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arxiv: 2603.08841 · v2 · pith:GVA6XRM3new · submitted 2026-03-09 · 🪐 quant-ph

Universal Non-stabilizerness Dynamics Across Quantum Phase Transitions

Andr\'as Grabarits , Adolfo del Campo This is my paper

Pith reviewed 2026-05-25 06:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-stabilizernessquantum magicstabilizer Rényi entropyPauli spectrumquantum phase transitionsslow drivingtransverse-field Ising modelKitaev model
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The pith

Stabilizer Rényi entropies and Pauli spectrum cumulants scale universally as power laws with driving rate in slow processes across quantum phase transitions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the dynamics of quantum magic, or non-stabilizerness, when a quantum system undergoes slow driving through a quantum phase transition. It establishes that stabilizer Rényi entropies and the cumulants of the Pauli spectrum display universal power-law scaling tied to the driving rate. The logarithmic Pauli spectrum approaches a Gaussian distribution, which means the underlying Pauli spectrum values follow a lognormal distribution. These scaling laws and distributional properties are verified exactly in the transverse-field Ising model and through approximations in long-range Kitaev models. A reader would care because this provides a way to predict and control a key computational resource in driven critical systems.

Core claim

In slow driving across quantum phase transitions, the stabilizer Rényi entropies and the cumulants of the Pauli spectrum exhibit universal power-law scaling with the driving rate. The logarithmic Pauli spectrum is asymptotically Gaussian, implying a lognormal distribution for the Pauli spectrum values. These findings are shown by exact calculations in the transverse-field Ising model and analytical approximations in long-range Kitaev models.

What carries the argument

Universal power-law scaling of stabilizer Rényi entropies and Pauli spectrum cumulants with driving rate, together with asymptotic Gaussianity of the logarithmic Pauli spectrum.

Load-bearing premise

The driving must be slow enough for the reported power-law scaling to emerge, and the transverse-field Ising and long-range Kitaev models must represent generic behavior across quantum phase transitions.

What would settle it

Driving the transverse-field Ising model slowly across its phase transition at different rates and measuring no power-law dependence in the stabilizer Rényi entropy or Pauli cumulants.

Figures

Figures reproduced from arXiv: 2603.08841 by Adolfo del Campo, Andr\'as Grabarits.

Figure 1
Figure 1. Figure 1: FIG. 1. Stabilizer R´enyi entropies versus [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. First three cumulants of the logarithmic Pauli spec [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Universal time evolution of quantum magic relative [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Pauli spectrum statistics in the LRKM ( [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Distribution of the logarithmic Pauli spectrum values in the TFIM following the predicted Gaussian character ( [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Cumulants of the logarithmic Pauli spectrum for [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Stabilizer R´enyi entropy for quantum magic, [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) Stabilizer R´enyi entropy for quantum magic, [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a) Stabilizer R´enyi entropy for quantum magic, [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a) Time evolution of the relative SRE in the LRKM in the dynamical scaling regime for short-range pairing, [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Stabilizer R´enyi entropy in the TFIM as a function of [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Stabilizer R´enyi entropies as a function of [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
read the original abstract

Quantum magic, or non-stabilizerness, is an important quantum resource that characterizes computational power beyond classically simulable Clifford operations and is therefore essential for achieving quantum advantage. While previous studies have explored non-stabilizerness dynamics in random circuits and under time-independent generators, here we extend the study of its universal dynamics to time-dependent driving across quantum phase transitions. In particular, we show that the stabilizer R\'enyi entropies and the cumulants of the Pauli spectrum exhibit universal power-law scaling with the driving rate in slow processes. Moreover, we show that the logarithmic Pauli spectrum is asymptotically Gaussian, implying a lognormal distribution for the Pauli spectrum values. Our results are explicitly demonstrated by exact results in the transverse-field Ising model and by analytical approximations in long-range Kitaev models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies non-stabilizerness (quantum magic) dynamics under slow time-dependent driving across quantum phase transitions. It claims that stabilizer Rényi entropies and cumulants of the Pauli spectrum exhibit universal power-law scaling with driving rate, and that the logarithmic Pauli spectrum becomes asymptotically Gaussian (implying lognormal Pauli-spectrum statistics). These statements are supported by exact calculations in the transverse-field Ising chain and analytical approximations in long-range Kitaev chains.

