U1 Phase Gate

A single-parameter diagonal gate that phases the one-state, implementable as a zero-cost virtual frame change.

Definition

The U1(λ) gate, equivalent to the phase gate P(λ), applies a phase e^{iλ} to |1⟩ and leaves |0⟩ unchanged. It is diagonal and is the θ = 0 case of U3. Special values give named gates: U1(π) = Z, U1(π/2) = S, U1(π/4) = T.

Matrix

Virtual-Z: the zero-cost gate

On superconducting hardware a Z-type phase can be applied by shifting the phase reference of every subsequent pulse rather than by emitting a physical pulse. This makes U1 effectively free and error-free — a software bookkeeping operation. Compilers push all diagonal single-qubit phase into virtual-Z gates and remove them from the pulse schedule.

The difference between U1 and RZ is a global phase: U1(λ) = e^{iλ/2}·RZ(λ). Global phase is unobservable on a single qubit, but on a control line it becomes a relative phase and must be tracked. See global phase.

import numpy as np
def u1(lam):
    return np.array([[1, 0], [0, np.exp(1j*lam)]])

Uses

Because diagonal gates commute, chains of U1 gates merge into one by summing angles, and they slide freely past control lines and other diagonal gates during optimization. The controlled version is exactly the controlled-phase gate CP. See controlled-phase.

The quantum Fourier transform, phase estimation, and Grover phase oracles are all built from U1/CP phases, so this humble gate underlies much of quantum algorithm design.