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--- quantum-x-gate [May 11, 2026 at 10:34] – yanevskiv
+++ quantum-x-gate [May 14, 2026 at 11:38] (current) – external edit 127.0.0.1
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+# Quantum X gate
+**Quantum X gate** (or **Pauli-X gate**, **quantum NOT gate**) is a quantum gate that is analogous to the classical "NOT" or "bit flip". For a single qubit, it applies a rotation around the X-axis by $\pi$ on the Bloch sphere. In matrix form, X gate is equivalently written as a Pauli matrix $\sigma_x$. This is why it's commonly called a Pauli-X gate.
+$$X = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}$$
+For states $\lvert0\rangle,\lvert 1\rangle$ on the computational axis (the Z-axis) it flips the state. This is why it's called a quantum analogue of "NOT" or "bit flip".
+$$X\lvert 0\rangle = \lvert 1\rangle\qquad X\lvert 1\rangle = \lvert 0\rangle$$
+The states $\lvert i\rangle, \lvert -i\rangle$ on the phase axis (the Y-axis) are flipped in the same way.
+$$X\lvert i\rangle = \lvert -i\rangle\qquad X\lvert -i\rangle = \lvert i\rangle$$
+The states $\lvert +\rangle, \lvert - \rangle$ on the hadamard axis (the X-axis) are eigenstates of $X$ meaning those states are unaffected by the operator.
+$$X\lvert +\rangle = \lvert + \rangle \qquad X\lvert -\rangle = \lvert -\rangle$$