√X†¶
√X† is the inverse of √X - applying √X then √X† (in either order) returns a qubit exactly to its starting state.
| Qubits | 1 |
| Parameters | None |
| Inverse | √X |
| Also called | SX-dagger, adjoint of SX |
What it does¶
√X† applies the complex-conjugate transpose of √X's matrix, undoing its effect.
Matrix¶
\[
\sqrt{X}^\dagger = \frac{1}{2}\begin{pmatrix} 1-i & 1+i \\ 1+i & 1-i \end{pmatrix}
\]
Action on basis states¶
- \( |0\rangle \rightarrow \dfrac{1}{2}\big[(1-i)|0\rangle + (1+i)|1\rangle\big] \)
- \( |1\rangle \rightarrow \dfrac{1}{2}\big[(1+i)|0\rangle + (1-i)|1\rangle\big] \)
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 | sxdg q[0]; |
| Qiskit | circuit.sxdg[q[0]] |
| Cirq | cirq.X(q[0])**-0.5 |
| Q# | - |