S-dagger gate
| Identifier |
Operator |
Example statement |
| Sdag |
\(S^\dagger\) |
Sdag q[0] |
Description
The S-dagger gate, i.e., the complex conjugate (inverse) of the S 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 inverse of the \(Z\) rotation: \(S^{\dagger} = Z^{-1/2}\).
Aliases
Also known as the Minus-Z90 or mZ90 gate.
Properties
Representation
\[\begin{align}
S^\dagger &= \left(\begin{matrix}
1 & 0 \\
0 & -i
\end{matrix}\right)
\end{align}\]
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 S-dagger gate is given by:
\[\begin{align}
S^\dagger &= R_\hat{\mathbf{n}}\left([0, 0, 1]^T, -\frac{\pi}{2}, -\frac{\pi}{4}\right) = e^{-i\frac{\pi}{4}} \cdot e^{i\frac{\pi}{4}\sigma_z}, \\
\\
S^\dagger &= \left(\begin{matrix}
1 & 0 \\
0 & -i
\end{matrix}\right).
\end{align}\]
In the Hadamard basis \(\{|+\rangle, |-\rangle\}\), the S-dagger gate \(S^\dagger_H\) is given by:
\[S^\dagger_H = HS^\dagger H = \frac{1}{2}\left(\begin{matrix}
1 - i & 1 + i \\
1 + i & 1 - i
\end{matrix}\right)=X^{-1/2}.\]
Operation examples
Standard basis
\[\begin{align}
S^\dagger\,|0\rangle &= |0\rangle \\
\\
S^\dagger\,|1\rangle &= -i|1\rangle \\
\end{align}\]
Hadamard basis
\[\begin{align}
S^\dagger\,|+\rangle &= \frac{1 - i}{2}|+\rangle + \frac{1 + i}{2}|-\rangle \\
\\
S^\dagger\,|-\rangle &= \frac{1 + i}{2}|+\rangle + \frac{1 - i}{2}|-\rangle \\
\end{align}\]