A gate for a Quantum Computer

Let

CNOT=(1000010000010010)\text{CNOT} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}

Example

CNOTa,b=a,ba\text{CNOT}\ket{a,b} = \ket{a,\, b \oplus a}

where \oplus is addition mod 2