Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates

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 $M\ge 3$ 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 ${\mathcal{U}}_{cu(2\otimes 2)}$ 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.