# X-gate (NOT gate) Definition $X=\begin{bmatrix} 0 & 1 \\ 1& 0\end{bmatrix}$ . also known as Pauli-X gate. analog to NOT gate: $X|0\rangle=|1\rangle$ and $X|1\rangle=|0\rangle$. If $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, then $X|\psi\rangle=\beta|0\rangle+\alpha|1\rangle$ It is Hermitian: $X^\dagger=X$ and unitary: $X^2=I$. for $\mu\in\{0,1\}$ and $|a\rangle$ is a computational basis, $X^{\mu_0} |a\rangle =|a\oplus\mu_0\rangle$ . This can be seen from the truth table of both sides: | a | $\mu_o$ | $X^{\mu_o}\|a\rangle$ | $\|a\oplus \mu_0\rangle$ | | --- | ------- | --------------------- | ------------------------ | | 0 | 0 | 0 | 0 | | 0 | 1 | 1 | 1 | | 1 | 0 | 1 | 1 | | 1 | 1 | 0 | 0 | #tech ## Created 2026-01-14 13:54