S†¶
S† ("S-dagger") is the inverse of the S gate: it multiplies \( |1\rangle \) by \( -i \) instead of \( +i \), undoing whatever S did.
| Qubits | 1 |
| Parameters | None |
| Inverse | S |
| Also called | S-dagger, adjoint of S; equal to P(−π/2) |
What it does¶
S† rotates the relative phase between \( |1\rangle \) and \( |0\rangle \) by \( -\pi/2 \). Applying S followed by S† (in either order) returns a state to exactly where it started.
Matrix¶
\[
S^\dagger = \begin{pmatrix} 1 & 0 \\ 0 & -i \end{pmatrix}
\]
Action on basis states¶
- \( |0\rangle \rightarrow |0\rangle \)
- \( |1\rangle \rightarrow -i\,|1\rangle \)
Example¶
H, then S, then S† on q[0] returns exactly to \( |{+}\rangle \), the same state plain H alone would give.
Other notations¶
Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.
| Language | Typical form |
|---|---|
| OpenQASM 2.0 | sdg q[0]; |
| Qiskit | circuit.sdg[q[0]] |
| Cirq | cirq.S(q[0])**-1 |
| Q# | Adjoint S(q[0]); |