quantum gate directory

The $U_2$ gate is the two-parameter member of the legacy OpenQASM 2 family $u_1, u_2, u_3$: a rotation with the angle frozen at $\pi/2$, sandwiched between two independent $z$ rotations:

$$ \begin{align*} U_2(\phi, \lambda) & = R_z(\phi) \, R_y(\pi/2) \, R_z(\lambda) \\ & = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -\mathrm{e}^{i\lambda} \\ \mathrm{e}^{i\phi} & \mathrm{e}^{i(\phi+\lambda)} \end{bmatrix} \end{align*} $$

Historically it exposed IBM's hardware calibration in gate form: the $R_y(\pi/2)$ is the one physical microwave pulse, and $\phi$, $\lambda$ are virtual $z$ rotations implemented as frame changes that cost nothing.

Special values

Properties

• The fixed-$\theta$ slice of $U_3$: $U_2(\phi, \lambda) = U_3(\pi/2, \phi, \lambda)$. The one-parameter sibling $u_1$ is simply the phase gate.
• Since $\phi$ and $\lambda$ are independent, $U_2$ generally includes a net $z$ rotation — unlike the phased rotation $R(\theta, \phi)$, whose frame change cancels. The two families overlap exactly on the $\lambda = -\phi$ slice.
• Any single-qubit unitary is expressible with two $U_2$ gates and virtual phases, which is why one calibrated $\pi/2$ pulse per gate was enough for universality.

Usage

• Legacy: OpenQASM 2 and early Qiskit compiled everything to $u_1, u_2, u_3$. Modern stacks target $\{R_z, \sqrt{X}, X\}$ or pulse-level natives instead, and Qiskit has deprecated the gate in favor of $U(\theta, \phi, \lambda)$. Kept here as reference material — the gate still appears throughout older papers, textbooks, and QASM files.