The magic cost of fermionic randomness
Abstract
Sampling-based quantum advantage raises a basic resource question: how much non-Gaussianity is needed to reproduce the random-matrix statistics often associated with quantum chaos? Passive fermionic linear optics preserves Gaussian structure, and even Haar-random mode mixing fails to anticoncentrate for Slater inputs. We show that overcoming this obstruction requires an extensive input resource. For any pure half-filled input state, the one-body defect ν, defined as the summed occupation variance of the natural orbitals, must scale linearly with system size. For products of bounded-size blocks on 4N modes, this condition is also sufficient, with normalized collision moment asymptotic to 2N/ν, compared with the Porter-Thomas value 2. For four-mode products, the full Porter-Thomas moment hierarchy is recovered precisely when the defect density tends to its maximum. Assuming average-case hardness, approximate classical sampling remains hard near maximal density, with an error tolerance that increases with the density. Two-copy number fluctuations measure ν directly, and the same quantity lower-bounds the combined cost of non-Gaussian resource modes and parity-preserving local gates in deterministic exact preparation. Fermionic randomness therefore carries an extensive magic cost, with a quantitative link between sampling statistics, preparation resources, and an experimentally accessible witness.
Related
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BibTeX
@misc{cao2026magic,
title = {{The magic cost of fermionic randomness}},
author = {Chenfeng Cao and Yifan Tang and Jens Eisert},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05093},
eprint = {2610.05093},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05093},
}
RIS
TY - UNPB
TI - The magic cost of fermionic randomness
AU - Chenfeng Cao
AU - Yifan Tang
AU - Jens Eisert
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05093
UR - https://arxiv.org/abs/2610.05093
AB - Sampling-based quantum advantage raises a basic resource question: how much non-Gaussianity is needed to reproduce the random-matrix statistics often associated with quantum chaos? Passive fermionic linear optics preserves Gaussian structure, and even Haar-random mode mixing fails to anticoncentrate for Slater inputs. We show that overcoming this obstruction requires an extensive input resource. For any pure half-filled input state, the one-body defect ν, defined as the summed occupation variance of the natural orbitals, must scale linearly with system size. For products of bounded-size blocks on 4N modes, this condition is also sufficient, with normalized collision moment asymptotic to 2N/ν, compared with the Porter-Thomas value 2. For four-mode products, the full Porter-Thomas moment hierarchy is recovered precisely when the defect density tends to its maximum. Assuming average-case hardness, approximate classical sampling remains hard near maximal density, with an error tolerance that increases with the density. Two-copy number fluctuations measure ν directly, and the same quantity lower-bounds the combined cost of non-Gaussian resource modes and parity-preserving local gates in deterministic exact preparation. Fermionic randomness therefore carries an extensive magic cost, with a quantitative link between sampling statistics, preparation resources, and an experimentally accessible witness.
ER -