pub fn SGate(radix: usize) -> UnitaryExpressionExpand description
The one-qudit S gate (Clifford level 2).
Defined as:
$$ S|k\rangle = e^{i\pi k^2 / d}|k\rangle $$
where $d$ is the number of levels. For qubits ($d = 2$) this reproduces the standard S gate $S = \text{diag}(1, i)$. Note that $S^2 = T^2\cdot T^2$ and $T^2 = S$, i.e. the T gate defined below is the square root of S.
This gate is Clifford (level 2 of the Clifford hierarchy): conjugation by S maps every Pauli to another Pauli.
For qutrits ($d = 3$), up to the global-phase convention of Yeh & van de Wetering (ζ⁸ with ζ = e^{2πi/9}), this agrees with their Definition 3.
References:
- Gheorghiu, V. (2014). Standard form of qudit stabilizer groups. Physics Letters A, 378(30–31), 2016–2021.
- Yeh, L. & van de Wetering, J. (2023). Completeness of the ZH-calculus.