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--- x-gate [June 09, 2026 at 22:53] – Ivan Janevski
+++ x-gate [June 15, 2026 at 15:23] (current) – Ivan Janevski
@@ Line 1: / Line 1: @@
-# X-gate
+# $X$ gate (NOT gate)
-**X-gate** (or **Pauli-X gate**, or **Quantum NOT gate**) is a quantum gate that flips $\lvert 0\rangle$ to $\lvert 1\rangle$, and $\lvert 1\rangle$ to $\lvert 0\rangle$, which is why it's called the quantum equivalent of NOT. It has the following 2x2 matrix representation.
+**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]].
-$$ X = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}$$
+$$X = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}$$
-On the Bloch sphere, it rotates the qubit state by $\pi$ radians i.e. 180 degrees around the X-axis. The state $\lvert 0\rangle$ is associated with the north pole, so it flips it to the south pole. The state $\lvert 1\rangle$ is associated with the south pole, so it flips it to the north pole.
+The gate flips the computational basis states and leaves the Hadamard basis states ($\lvert +\rangle$, $\lvert -\rangle$) unchanged, since those are its eigenstates:
-If you apply the gate twice, you will flip the state twice, meaning you go back to where you started. This means. This property is called the "involution" property of X-gate. Other Pauli gates also have this property
+$$X\lvert 0\rangle = \lvert 1\rangle \qquad X\lvert 1\rangle = \lvert 0\rangle$$
-$$X^2 = I$$
+$$X\lvert +\rangle = \lvert +\rangle \qquad X\lvert -\rangle = \lvert -\rangle$$
+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)]]