rotations 1 qubit 3 parameters
U3
General single-qubit gate parameterized by three Euler angles.
The $U_3$ gate is a universal single-qubit rotation. It can be decomposed as $U_3(\theta, \phi, \lambda) = R_z(\phi) R_y(\theta) R_z(\lambda)$.
$$ U_3(\theta, \phi, \lambda) = \begin{bmatrix} \cos\frac{\theta}{2} & -e^{i\lambda}\sin\frac{\theta}{2} \\ e^{i\phi}\sin\frac{\theta}{2} & e^{i(\phi+\lambda)}\cos\frac{\theta}{2} \end{bmatrix} $$
Properties
- Generates all of $\mathsf{U}(2)$ up to a global phase.
- Equivalent to an Euler-angle decomposition around $z$ and $y$ axes.
Usage
- Standard single-qubit parameterization in compilation and variational circuits.
- Maps directly to hardware-native rotations on many platforms.
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 ($\theta = 2\pi t,\ \phi = 4\pi t,\ \lambda = 6\pi t$).
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.UGate ⓘ |
| PennyLane | pennylane.U3 |
| Cirq | — ⓘ |
| Q# | — |
| PyQuil | pyquil.simulation.matrices.U ⓘ |
| Braket | braket.circuits.gates.U |
| BQSKit | bqskit.ir.gates.U3Gate |
| Qibo | qibo.gates.U3 ⓘ |
| pytket | pytket.circuit.OpType.U3 ⓘ |
| Stim | — |
| OpenQASM | stdgates.inc: u3 ⓘ |