# Y gate
**Y gate** (or **Pauli-Y gate**) is a single-qubit gate represented by the second Pauli matrix. It combines the actions of the [[x-gate|X gate]] (bit flip) and the [[z-gate|Z gate]] (phase flip), and additionally multiplies the amplitude by $i$. It is one of the three [[pauli-gates|Pauli gates]].

$$Y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}$$

The gate maps the computational basis states with an accompanying imaginary factor, and its eigenstates are the $Y$-axis states on the [[bloch-sphere|Bloch sphere]]:

$$Y\lvert 0\rangle = i\lvert 1\rangle \qquad Y\lvert 1\rangle = -i\lvert 0\rangle$$
$$Y\lvert i\rangle = \lvert i\rangle \qquad Y\lvert{-i}\rangle = \lvert{-i}\rangle$$

On the Bloch sphere, $Y$ corresponds to a $\pi$ rotation about the $y$-axis. Like all Pauli gates it is Hermitian, unitary, and satisfies $Y^2 = I$. The $Y$ gate can be decomposed as $Y = iXZ$. In quantum error correction, a $Y$ error on a qubit means both a bit flip and a phase flip occurred simultaneously.

## List of code implementations

- [[y-gate-qiskit|Y gate (Qiskit)]]
- [[y-gate-custatevec|Y gate (cuStateVec)]]
- [[y-gate-cudaq|Y gate (CUDA-Q)]]