SWAP test and Hong-Ou-Mandel effect are equivalent

We show that the Hong-Ou-Mandel effect from quantum optics is equivalent to the SWAP test, a quantum information primitive which compares two arbitrary states. We first derive a destructive SWAP test that does not need the ancillary qubit that appears in the usual quantum circuit. Then we study the Hong-Ou-Mandel effect for two photons meeting at a beam splitter and prove it is, in fact, an optical implementation of the destructive SWAP test. This result offers both an interesting simple realization of a powerful quantum information primitive and an alternative way to understand and analyze the Hong-Ou-Mandel effect.