Symmetric functions of qubits in an unknown basis

TL;DR

This paper demonstrates how to reconstruct the Hamming weight of a permuted and locally transformed n-qubit state with high probability, enabling the computation of symmetric functions, generalizing the swap test.

Contribution

It introduces algorithms for determining the Hamming weight and symmetric functions of qubits under unknown local unitaries and permutations, extending the swap test methodology.

Findings

Successfully reconstructs |x| with high probability

Computes symmetric functions like parity efficiently

Algorithms are near-optimal in performance

Abstract

Consider an n qubit computational basis state corresponding to a bit string x, which has had an unknown local unitary applied to each qubit, and whose qubits have been reordered by an unknown permutation. We show that, given such a state with Hamming weight |x| at most n/2, it is possible to reconstruct |x| with success probability 1 - |x|/(n-|x|+1), and thus to compute any symmetric function of x. We give explicit algorithms for computing whether or not |x| is at least t for some t, and for computing the parity of x, and show that these are essentially optimal. These results can be seen as generalisations of the swap test for comparing quantum states.