Pauli-Y gate
| Identifier | Operator | Example statement |
|---|---|---|
| Y | \(Y\) | Y q[0] |
Description
The Pauli-Y, or Y, gate is an anti-clockwise rotation of \(\pi\) [rad] about the \(\hat{\mathbf{y}}\)-axis and a global phase of \(\pi/2\) [rad].
Properties
Representation
\[Y = \sigma_y = \sigma_2 = \left(\begin{matrix}
0 & -i \\
i & 0
\end{matrix}\right)\]
Any single-qubit operation in \(U(2)\) (including global phase) can be expressed by 5 parameters in the canonical representation \(R_\hat{\mathbf{n}}\)
\[R_\hat{\mathbf{n}}\left([n_x, n_y, n_z]^T, \theta, \phi\right) = e^{i\phi} \cdot e^{-i\frac{\theta}{2}\left(n_x\cdot\sigma_x + n_y\cdot\sigma_y + n_z\cdot\sigma_z\right)},\]
where \(\hat{\mathbf{n}}=[n_x, n_y, n_z]^T\) denotes the axis of rotation, \(\theta\in(-\pi, \pi]\) the angle of rotation [rad], and \(\phi\in[0,2\pi)\) the global phase angle [rad].
The Pauli-Y gate is given by:
\[\begin{align}
Y &= R_\hat{\mathbf{n}}\left([0, 1, 0]^T, \pi, \frac{\pi}{2}\right) = e^{i\frac{\pi}{2}} \cdot e^{-i\frac{\pi}{2}\sigma_y}, \\
\\
Y &= \left(\begin{matrix}
0 & -i \\
i & 0
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the Pauli-Y gate \(Y_H\) is given by:
\[Y_H = HYH = \left(\begin{matrix}
0 & i \\
-i & 0
\end{matrix}\right)=-Y=Y^T.\]
Operation examples
Standard basis
\[\begin{align}
Y\,|0\rangle &= i|1\rangle \\
\\
Y\,|1\rangle &= -i|0\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
Y\,|+\rangle &= -i|-\rangle \\
\\
Y\,|-\rangle &= i|+\rangle
\end{align}\]