Optimal Qubit Purification and Unitary Schur Sampling via Random SWAP Tests

TL;DR

This paper introduces a simple, efficient protocol using random SWAP tests that achieves optimal qubit purification and Schur sampling, matching the performance of complex transforms with elementary operations.

Contribution

The authors demonstrate that random SWAP tests can perform optimal qubit purification and invariant state sampling, simplifying previous methods based on the Schur transform.

Findings

Achieves optimal fidelity comparable to the Schur transform.

Requires approximately n log n SWAP tests for effective purification.

Provides a lossless, elementary method for permutation-invariant quantum information tasks.

Abstract

The goal of qubit purification is to combine multiple noisy copies of an unknown pure quantum state to obtain one or more copies that are closer to the pure state. We show that a simple protocol based solely on random SWAP tests achieves the same fidelity as the Schur transform, which is optimal. This protocol relies only on elementary two-qubit SWAP tests, which project a pair of qubits onto the singlet or triplet subspaces, to identify and isolate singlet pairs, and then proceeds with the remaining qubits. For a system of $n$ qubits, we show that after approximately $T \approx n \ln n$ random SWAP tests, a sharp transition occurs: the probability of detecting any new singlet decreases exponentially with $T$. Similarly, the fidelity of each remaining qubit approaches the optimal value given by the Schur transform, up to an error that is exponentially small in $T$. More broadly, this protocol achieves what is known as weak Schur sampling and unitary Schur sampling with error $\epsilon$, after only $2n \ln(n \epsilon^{-1})$ SWAP tests. That is, it provides a lossless method for extracting any information invariant under permutations of qubits, making it a powerful subroutine for tasks such as quantum state tomography and metrology.