quantum gate directory
Arbitrary Rotation
Symbol
$R_{\mathbf{n}}$
Description:
Rotation by angle $\theta$ about the Bloch-sphere axis $\vec{n}$.
Alternate notations:
- $\mathrm{Rot}(\vec{n}, \theta)$
- $R_{\vec{n}}(\theta)$
SDK Support
| SDK | Name |
|---|---|
| Qiskit | — |
| PennyLane |
pennylane.Rot
|
| Cirq | — |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
The $\mathrm{Rot}(\vec{n}, \theta)$ gate performs a rotation by angle $\theta$ about the unit vector $\vec{n} = (n_x, n_y, n_z)$ on the Bloch sphere.
$$ \mathrm{Rot}(\vec{n}, \theta) = e^{-i\frac{\theta}{2}(n_x X + n_y Y + n_z Z)} $$
$$ \mathrm{Rot}(\vec{n}, \theta) = \begin{bmatrix} \cos\frac{\theta}{2} - i n_z \sin\frac{\theta}{2} & (-i n_x - n_y)\sin\frac{\theta}{2} \\ (-i n_x + n_y)\sin\frac{\theta}{2} & \cos\frac{\theta}{2} + i n_z \sin\frac{\theta}{2} \end{bmatrix} $$
Properties
- Recovers $R_x$, $R_y$, and $R_z$ when $\vec{n}$ is aligned with the axes.
- Generates all single-qubit unitaries up to global phase.
Usage
- Arbitrary axis rotations in pulse-level control.
- Compact parameterization for variational circuits and compilation.
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