U3 universal single-qubit gate

Definition

$$U_3(\theta,\varphi,\lambda) = \begin{pmatrix} \cos(\theta/2) & -e^{i\lambda}\sin(\theta/2) \\ e^{i\varphi}\sin(\theta/2) & e^{i(\varphi+\lambda)}\cos(\theta/2) \end{pmatrix}$$

Every $2\times 2$ unitary can be written, up to an overall global phase, as some $U_3(\theta,\varphi,\lambda)$: $\theta$ sets the polar angle on the Bloch sphere and $\varphi,\lambda$ set the two phases needed to fix the axis and the relative phase.

Euler-angle decomposition

$$U_3(\theta,\varphi,\lambda) = e^{i(\varphi+\lambda)/2}\, R_Z(\varphi)\, R_Y(\theta)\, R_Z(\lambda)$$

This is the ZYZ Euler decomposition used by every gate-synthesis pass that targets a native $\{R_Z, R_Y\}$ or $\{R_Z, R_X\}$ gate set: any single-qubit unitary is first written as $U_3$, then compiled into three native rotations plus a tracked global phase.

Special cases

$$U_3(\pi,0,\pi) = X, \qquad U_3\!\left(\tfrac{\pi}{2},0,\pi\right) \doteq H, \qquad U_3(\theta,0,0) \doteq R_Y(\theta), \qquad U_3(0,0,\lambda) = P(\lambda)$$

(the $\doteq$ marks equality up to an unobservable global phase). Because it subsumes every other single-qubit gate in this catalog as a special case, $U_3$ is the natural "universal" record to compare them against.

Practical note

Modern Qiskit exposes this gate as the generic $U$ gate (qc.u(theta, phi, lam, qubit)) — the historical name u3 referred to the same three-parameter definition before Qiskit's gate-naming cleanup.