Pauli spectrum and nonstabilizerness of random fermionic Gaussian states
Abstract
We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. For the Haar ensemble of Majorana Gaussian states we derive closed finite-N expressions for the average stabilizer purities and reconstruct the Pauli spectrum exactly, as a mixture of products of beta-distributed variables resolved by Majorana weight. The stabilizer entropies, and their filtered versions, freeze at Rényi index qc=2: for q>2 the magic density is 1/(q−1), controlled by rare weight-two Majorana strings. Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer q as coefficient integrals over SU(2q), with dimension independent of N, and evaluate the q=2 case exactly. We establish that the frozen density describes typical states in the Majorana and half-filled number-conserving ensembles and relate these ensembles to the corresponding SYK2 ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction κ<κ⋆G≃0.7613 of their modes in the thermodynamic limit, remaining magical even beyond the 2/3 threshold for Haar-random states in the full Hilbert space.
Related
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BibTeX
@misc{turkeshi2026pauli,
title = {{Pauli spectrum and nonstabilizerness of random fermionic Gaussian states}},
author = {Xhek Turkeshi and Piotr Sierant and Poetri Sonya Tarabunga},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15739},
eprint = {2609.15739},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15739},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pauli spectrum and nonstabilizerness of random fermionic Gaussian states
AU - Xhek Turkeshi
AU - Piotr Sierant
AU - Poetri Sonya Tarabunga
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15739
UR - https://arxiv.org/abs/2609.15739
AB - We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. For the Haar ensemble of Majorana Gaussian states we derive closed finite-N expressions for the average stabilizer purities and reconstruct the Pauli spectrum exactly, as a mixture of products of beta-distributed variables resolved by Majorana weight. The stabilizer entropies, and their filtered versions, freeze at Rényi index q_c=2: for q>2 the magic density is 1/(q−1), controlled by rare weight-two Majorana strings. Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer q as coefficient integrals over SU(2q), with dimension independent of N, and evaluate the q=2 case exactly. We establish that the frozen density describes typical states in the Majorana and half-filled number-conserving ensembles and relate these ensembles to the corresponding SYK_2 ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction κ<κ_⋆^G≃0.7613 of their modes in the thermodynamic limit, remaining magical even beyond the 2/3 threshold for Haar-random states in the full Hilbert space.
ER -