Significance. If the reported scaling and Gaussianity are robust beyond the specific models, the work would extend non-stabilizerness studies from random circuits and static Hamiltonians to driven critical systems, potentially connecting to Kibble-Zurek-type universality. The exact TFIM results constitute a concrete, verifiable strength; the analytic approximations in the Kitaev family add further support within the same universality class.

major comments (2)
  1. [Demonstrations (TFIM exact results and Kitaev approximations)] The central claim of universality (title, abstract, and concluding statements) rests exclusively on exact results for the 1D transverse-field Ising model and analytic approximations for long-range Kitaev chains, both integrable and in the same universality class. No renormalization-group argument, general derivation, or numerical checks in non-integrable or higher-dimensional systems are provided; this directly affects whether the power-law exponents and Gaussianity can be regarded as universal.
  2. [Slow-process regime statements] The slow-driving regime required for the power-law scaling to emerge is invoked without quantitative bounds on the driving rate or error estimates showing when the reported scaling crosses over to faster-driving behavior. This assumption is load-bearing for the scaling claims but is not delimited in the derivations.
minor comments (2)
  1. [Introduction and methods] Notation for the stabilizer Rényi entropies and Pauli-spectrum cumulants should be introduced with explicit definitions and consistent symbols before the scaling statements.
  2. [Figures showing power-law scaling] Figure captions and axis labels for the scaling plots should include the precise driving-rate range and system sizes used, to allow direct comparison with the analytic expressions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we respond point-by-point to the major comments and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Demonstrations (TFIM exact results and Kitaev approximations)] The central claim of universality (title, abstract, and concluding statements) rests exclusively on exact results for the 1D transverse-field Ising model and analytic approximations for long-range Kitaev chains, both integrable and in the same universality class. No renormalization-group argument, general derivation, or numerical checks in non-integrable or higher-dimensional systems are provided; this directly affects whether the power-law exponents and Gaussianity can be regarded as universal.

    Authors: We acknowledge that the demonstrations are confined to models in the Ising universality class. The power-law scaling originates from the Kibble-Zurek mechanism that governs defect production across the critical point, which is expected to be universal within this class independent of integrability. The asymptotic Gaussianity of the logarithmic Pauli spectrum follows from the central-limit behavior of the independent critical-mode contributions. We agree that explicit verification in non-integrable or higher-dimensional systems would strengthen the claim of broader universality. We will revise the abstract, title, and conclusions to qualify the universality statement as applying within the Ising class and to discuss the KZ-based reasoning for its expected generality. revision: partial

  2. Referee: [Slow-process regime statements] The slow-driving regime required for the power-law scaling to emerge is invoked without quantitative bounds on the driving rate or error estimates showing when the reported scaling crosses over to faster-driving behavior. This assumption is load-bearing for the scaling claims but is not delimited in the derivations.

    Authors: We thank the referee for highlighting this point. In the revised manuscript we will add explicit quantitative bounds on the driving rate, derived from the Kibble-Zurek scaling and the closing of the gap at criticality, together with estimates of the crossover to faster-driving regimes and associated error estimates. revision: yes

Circularity Check

0 steps flagged

No circularity; scaling and Gaussianity are computed outcomes in specific models

full rationale

The paper states that stabilizer Rényi entropies and Pauli-spectrum cumulants exhibit universal power-law scaling, and that the log-Pauli spectrum is asymptotically Gaussian, with these statements framed as results demonstrated by exact calculations in the transverse-field Ising model and analytical approximations in long-range Kitaev models. No equations or claims reduce a prediction to a fitted input by construction, no self-citation is invoked as load-bearing justification for uniqueness or ansatz, and the derivation chain consists of model-specific computations rather than self-referential definitions. The scope of universality is a separate question of evidence strength, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption that the transverse-field Ising model and long-range Kitaev models are representative for extracting universal non-stabilizerness scaling under slow driving; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The transverse-field Ising model and long-range Kitaev models capture the universal non-stabilizerness dynamics across quantum phase transitions
    Invoked when the abstract states that results are demonstrated by exact results in the Ising model and analytical approximations in Kitaev models.

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discussion (0)

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