rotations 1 qubit 3 parameters
Arbitrary Rotation
Rotation by angle $\theta$ about the Bloch-sphere axis $\vec{n}$.
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.
$$ \mathrm{Rot}(\vec{n}, \theta) = e^{-i\frac{\theta}{2}(n_x X + n_y Y + n_z Z)} $$
Writing $c = \cos\frac{\theta}{2}$ and $s = \sin\frac{\theta}{2}$, the matrix is
$$ \mathrm{Rot}(\vec{n}, \theta) = \begin{bmatrix} c - i n_z s & (-i n_x - n_y)\, s \\ (-i n_x + n_y)\, s & c + i n_z s \end{bmatrix} $$
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.
The unitary, drawn
Every entry of the matrix as a disc. The area is the transition probability $|U_{jk}|^2$; the colour and the hand are the phase. Rows are outputs $\langle j|$, columns inputs $|k\rangle$, in the computational basis. Each parameter has a slider; play runs through the family along the path of the Quirk link ($\theta = 2\pi t,\ \mathbf{n} = (\sqrt{2}\cos 2\pi t,\ \sqrt{2}\sin 2\pi t,\ 1)/\sqrt{3}$).
Phase
Disc area $= |U_{jk}|^2$. Hover an entry for its value, or take the JSON.
SDK support
How to call this gate in each framework. Hover a ⓘ for a note; the support matrix compares every gate at once.
| SDK | Name |
|---|---|
| Qiskit | qiskit.circuit.library.RVGate ⓘ |
| PennyLane | pennylane.Rot ⓘ |
| Cirq | — ⓘ |
| Q# | Std.Intrinsic.R ⓘ |
| PyQuil | — |
| Braket | — ⓘ |
| BQSKit | bqskit.ir.gates.PauliGate ⓘ |
| Qibo | — ⓘ |
| pytket | — ⓘ |
| Stim | — |
| OpenQASM | — |