Anti-symmetrization is cheap and useful
Abstract
We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-r qudit state ρ=∑i=1r λi |φi⟩ ⟨φi|, prepare a copy of the state proportional to Πantir (|φ1⟩⊗ …⊗|φr⟩), with Πantir the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses Õ(r / λmin) copies of ρ, where λmin is the minimum non-zero eigenvalue of ρ. This algorithm works even in the streaming setting with small working memory, and it uses only simple quantum operations like controlled-SWAPs and constantly many layers of single-qubit gates. As an application of this streaming anti-symmetrization procedure, we give a reduction from gapped mixed-state one-way state generators to pure-state one-way state generators. Then, using tools from Schur-Weyl duality, we give and analyze a second (non-streaming) anti-symmetrization procedure that uses O(r / λmin) copies of ρ, which then serves as a building block in new algorithms for weak Schur sampling and unitary Schur sampling, outperforming existing approaches in some regimes of number of copies n and local dimension d, as well as in an algorithm for optimal purity amplification for qudit states.
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BibTeX
@misc{watts2026antisymmetrization,
title = {{Anti-symmetrization is cheap and useful}},
author = {Adam Bene Watts and John Bostanci and Matthias C. Caro and Daniel Grier and Sihan Liu and Gaurav Mahajan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06737},
eprint = {2610.06737},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06737},
}
RIS
TY - UNPB
TI - Anti-symmetrization is cheap and useful
AU - Adam Bene Watts
AU - John Bostanci
AU - Matthias C. Caro
AU - Daniel Grier
AU - Sihan Liu
AU - Gaurav Mahajan
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06737
UR - https://arxiv.org/abs/2610.06737
AB - We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-r qudit state ρ=∑_i=1^r λ_i |φ_i⟩ ⟨φ_i|, prepare a copy of the state proportional to Π_anti^r (|φ_1⟩⊗ …⊗|φ_r⟩), with Π_anti^r the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses Õ(r / λ_min) copies of ρ, where λ_min is the minimum non-zero eigenvalue of ρ. This algorithm works even in the streaming setting with small working memory, and it uses only simple quantum operations like controlled-SWAPs and constantly many layers of single-qubit gates. As an application of this streaming anti-symmetrization procedure, we give a reduction from gapped mixed-state one-way state generators to pure-state one-way state generators. Then, using tools from Schur-Weyl duality, we give and analyze a second (non-streaming) anti-symmetrization procedure that uses O(r / λ_min) copies of ρ, which then serves as a building block in new algorithms for weak Schur sampling and unitary Schur sampling, outperforming existing approaches in some regimes of number of copies n and local dimension d, as well as in an algorithm for optimal purity amplification for qudit states.
ER -