S gate
| Identifier | Operator | Example statement |
|---|---|---|
| S | \(S\) | S q[0] |
Description
The S gate or Phase gate is an anti-clockwise rotation of \(\pi / 2\) [rad] about the \(\hat{\mathbf{z}}\)-axis and a global phase of \(\pi / 4\) [rad].
It is equal to half the \(Z\) rotation: \(S = Z^{1/2}\).
Aliases
Also known as the Z90 gate.
Properties
Representation
\[\begin{align}
S &= \left(\begin{matrix}
1 & 0 \\
0 & i
\end{matrix}\right)
\end{align}\]
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 Phase gate is given by:
\[\begin{align}
S &= R_\hat{\mathbf{n}}\left([0, 0, 1]^T, \frac{\pi}{2}, \frac{\pi}{4}\right) = e^{i\frac{\pi}{4}} \cdot e^{-i\frac{\pi}{4}\sigma_z}, \\
\\
S &= \left(\begin{matrix}
1 & 0 \\
0 & i
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the Phase gate \(S_H\) is given by:
\[S_H = HSH = \frac{1}{2}\left(\begin{matrix}
1 + i & 1 - i \\
1 - i & 1 + i
\end{matrix}\right)=X^{1/2}.\]
Operation examples
Standard basis
\[\begin{align}
S\,|0\rangle &= |0\rangle \\
\\
S\,|1\rangle &= i|1\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
S\,|+\rangle &= \frac{1 + i}{2}|+\rangle + \frac{1 - i}{2}|-\rangle \\
\\
S\,|-\rangle &= \frac{1 - i}{2}|+\rangle + \frac{1 + i}{2}|-\rangle \\
\end{align}\]