 4 clearly illustrates that the performance of the proposed MMSE precoding scheme under Poisson noise is significantly superior to that of the ZF coding scheme under the AWGN model. Specifically, when the transmitted signal energy per bit is \(E_{b}=-163\) dBJ, the MSE of the ZF scheme is less than \(10^{-1}\), while the MSE of our proposed MMSE scheme can reach \(10^{-3}\). This improvement arises because the ZF scheme fails to capture the correlation between the noise and signal in the photon-counting system. Consequently, even with our proposed IRS configuration scheme, the MSE performance of the ZF method can only achieve approximately \(10^{-2}\). On the other hand, when compared to the randomized IRS configuration scheme, our proposed IRS configuration scheme significantly reduces the MSE of the system, highlighting the effectiveness of our IRS design.