Controlled NOT(CNOT) Gate

Definition¹

The following vector-valued Boolean function is called the $\operatorname{CNOT}$ gate.

$$ \operatorname{CNOT} : \left\{ 0, 1 \right\}^{2} \to \left\{ 0, 1 \right\}^{2} $$

$$ \operatorname{CNOT} (a,b) = (a, a \oplus b) $$

• It is also known as the Feynman gate. ²

Description

The specific calculations of the input and output of the $\operatorname{CNOT}$ gate are as follows.

$$ \begin{align*} \operatorname{CNOT} (0,0) &= (0, 0 \oplus 0) = (0, 0) \\ \operatorname{CNOT} (0,1) &= (0, 0 \oplus 1) = (0, 1) \\ \operatorname{CNOT} (1,0) &= (1, 1 \oplus 0) = (1, 1) \\ \operatorname{CNOT} (1,1) &= (1, 1 \oplus 1) = (1, 0) \end{align*} $$

Looking at the table above, it is easy to see that $\operatorname{CNOT}$ is a reversible function and that composing $\operatorname{CNOT}$ twice results in an identity function.

$$ \operatorname{Id} = \operatorname{CNOT} \circ \operatorname{CNOT} $$

If only the second value of the output is considered, it is similar to the $\text{XOR}$ gate, hence it is also referred to as the reversible $\text{XOR}$ gate..

부울 함수	기호	진리표
$\operatorname{CNOT}$		입력 출력 $a$ $b$ $a$ $a \oplus b$ $0$ $0$ $0$ $0$ $0$ $1$ $0$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ $1$ $0$