# Y gate
**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.