  Quantum circuit model is the most popular paradigm for implementing complex
quantum computation. Based on Cartan decomposition, we show that $2(N-1)$
generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the
$y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which
are defined in this paper are sufficient to simulate a controlled-unitary gate
$\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on $\mathbb{C}^2\otimes
\mathbb{C}^N$. In the scenario of the unitary gate $\mathcal{U}_{cd(M\otimes
N)}$ with $M\geq3$ that is locally equivalent to a diagonal unitary on
$\mathbb{C}^M\otimes \mathbb{C}^N$, $2M(N-1)$ GCX gates and $2M(N-1)+10$
single-partite $y$- and $z$-rotation-types are required to simulate it. The
quantum circuit for implementing $\mathcal{U}_{cu(2\otimes N)}$ and
$\mathcal{U}_{cd(M\otimes N)}$ are presented. Furthermore, we find
$\mathcal{U}_{cu(2\otimes2)}$ with $A$ controlling has Schmidt rank two, and in
other cases the diagonalized form of the target unitaries can be expanded in
terms of specific simple types of product unitary operators.
