A global phase is a common complex phase factor multiplying every component of a quantum state. Because it does not change measurement outcomes, a single-qubit experiment cannot distinguish X from −X. section 5.1.1
A controlled unitary can make the phase relative: prepare a suitable superposition so that only some components acquire the phase, then distinguish the resulting states through measurement. section 5.1.1
Distinguishing X from −X
Prepare two qubits in |++⟩ by applying H to each, apply the unknown gate as a controlled operation, and measure. The controlled X and controlled −X operations produce distinguishable outcomes because their phases affect different components. section 5.1.1
