Gates API Reference
All gates are imported from qutritium.gates:
from qutritium.gates import H3, X01, Rx01, CSUM
Every gate inherits from qutritium.gates.Gate and provides:
.matrix() → NDArray[np.complex128] — the unitary matrix (3×3 or 9×9).label → str — gate name.num_qutrits → int — 1 or 2.params → tuple[float, ...] — gate parameters (empty for fixed gates).inverse() → Gate — the conjugate transpose gate.is_unitary() → bool — verify unitarity
Single-qutrit fixed gates
All matrices are 3×3 unitary.
Subspace Pauli-X
| Gate |
Action |
Matrix |
X01() |
\(\|0\rangle \leftrightarrow \|1\rangle\) |
\(\lambda_1 + \|2\rangle\langle 2\|\) |
X02() |
\(\|0\rangle \leftrightarrow \|2\rangle\) |
\(\lambda_4 + \|1\rangle\langle 1\|\) |
X12() |
\(\|1\rangle \leftrightarrow \|2\rangle\) |
\(\lambda_6 + \|0\rangle\langle 0\|\) |
Subspace Pauli-Y
| Gate |
Generator |
Y01() |
\(\lambda_2 + \|2\rangle\langle 2\|\) |
Y02() |
\(\lambda_5 + \|1\rangle\langle 1\|\) |
Y12() |
\(\lambda_7 + \|0\rangle\langle 0\|\) |
Subspace Pauli-Z
| Gate |
Matrix |
Z01() |
\(\mathrm{diag}(1, -1, 1)\) |
Z02() |
\(\mathrm{diag}(1, 1, -1)\) |
Z12() |
\(\mathrm{diag}(1, 1, -1)\) |
Other fixed gates
| Gate |
Description |
I3() |
3×3 identity |
XPlus() |
Cyclic shift \(\|i\rangle \to \|i{+}1 \bmod 3\rangle\). \(X_+^3 = I\) |
XMinus() |
Inverse cyclic shift. \((X_+)^\dagger\) |
H3() |
Qutrit Hadamard / DFT \(F_3/\sqrt{3}\) |
S3() |
\(\mathrm{diag}(1, 1, \omega)\). \(S^3 = I\) |
T3() |
\(\mathrm{diag}(1, \omega^{1/3}, \omega^{-1/3})\). \(T^9 = I\) |
UFT() |
Fourier-related gate \(U_{FT}\) |
Single-qutrit parametric gates
Subspace rotations
Convention: \(R_{\text{axis},ij}(\theta) = \exp(-i\theta/2 \cdot \sigma_{\text{axis},ij})\).
| Gate |
Parameters |
Subspace |
Rx01(θ) |
1 |
{\(\|0\rangle\), \(\|1\rangle\)} |
Rx02(θ) |
1 |
{\(\|0\rangle\), \(\|2\rangle\)} |
Rx12(θ) |
1 |
{\(\|1\rangle\), \(\|2\rangle\)} |
Ry01(θ) |
1 |
{\(\|0\rangle\), \(\|1\rangle\)} |
Ry02(θ) |
1 |
{\(\|0\rangle\), \(\|2\rangle\)} |
Ry12(θ) |
1 |
{\(\|1\rangle\), \(\|2\rangle\)} |
Rz01(φ) |
1 |
{\(\|0\rangle\), \(\|1\rangle\)} |
Rz02(φ) |
1 |
{\(\|0\rangle\), \(\|2\rangle\)} |
Rz12(φ) |
1 |
{\(\|1\rangle\), \(\|2\rangle\)} |
Generalized rotations
\(g_{ij}(\theta, \phi) = \exp\bigl(-i\tfrac{\theta}{2}(\cos\phi\, X_{ij} + \sin\phi\, Y_{ij})\bigr)\)
Native gate in trapped-ion implementations (Ringbauer et al., Nat. Phys. 18, 1053, 2022).
| Gate |
Parameters |
G01(θ, φ) |
2 |
G02(θ, φ) |
2 |
G12(θ, φ) |
2 |
Diagonal phase gate
| Gate |
Parameters |
Matrix |
Ud(φ₁, φ₂, φ₃) |
3 |
\(\mathrm{diag}(e^{i\phi_1}, e^{i\phi_2}, e^{i\phi_3})\) |
Two-qutrit gates
All matrices are 9×9 unitary.
| Gate |
Action |
Reference |
CSUM() |
\(\|c, t\rangle \to \|c, (t{+}c) \bmod 3\rangle\) |
Wang et al. (2020) |
CSUMDag() |
\(\|c, t\rangle \to \|c, (t{-}c) \bmod 3\rangle\) |
CSUM inverse |
CPhase() |
\(\|c, t\rangle \to \omega^{ct}\|c, t\rangle\) |
Qutrit CZ analogue |
CNOT3() |
Legacy CNOT from v0.0.1 |
Equivalent to CSUM |
SWAP3() |
\(\|a, b\rangle \to \|b, a\rangle\) |
Self-inverse |
References
- Bertlmann, R. A. & Krammer, P. (2008). arXiv:0806.1174 — Gell-Mann matrices \(\lambda_1\)–\(\lambda_8\)
- Ringbauer, M. et al. (2022). Nat. Phys. 18, 1053 — Generalized rotations, T3 gate
- Wang, Y. et al. (2020). Front. Phys. 8, 589504 — CSUM, qutrit gate compilation
- Vitanov, N. V. (2012). Phys. Rev. A 85, 032331 — SU(3) decomposition