Global Phase Gate

Multiplies the entire state by a phase factor, invisible on its own but consequential when placed under a control.

Definition

The global phase gate multiplies the whole quantum state by a complex number e^{iγ} of unit modulus. On its own it changes no measurement outcome and no expectation value: |ψ⟩ and e^{iγ}|ψ⟩ are physically identical states. It is often omitted from circuit diagrams entirely.

Matrix

When global phase becomes physical

A global phase turns into an observable relative phase the moment the gate is controlled. Conditioning a global-phase-γ operation on a qubit applies e^{iγ} only when that control is |1⟩ — which is exactly a phase gate on the control. This is why the leading factor in the ABC decomposition of controlled-U cannot be discarded.

• Standalone global phase: unobservable, safe to drop
• Controlled global phase: becomes a P(γ) phase gate on the control
• Relative phase between two branches of a superposition: always observable via interference

import numpy as np
def global_phase(gamma, n=1):
    return np.exp(1j*gamma)*np.eye(2**n)

Bookkeeping in compilers

Simulators and compilers track global phase carefully even though it is invisible, because a subroutine may later be controlled or embedded in a larger interference. The distinction between U1 and RZ, or between a gate and its up-to-phase equivalent, is exactly a global-phase difference. See U1 phase and controlled-U.

In gate-identity proofs, two circuits are called equal only up to global phase unless the phase is explicitly required, so stating whether phase matters is part of a correct equivalence.