Symmetric functions of qubits in an unknown basis

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.