CNOT Gate

The CNOT gate is two-qubit operation, where the first qubit is usually referred to as the control qubit and the second qubit as the target qubit. Expressed in basis states, the CNOT gate:

• leaves the control qubit unchanged and performs a Pauli-X gate on the target qubit when the control qubit is in state $|1\rangle$;
• leaves the target qubit unchanged when the control qubit is in state $|0\rangle$.

$\ CNOT = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & 1 & 0 \end{pmatrix}$