Controlled-U Gate
The controlled-U gate, just like the CNOT Gate, applies some Quantum Gate $U$ to the right Qubit if the left qubit is $1$ 1 : $$\displaylines{
CU\ket{00} = \ket{00}, \
CU\ket{01} = \ket{01}, \
CU\ket{10} = \ket{1} \otimes U\ket{0}, \
CU\ket{11} = \ket{1} \otimes U\ket{1}.
}$$ Generally speaking, if $U$ is a $2\times 2$ matrix: $$U = \begin{pmatrix} a & c \\ b & d\end{pmatrix},$$ then: $$\begin{aligned}
CU\ket{00} &= \ket{00}, \
CU\ket{01} &= \ket{01}, \
CU\ket{10} &= \ket{1} \otimes (a\ket0 + b\ket1) = a\ket{10} + b\ket{11}, \
CU\ket{11} &= \ket{1} \otimes (c\ket0 + d\ket1) = c\ket{10} + d\ket{11}.
\end{aligned}$$
Then, the matrix representation of $CU$ is : $$CU = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & a & b \\ 0 & 0 & c & d \end{pmatrix}.$$
# Examples
- controlled-Z
- controlled-phase