are given as $$\begin{aligned}
 R_{\rm C}(\phi,\theta) 
&=& R(\phi,\theta)\otimes \sigma_0 \otimes \sigma_0, \\
 Z_{\rm C}(\theta) 
&=& Z(\theta)\otimes \sigma_0 \otimes \sigma_0,\end{aligned}$$ where $ R(\phi,\theta)= e^{-i \theta(\sigma_x \cos \phi 
+ \sigma_y\sin \phi)/2}$ and $Z(\theta) = e^{-i \theta \sigma_z/2}$. $\theta $ in $R(\theta,\phi)$ is the rotation angle and $\phi$ defines the rotation axis in the $xy$-plane from the $x$-axis, while $\theta$ in $Z(\theta)$ is the rotation angle about the $z$-axis. The rotation operations on CSs can be implemented simultaneously thanks to the symmetry of the molecular structure and are given as $$\begin{aligned}
  R_{\rm S}(\phi,\theta) 
&=& \sigma_0 \otimes R(\phi,\theta)\otimes R(\phi,\theta), \\
 Z_{\rm S}(\theta) 
&=& \sigma_0 \otimes Z(\theta)\otimes  Z(\theta). \end{aligned}$$ The Hadamard gate on CC was effectively implemented with an $R_{\rm C}(\pi/2,-\pi/2)$, while the pseudo CNOT gate was realized as follows $$\begin{aligned}
 {\rm CNOT} 
&=& e^{ -i \frac{\pi}{4}} Z_{\rm C}(-\frac{\pi}{2}) Z_{\rm S}(-\frac{\pi}{2}) 
R_{\rm C}(0, \frac{\pi}{2})  U_{\rm E} R_{\rm S}(\frac{\pi}{2}, \frac{\pi}{2}) 
\nonumber \\
&=&  
{\small  \left(
\begin{array}{cccccccc}
 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0  \\
 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0  \\
 0 & 