quantum gate directory
U3
Symbol
$U_3$
Description:
General single-qubit gate parameterized by three Euler angles.
Alternate notations:
- $U_3(\theta, \phi, \lambda)$
- $\mathrm{U3}(\theta, \phi, \lambda)$
- $U(\theta, \phi, \lambda)$
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.UGate
ⓘ
|
| PennyLane |
pennylane.U3
|
| Cirq | — |
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
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.
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