# $X$ gate (NOT gate)
**X gate** (or **Pauli-X gate**, or **quantum NOT gate**) is a single-qubit gate that flips $\lvert 0\rangle$ to $\lvert 1\rangle$ and $\lvert 1\rangle$ to $\lvert 0\rangle$, making it the quantum analog of a classical NOT gate. It is one of the three [[pauli-gates|Pauli gates]].

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

The gate flips the computational basis states and leaves the Hadamard basis states ($\lvert +\rangle$, $\lvert -\rangle$) unchanged, since those are its eigenstates:

$$X\lvert 0\rangle = \lvert 1\rangle \qquad X\lvert 1\rangle = \lvert 0\rangle$$
$$X\lvert +\rangle = \lvert +\rangle \qquad X\lvert -\rangle = \lvert -\rangle$$

On the [[bloch-sphere|Bloch sphere]], $X$ corresponds to a rotation of $\pi$ radians about the $x$-axis. Applying the gate twice returns the qubit to its original state — this is the involution property, $X^2 = I$, shared by all Pauli gates. The $X$ gate appears in virtually every quantum algorithm as the quantum equivalent of setting or flipping a bit.

## List of code implementations

- [[x-gate-qiskit|X gate (Qiskit)]]
- [[x-gate-custatevec|X gate (cuStateVec)]]
- [[x-gate-cudaq|X gate (CUDA-Q)]]