U2 Gate

A two-parameter single-qubit gate fixing the polar angle at ninety degrees, the workhorse for superposition creation.

Definition

The U2(φ, λ) gate is the special case of U3 with θ = π/2. It creates an equal-weight superposition with adjustable relative phases, so it generalizes the Hadamard. Historically it was a hardware-friendly primitive because it required exactly one physical microwave pulse of fixed area.

Matrix

Setting φ = 0 and λ = π gives the Hadamard. Any U2 turns a basis state into a balanced superposition; the phases φ and λ steer where on the equator of the Bloch sphere the resulting state lands.

The pulse-count hierarchy

In the older IBM basis gate model the single-qubit gates were ranked by physical pulses: U1 needed zero pulses (a virtual frame change), U2 needed one, and U3 needed two. Compilers therefore preferred U2 over U3 wherever the polar angle happened to be π/2, saving a pulse.

import numpy as np
def u2(phi, lam):
    r = 1/np.sqrt(2)
    return r*np.array([[1, -np.exp(1j*lam)],
                       [np.exp(1j*phi), np.exp(1j*(phi+lam))]])

Modern status

Contemporary hardware often exposes only virtual-Z plus a single fixed √X (SX) pulse; U2 and U3 are then synthesized from these. But U2 remains a clean conceptual label for equal-superposition-with-phase operations and is still emitted by many circuit libraries. See U3 and native gate sets.