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--- y-gate [May 25, 2026 at 10:23] – Ivan Janevski
+++ y-gate [May 25, 2026 at 13:55] (current) – external edit 127.0.0.1
@@ Line 1: / Line 1: @@
-# Y-gate
+# Y gate
-**Y-gate** (or **Pauli-Y gate**, **phase gate**) is a single qubit quantum gate. It is represented by the following 2x2 matrix.
+**Y gate** (or **Pauli-Y gate**) is a single-qubit quantum gate represented by the second Pauli matrix. It combines the actions of the X gate (bit flip) and the Z gate (phase flip), and also introduces a factor of $i$.
 $$Y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}$$
+The gate maps the computational basis states as $Y\lvert 0\rangle = i\lvert 1\rangle$ and $Y\lvert 1\rangle = -i\lvert 0\rangle$. On the [[bloch-sphere]], the Y gate corresponds to a $\pi$ rotation about the $y$-axis. Like the X and Z gates, $Y$ is Hermitian ($Y^\dagger = Y$) and unitary ($Y^\dagger Y = I$), and it satisfies $Y^2 = I$.
+## Relation to X and Z gates
+The Y gate can be decomposed as $Y = iXZ$. The eigenstates of the Y gate are the [[i-state|plus-i state]] $\lvert i\rangle$ (eigenvalue $+1$) and the [[minus-i-state|minus-i state]] $\lvert -i\rangle$ (eigenvalue $-1$). Together with the X and Z Pauli gates, the Y gate generates the Pauli group and forms the basis of quantum error correction, where X, Y, and Z errors must be detected and corrected.