Table of Contents
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 (bit flip) and the Z gate (phase flip), and additionally multiplies the amplitude by $i$. It is one of the three 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:
$$Y\lvert 0\rangle = i\lvert 1\rangle \qquad Y\lvert 1\rangle = -i\lvert 0\rangle$$ $$Y\lvert i\rangle = \lvert i\rangle \qquad Y\lvert{-i}\rangle = \lvert{-i}\rangle$$
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.