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Pauli spectrum and nonstabilizerness of random fermionic Gaussian states

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Xhek Turkeshi, Piotr Sierant, Poetri Sonya Tarabunga

Quantum Physics · Submitted 14 Sep 2026 · 111 references

Pith's verdict: First exact Pauli spectrum for Gaussian states, freezing transition, and clear sampling caveat—rigorous and important.

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 15, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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arXiv ID2609.15739 (version 1)
Pith ID2VOOEGV2, Pith's permanent record of this paper and its review
CitationPiotr Sierant, Poetri Sonya Tarabunga, Xhek Turkeshi. "Pauli spectrum and nonstabilizerness of random fermionic Gaussian states." arXiv:2609.15739 (2026). https://pith.science/paper/2VOOEGV2
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The paper

The paper claims

Random fermionic Gaussian states have a hierarchy of Pauli expectation values that causes stabilizer enropies to freeze at a critical Rényi index, with implications for sampling and magic under partial trace.

The reading

This paper studies the magic (nonstabilizerness) of random fermionic Gaussian states, which are the simplest class of quantum many-body states beyond product states. Magic quantifies how difficult it is to simulate a quantum state with classical computers. The authors exactly compute the full distribution of Pauli expectation values (the Pauli spectrum) for these states, something previously only approximated. They find that the stabilizer entropy, a standard magic measure, freezes at a critical value of the Rényi index. This means that for large system sizes, the magic per mode is fixed at a number les than one, instead of growing. They also show that when you discard a fraction of the quantum modes, the reduced state remains magical up to a larger fraction than for generic random states (which are the hardest to simulate). This is counterintuitive: Gaussian states are classically simulable, yet they keep magic better. The paper also explains why some numerical estimates of magic for these states give wrong answers if they use too few samples.
Extended reading notes

Core claim

The filtered stabilizer entropy density of random Maiorana Gaussian states freezes: it equals 1 for q≤2 and 1/(q-1) for q>2 (eq 2.4). The exact ensemble-averaged Pauli spectrum is a mixture of products of independent beta-distributed variables (eq 4.11).

Load-bearing premise

The subleading corrections for number-conserving ensembles (section 6.4) assume that the leading contribution to the filtered stabilizer purity at q>2 comes from fixed-weight boundary sectors; the paper acknowledges this as an unproven assumption for growing-weight sectors. However, the leading exponential rate is proved independently in section 8 for half-filling via quenched averages, so the central claim does not depend on this assumption.

Original 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 R\'enyi 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 $\mathrm{SU}(2q)$, 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 $\mathrm{SYK}_2$ ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction $\kappa<\kappa_\star^{\rm G}\simeq0.7613$ of their modes in the thermodynamic limit, remaining magical even beyond the $2/3$ threshold for Haar-random states in the full Hilbert space.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,956 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.87 · human participants 0.01 · controlled comparison 0.44 · randomized assignment 0.05 · states quantitative result 0.99 · compares with prior results 0.85 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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 R\'enyi 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 $\mathrm{SU}(2q)$, 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 $\mathrm{SYK}_2$ ground states. Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction $\kappa<\kappa_\star^{\rm G}\simeq0.7613$ of their modes in the thermodynamic limit, remaining magical even beyond the $2/3$ threshold for Haar-random states in the full Hilbert space.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We characterize the magic of random fermionic Gaussian states through their Pauli spectrum and their stabilizer entropies. contribution 1.00 0.93 0.12 asserted 0.99
s1 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. result 0.74 0.96 0.07 asserted 1.00
s2 The stabilizer entropies, and their filtered versions, freeze at R\'enyi index $q_c=2$: for $q>2$ the magic density is $1/(q-1)$, controlled by rare weight-two Majorana strings. result 1.00 0.91 0.19 asserted 1.00
s3 Resolving these contributions requires exponentially many characteristic samples, making direct sampling estimates of the stabilizer entropies inefficient in this frozen regime. limitation 0.52 0.76 0.04 asserted 1.00
s4 For Haar-random Gaussian states at fixed filling, we express the averaged stabilizer purities at positive integer $q$ as coefficient integrals over $\mathrm{SU}(2q)$, with dimension independent of $N$, and evaluate the $q=2$ case exactly. method 0.48 0.96 0.06 asserted 1.00
s5 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 $\mathrm{SYK}_2$ ground states. result 0.80 0.94 0.13 asserted 1.00
s6 Finally, we show that typical fermionic Gaussian states retain certified magic after discarding any fixed fraction $\kappa<\kappa_\star^{\rm G}\simeq0.7613$ of their modes in the thermodynamic limit, remaining magical even beyond the $2/3$ threshold for Haar-random states in the full Hilbert space. result 0.99 0.95 0.52 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Squared Pauli spectrum of the Majorana ensemble at N = 8, conditioned on nontrivial even strings and plotted as z = − log2 Y . Symbols: every such Pauli expectation evaluated from the covariance matrix in each of 1024 independently drawn states, using Wick’s theorem. Dashed curve: the analytical density eq. (4.15), with the sector density from eq. (B.2). In figure 1, we retain only the 2 2N−1 − 2 nontrivial even strings and normalize their distribution to one. The plotted density of z = − log2 Y is therefore gN (z) = (ln 2) 2−z 2 2N−1 − 2 N X−1 k=1
Figure 2. Filtered magic density in the Majorana ensemble. Solid curves: exact annealed result eq. (4.5) at N = 4, 6, 8, 10, 12, 32, 128. Circles and crosses: annealed and quenched averages at N = 4, 6, 8, 10, 12, respectively, calculated from all nontrivial even strings at integer q. Error bars are standard errors across states, propagated through the logarithm for the annealed values. The larger sizes show the approach to the dashed thermodynamic limit eq. (5.10). 5.2 High-q asymptotics of the filtered SRE For q > 2, the unfiltered stabilizer purity in eq. (5.4) is asymptotically saturated…
Figure 3. Signed Pauli spectrum of half-filled GUE SYK2 ground states, conditioned on nontrivial even strings. Symbols: all such Pauli expectations at N = 8, 10, 12, evaluated from the occupied￾orbital covariance matrices using Wick’s theorem. Dashed curve: the analytical bulk approximation eq. (7.13), obtained by averaging Gaussian amplitudes over an effective exponential variance, with bN fixed by Parseval’s identity. Both axes are rescaled by bN . The corresponding filtered moments are eζ exp q = Γ(2q + 1) 2 q  D − 2 nN q−1 , q > 0. (7.15) The low-weight analysis predicts enhanced

Figures reproduced from arXiv:2609.15739 by the authors.

References

Discussion

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