Unitary Schur Sampling of Qudits via Random SWAP Tests: Hunt for Antisymmetry
Abstract
Schur sampling is a fundamental primitive for extracting permutation-invariant information from many-body quantum systems and has broad applications in quantum information science. Circuit-based implementations typically rely on coherent representation-theoretic operations such as Clebsch-Gordan transforms, generalized phase estimation, or quantum Fourier transforms over the symmetric group. We show that unitary Schur sampling of arbitrary permutation-invariant mixed states on n d-dimensional qudits can instead be implemented using only pairwise SWAP tests, namely, two-qudit projective measurements onto the symmetric and antisymmetric subspaces. Our protocol achieves error ε in diamond distance using O(nmin{n,d}3log2(n/ε)) random SWAP tests. The central idea is to repeatedly identify and extract the largest antisymmetric subsystem. This turns unitary Schur sampling into a search for antisymmetry and gives the algorithm a natural interpretation as a stochastic traversal of a Young diagram. The protocol preserves the SU(d)-irrep state while preparing a canonical pure state in the multiplicity subsystem. As an application, we consider quantum purity amplification (QPA), in which multiple noisy copies of a pure quantum state are combined to produce a state of higher purity. We show that the optimal purified state can be obtained by retaining a single designated output qudit and discarding the rest.
Related
- Robustness of quantum spectrum estimation: weak Schur sampling under noisy inputs
- Averages of traces and ratios of characteristic polynomials of random unitary matrices
- Edgeworth expansions of extreme eigenvalue distributions for random unitary ensembles
- The Lq-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths
- Random unitary circuits with constant spectral gap
BibTeX
@misc{brahmachari2026unitary,
title = {{Unitary Schur Sampling of Qudits via Random SWAP Tests: Hunt for Antisymmetry}},
author = {Shrigyan Brahmachari and David Jakab and Henry D. Pfister and Iman Marvian},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02103},
eprint = {2610.02103},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02103},
}
RIS
TY - UNPB
TI - Unitary Schur Sampling of Qudits via Random SWAP Tests: Hunt for Antisymmetry
AU - Shrigyan Brahmachari
AU - David Jakab
AU - Henry D. Pfister
AU - Iman Marvian
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02103
UR - https://arxiv.org/abs/2610.02103
AB - Schur sampling is a fundamental primitive for extracting permutation-invariant information from many-body quantum systems and has broad applications in quantum information science. Circuit-based implementations typically rely on coherent representation-theoretic operations such as Clebsch-Gordan transforms, generalized phase estimation, or quantum Fourier transforms over the symmetric group. We show that unitary Schur sampling of arbitrary permutation-invariant mixed states on n d-dimensional qudits can instead be implemented using only pairwise SWAP tests, namely, two-qudit projective measurements onto the symmetric and antisymmetric subspaces. Our protocol achieves error ε in diamond distance using O(nmin{n,d}^3log^2(n/ε)) random SWAP tests. The central idea is to repeatedly identify and extract the largest antisymmetric subsystem. This turns unitary Schur sampling into a search for antisymmetry and gives the algorithm a natural interpretation as a stochastic traversal of a Young diagram. The protocol preserves the SU(d)-irrep state while preparing a canonical pure state in the multiplicity subsystem. As an application, we consider quantum purity amplification (QPA), in which multiple noisy copies of a pure quantum state are combined to produce a state of higher purity. We show that the optimal purified state can be obtained by retaining a single designated output qudit and discarding the rest.
ER -