SWAP Test for an Arbitrary Number of Quantum States

TL;DR

This paper introduces a recursive quantum algorithm that generalizes the SWAP test to efficiently measure overlaps among multiple quantum states using minimal resources.

Contribution

It presents a novel recursive circuit design for the multi-state SWAP test, reducing resource requirements and enabling simultaneous overlap measurements.

Findings

Requires O(m) controlled-swap gates for m states

Uses O(log m) ancillary qubits

Allows simultaneous measurement of multiple state overlaps

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 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.