A promise problem to decide whether two given pure quantum states are close or far apart
A standard tool is the swap test, which receives two unentangled pure states |φ1⟩ and |φ2⟩ and answers “equal” with probability (1+|⟨φ1|φ2⟩|2)/2 and “not equal” with probability (1−|⟨φ1|φ2⟩|2)/2.  See [BCWW01] (Figure 1 shows the circuit).

[BCWW01] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf.  Quantum fingerprinting.  Physical Review Letters, 87, article 167902, Sept. 2001.  DOI: 10.1103/PhysRevLett.87.167902.  A preprint version: arXiv:quant-ph/0102001v1.