Pauli-X operator

Pauli-$X$ is the bit-flip operator $X = \begin{pmatrix}0&1\\1&0\end{pmatrix}$, the quantum analogue of a classical NOT gate. On computational basis states it acts exactly as a truth table: $X|0\rangle=|1\rangle$ and $X|1\rangle=|0\rangle$, which is the identity this record's basis-state simulation checks directly rather than inferring from a general formula. $X$ is Hermitian and unitary with $X^2=I$, so it is its own inverse, and its eigenstates are $|+\rangle=\frac{1}{\sqrt2}(|0\rangle+|1\rangle)$ and $|-\rangle=\frac{1}{\sqrt2}(|0\rangle-|1\rangle)$ with eigenvalues $+1$ and $-1$. Geometrically, $X$ is a $\pi$ rotation of the Bloch sphere about the $x$-axis. Because $X$, together with $Y$ and $Z$, generates the full single-qubit Pauli group ($XY=iZ$ and cyclic permutations), it anchors error models, stabilizer codes, and controlled two-qubit gates such as CNOT.