history and has been applied to two-point functions in a variety of contexts. More recently, applications to three-point functions have shown their effectiveness for form factor calculations [@ETMC; @RBC]. We consider the use of $Z(2) \otimes Z(2)$ random noisy source as introduced in [@Z2_3].

Consider a set of random sources whose real and imaginary components are randomly chosen from $Z(2)$ for each site, color and spin, $$\label{eq:Impr2}
\{\eta^{(n)}(x)_{a\alpha} \in Z(2) \otimes Z(2) | n = 1...N\}.$$ This set has the property that in the limit $N \rightarrow \infty$ $$\label{eq:Impr3}
\langle\eta_{a\alpha}^{(n)}(x) \eta_{b\beta}^{\dagger(n)}(y)\rangle_n = 
\frac{1}{N}\sum_{n=1}^N \eta_{a\alpha}^{(n)}(x) \eta_{b\beta}^{\dagger(n)}(y) \rightarrow \delta_{xy}\delta_{ab}\delta_{\alpha\beta}.$$ To use this kind of source in calculating correlators, one can choose it to be a set of random wall source located at $t_0$, $$\begin{aligned}
\label{eq:Impr4}
\eta_{a\alpha}^{(n)}(\vec{x},t|t_0) \in Z(2) \otimes Z(2) &|& t = t_0 \nonumber\\
= 0 &|& t \neq t_0,\end{aligned}$$ $$\label{eq:Impr5}
\langle\eta_{a\alpha}^{(n)}(\vec{x},t|t_0) \eta_{b\beta}^{\dagger(n)}(\vec{y},t|t_0)\rangle_n = \delta_{xy}\delta_{ab}\delta_{\alpha\beta},
 \qquad N \rightarrow \infty.$$ Making use of (\[eq:Impr5\]) to rewrite the pion two-point function at zero momentum as, $$\begin{aligned}
\label{eq:Impr6}
\nonumber
C(\tau;\vec{0}) &=& \sum_{\vec{x},\vec{y}} tr \left( D^{-1}(\vec{y},t; \vec{x},t_0) D^{-1\dagger}(\vec{y},t; \vec{x},t_0) \right) \\\nonumber
&=& \sum_{\vec{x},\vec{y},\vec{z}} 
\Big( D^{-1}_{a\alpha,b\beta}(\vec{y},t; \vec{x},t_0) \left[ \delta_{xz}\delta_{bc}\delta_{\beta\kappa} \right] 
D^{-1\dagger}_{c\kappa,a\alpha}(\vec{y},t; \vec{z},t_0) \Big)\\
&=& \sum_{\vec{y}} \left< \psi^{(n)}(\vec{y},t|t_0) \psi^{\dagger(n)}(\vec{y},t|t_0) \right>_n,\end{aligned}$$ where $\psi^{(n)}$ is the solution vector of 