SWAP Test for an Arbitrary Number of Quantum States

Understand

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.