SWAP Test for an Arbitrary Number of Quantum States

We develop a recursive algorithm to generalize the quantum SWAP test for an arbitrary number m of quantum states requiring O(m) controlled-swap (CSWAP) gates and O(log m) ancillary qubits. We construct a quantum circuit able to simultaneously measure overlaps of m arbitrary pure states. Our construction relies on a pairing unitary that generates a superposition state where every pair of input states is labelled by a basis state formed by the ancillaries.