## 3 Problem Formulation  
The MSE between the transmitted and received signals is often used as an optimization indicator for BER performance, and MSE-based transceiver design has been widely studied in MIMO systems [38]. Based on this, we derive the MSE expression for MU photon-counting systems and construct an optimization problem to minimize the MSE under power constraints and optical IRS configuration constraints. Based on the definition of MSE, we give the expression of user \(k\),  
\[\text{MSE}_{k}\triangleq\mathbb{E}\Big{[}\big{(}y_{k}-u_{k}\big{)}^{2}\Big{]}= \mathbb{E}\Big{(}y_{k}^{2}\Big{)}+\mathbb{E}\Big{(}u_{k}^{2}\Big{)}-2\mathbb{E }(y_{k}u_{k}), \tag{18}\]  
where \(y_{k}\) is the received signal, and \(u_{k}\) is the original data. Note that the expectation is taken over variables \(y_{k}\) and \(u_{k}\). Differing from the AWGN model, which is independent of the signal, the MSE expression can be directly obtained. In the photon-counting communication system under Poisson shot noise, we need to use the distribution of the transmitted signal \(u_{k}\) and statistical information, such as the expectation of the received signal \(y_{k}\), to indirectly derive the MSE expression. Without a loss of generality, we assume that \(u_{k}=\pm 1\), i.e., \(\Pr(u_{k}=-1)=\Pr(u_{k}=1)=1/2,\forall k\in\mathcal{K}\)[39]. Obviously, we can get  
\[\mathbb{E}\Big{(}u_{k}^{2}\Big{)}=1\times\Pr(u_{k}=-1)+1\times\Pr(u_{k}=1)=1. \tag{19}\]  
Regarding the solution of \(\mathbb{E}\big{(}y_{k}^{2}\big{)}\), we can use the conditional expectation method to achieve it. Since \(y_{k}\) follows the Poisson distribution, we can use the property of the Poisson distribution, \(\mathbb{E}(X)=\text{Var}(X)\), and get  
\[\mathbb{E}\Big{(}y_{k}^{2}\Big{)}=\sum_{\mathbf{u}\in\mathcal{U}}\mathbb{E} \Big{(}y_{k}^{2}|\mathbf{u}\Big{)}\Pr(\mathbf{u})=\sum_{\mathbf{u}\in\mathcal{ U}}\Big{\{}\big{[}\mathbb{E}(y_{k}|\mathbf{u})\big{]}^{2}+\mathbb{E}(y_{k}| \mathbf{u})\Big{\}}\Pr(\mathbf{u}), \tag{20}\]  
where the set \(\mathcal{U}_{0}\) is used to represent all possible values of \(\mathbf{u}\), expressed as \(\mathcal{U}_{0}=\{\mathbf{u}^{(i)}\in\{-1,1\}|i=1,\cdots,2^{K}\}\). Combining Equation (17) and expanding the above formula, we get  
\[ \begin{split}&\mathbb{E}\Big{(}y_{k}^{2}\Big{)}=\sum_{\mathbf{u} \in\mathcal{U}}\bigg{\{}\Big{(}\eta\mathbf{g}_{k}^{\mathsf{T}}\mathbf{Wu}+ \gamma_{k}\Big{)}^{2}+\Big{(}\eta\mathbf{g}_{k}^{\mathsf{T}}\mathbf{Wu}+ \gamma_{k}\Big{)}\bigg{\}}\Pr(\mathbf{u})\\ &=\frac{1}{2^{K}}\sum_{i=1}^{2^{K}}\Big{(}\eta\mathbf{g}_{k}^{ \mathsf{T}}\mathbf{Wu}^{(i)}\Big{)}^{2}+(2Y_{k}+1)\eta\mathbf{g}_{k}^{\mathsf{ T}}\mathbf{W}\underbrace{\sum_{i=1}^{2^{K}}\mathbf{u}^{(i)}}_{=0}+Y_{k}^{2}+Y_{k} \\ &=\frac{1}{2^{K}}\eta^{2}\mathbf{g}_{k}^{\mathsf{T}}\underbrace{ \mathbf{W}\left(\sum_{i=1}^{2^{K}}\mathbf{u}^{(i)}\Big{[}\mathbf{u}^{(i)} \Big{]}^{T}\right)}_{=\mathbf{I}_{K}}\mathbf{W}^{\mathsf{T}}\mathbf{g}_{k}+ \eta^{2}\mathbf{g}_{k}^{\mathsf{T}}\mathbf{D}\mathbf{D}^{\mathsf{T}}g_{k}+ \eta(2n_{b}+1)\mathbf{g}_{k}^{\mathsf{T}}\mathbf{D}+n_{b}^{2}+n_{b}\\ &=\eta^{2}\mathbf{g}_{k}^{\mathsf{T}}\mathbf{W}\mathbf{W}^{\mathsf{ T}}\mathbf{g}_{k}+\eta^{2}\mathbf{g}_{k}^{\mathsf{T}}\mathbf{D}\mathbf{D}^{ \mathsf{T}}g_{k}+\eta(2n_{b}+1)\mathbf{g}_{k}^{\mathsf{T}}\mathbf{D}+n_{b}^{2} +n_{b}.\end{split}  \tag{21}\]  
where \(\mathbf{I}_{K}\in\mathbb{R}^{K\times K}\) represents the \(\text{K}\times\text{K}\) dimensional identity matrix. Similarly, we can expand \(\mathbb{E}(y_{k}u_{k})\) into  
\[ \begin{split}\mathbb{E}(y_{k}u_{k})&=\sum_{u_{k} \in\{\pm 1\}}\mathbb{E}(y_{k}u_{k}|u_{k})\Pr(u_{k})\\ &=\frac{1}{2}\mathbb{E}(y_{k}|u_{k}=1)-\frac{1}{2}\mathbb{E}(y_{k} |u_{k}=-1)\end{split}  \tag{22}\]According to ([19], Lemma 2), we can get the conclusion of (22), as given by  
\[\mathbb{E}(y_{k}u_{k})=\eta\mathbf{g}_{k}^{T}\mathbf{w}_{k}, \tag{23}\]  
By combining Equations (19), (20) and (22), we obtain the MSE expression for user \(k\). Summing the MSE of all \(K\) users yields the overall system MSE. \[