Quantum CNOT Gate

Definition¹

(From the definition of a classical $\operatorname{CNOT}$ gate) A quantum $\operatorname{CNOT}$ gate for $2$qubits $\ket{a, b} = \ket{a} \otimes \ket{b}$ is defined as follows.

$$ \begin{align*} \operatorname{CNOT}_{q} : (\mathbb{C}^{2})^{\otimes 2} &\to (\mathbb{C}^{2})^{\otimes 2} \\ \ket{a, b} &\mapsto \ket{a, a \oplus b},\quad \forall a,b \in \left\{ 0, 1 \right\} \end{align*} $$

$$ \operatorname{CNOT}_{q} (\ket{a} \otimes \ket{b}) = \ket{a} \otimes \ket{a \oplus b} $$

Here, $(\mathbb{C}^{2})^{\otimes 2}$ is the tensor product of vector spaces, $\ket{a} \otimes \ket{b}$ is the product vector, and $\oplus$ is the exclusive or.

Description

In quantum circuits, the logical negation is a Pauli $X$ gate, hence it is also called the Controlled Pauli X gate.

The specific input and output of $\operatorname{CNOT}_{q}$ are as follows.

$$ \operatorname{CNOT}_{q} (\ket{00}) = \ket{0, 0 \oplus 0} = \ket{00} \\[0.5em] \operatorname{CNOT}_{q} (\ket{01}) = \ket{0, 0 \oplus 1} = \ket{01} \\[0.5em] \operatorname{CNOT}_{q} (\ket{10}) = \ket{1, 1 \oplus 0} = \ket{11} \\[0.5em] \operatorname{CNOT}_{q} (\ket{11}) = \ket{1, 1 \oplus 1} = \ket{10} $$

The matrix representation is as follows.

$$ \operatorname{CNOT}_{q} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} $$