SWAP test for an arbitrary number of quantum states

Abstract

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 $$|\langle \phi _i, \phi _j\rangle |^2$$ of m arbitrary pure states $${|{\phi _1\ldots \phi _m}\rangle }$$ . Our construction relies on a pairing unitary that generates a superposition state where every pair of input states is labeled by a basis state formed by the ancillaries. By implementing a simple genetic algorithm, we give numerical evidence indicating that our method of labeling each pair of inputs using CSWAP gates is optimal up to $$m=8$$ . Potential applications of the new circuits in the context of quantum machine learning are discussed.