The unitary coupled cluster ansatz is $$ \begin{align} \vert\Psi\rangle=e^{T-T^{\dagger}}\vert\Phi\rangle, \tag{3} \end{align} $$ and $$ \begin{align} \vert\Psi\rangle=\exp{(T_1-T_1^{\dagger})}\vert\Phi\rangle, \tag{4} \end{align} $$ where \( \vert\Phi\rangle \) is a Fock state and \( T=\sum_{k=1}^AT_k \).