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arxiv: 2606.13606 · v1 · pith:FIVZSOTFnew · submitted 2026-06-11 · 🪐 quant-ph · cond-mat.mes-hall· cond-mat.stat-mech

Diffusive Dynamics of Nonstabilizerness

Zhenyu Xiao , Shinsei Ryu This is my paper

Pith reviewed 2026-06-27 06:23 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallcond-mat.stat-mech
keywords nonstabilizernessstabilizer Rényi entropydiffusive dynamicsU(1) symmetryrandom circuitstensor networksmany-body dynamics
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The pith

Nonstabilizerness approaches its random-state value diffusively with a 1/t gap closure in symmetric systems.

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

This paper shows that in one-dimensional quantum circuits with U(1) symmetry, the stabilizer Rényi entropy, which quantifies nonstabilizerness, approaches the value for fully random states with a gap that decreases as 1/t at late times. The dynamics are computed using a four-replica tensor network in the thermodynamic limit. The same 1/t scaling appears in a nonintegrable Ising chain with energy conservation. Together with a hydrodynamic argument, this places the late-time nonstabilizerness dynamics in a diffusive universality class. This matters for understanding how quantum resources are built up under symmetry constraints in many-body systems.

Core claim

The disorder-averaged dynamics of the stabilizer Rényi entropy in U(1)-symmetric one-dimensional random circuits is captured by a four-replica tensor network evaluated via S4-adapted iTEBD, revealing that the entropy gap to the random-state value closes as 1/t at late times and thereby identifying a diffusive universality class for nonstabilizerness; the identical scaling is verified in an energy-conserving nonintegrable Ising chain.

What carries the argument

Four-replica tensor network for the averaged dynamics of the stabilizer Rényi entropy, evaluated in the thermodynamic limit with S4-adapted iTEBD, interpreted via a hydrodynamic argument for the diffusive scaling.

If this is right

  • The generation of nonstabilizerness under U(1) symmetry is controlled by diffusion at long times.
  • The universality class is independent of microscopic details as long as the symmetry is present.
  • This scaling provides a concrete way to estimate how close Hamiltonian dynamics gets to maximal nonstabilizerness.
  • The framework extends the hydrodynamic perspective to nonstabilizerness generation in addition to other quantities.
  • Approximate Haar-random states can be reached via symmetry-preserving dynamics with this known relaxation rate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Symmetry may generically slow the buildup of nonstabilizerness relative to symmetry-broken cases.
  • Similar diffusive scaling could appear for other measures of quantum resources in conserved-charge systems.
  • Extensions to higher dimensions or different symmetries would test the robustness of this universality class.
  • The results suggest designing circuits or Hamiltonians to achieve target nonstabilizerness levels in finite time.

Load-bearing premise

The hydrodynamic argument used to link the observed 1/t scaling to diffusive behavior.

What would settle it

Finding that the stabilizer Rényi entropy gap decays with a power other than -1 or exponentially in the late-time regime of the U(1)-symmetric random circuits.

Figures

Figures reproduced from arXiv: 2606.13606 by Shinsei Ryu, Zhenyu Xiao.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Disorder-averaged four-replica transfer network for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) U(1) random circuit: inverse gap ∆ [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

Symmetries shape the quantum-information dynamics of many-body systems, but their effect on nonstabilizerness, the resource complementary to entanglement, is less understood. We compute the stabilizer R\'enyi entropy, a measure of nonstabilizerness, in $\mathrm{U}(1)$-symmetric one-dimensional random circuits. The disorder-averaged dynamics is captured by a four-replica tensor network, which we evaluate by $S_4$-adapted infinite time-evolving block decimation (iTEBD) directly in the thermodynamic limit. Together with a hydrodynamic argument, our results identify a diffusive universality class for the late-time approach of nonstabilizerness to its random-state value, with the stabilizer R\'enyi entropy gap closing as $1/t$. The same scaling is verified in an energy-conserving nonintegrable Ising chain. More broadly, our framework provides a hydrodynamic perspective on nonstabilizerness generation and offers insight into the design of approximate Haar-random states in Hamiltonian dynamics.

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 / 1 minor

Summary. The paper computes the stabilizer Rényi entropy (SRE) in U(1)-symmetric 1D random circuits using a four-replica tensor network evaluated by S4-adapted iTEBD in the thermodynamic limit. Combined with a hydrodynamic argument, it claims that nonstabilizerness approaches its random-state value in a diffusive universality class, with the SRE gap closing as 1/t; the same scaling is verified numerically in an energy-conserving nonintegrable Ising chain. The framework is presented as providing a hydrodynamic perspective on nonstabilizerness generation.

Significance. If the hydrodynamic mapping is valid, the work identifies a concrete diffusive universality class for late-time nonstabilizerness relaxation tied to U(1) charge diffusion, extending hydrodynamic ideas to a resource complementary to entanglement. Strengths include the direct thermodynamic-limit iTEBD treatment of the replica network and the independent verification in a Hamiltonian system; these establish the 1/t scaling itself as a robust numerical result.

major comments (2)
  1. [Hydrodynamic argument (following the iTEBD results)] The identification of a diffusive universality class rests on the hydrodynamic argument that maps the SRE gap closure to U(1) charge diffusion (invoked in the abstract and used to interpret the iTEBD data). Without an explicit derivation showing the coupling of the SRE operator to the diffusive mode—e.g., via the effective hydrodynamic equations or the four-replica tensor network structure—any mismatch in the assumed coupling would alter the predicted power law, undermining the universality-class claim.
  2. [Section describing the four-replica tensor network and iTEBD implementation] The abstract states that the four-replica tensor network encodes the disorder-averaged U(1)-symmetric circuit dynamics, yet the manuscript does not provide the explicit contraction rules or the S4 symmetry adaptation that would allow independent verification that the 1/t scaling is not an artifact of the replica construction.
minor comments (1)
  1. [Introduction / abstract] Notation for the stabilizer Rényi entropy gap and its relation to the random-state value should be defined with an equation number at first use to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism. The two major comments identify areas where the manuscript can be strengthened by providing more explicit technical details. We address each point below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Hydrodynamic argument (following the iTEBD results)] The identification of a diffusive universality class rests on the hydrodynamic argument that maps the SRE gap closure to U(1) charge diffusion (invoked in the abstract and used to interpret the iTEBD data). Without an explicit derivation showing the coupling of the SRE operator to the diffusive mode—e.g., via the effective hydrodynamic equations or the four-replica tensor network structure—any mismatch in the assumed coupling would alter the predicted power law, undermining the universality-class claim.

    Authors: We agree that the hydrodynamic argument is presented at a conceptual level in the current version and that an explicit derivation of the coupling would strengthen the universality-class claim. The argument relies on the fact that, in the four-replica space, the SRE operator is the lowest-dimension operator that is odd under the diagonal U(1) action while being invariant under the S4 permutation symmetry; this forces it to couple to the conserved charge density whose dynamics are diffusive. In the revision we will add a dedicated subsection deriving the effective hydrodynamic equations for the replica-averaged SRE gap, showing that the leading relaxation is controlled by the charge diffusion constant. This addition will make the power-law prediction fully traceable to the symmetry and conservation laws. revision: partial

  2. Referee: [Section describing the four-replica tensor network and iTEBD implementation] The abstract states that the four-replica tensor network encodes the disorder-averaged U(1)-symmetric circuit dynamics, yet the manuscript does not provide the explicit contraction rules or the S4 symmetry adaptation that would allow independent verification that the 1/t scaling is not an artifact of the replica construction.

    Authors: We accept that the current manuscript describes the construction at a high level without spelling out the gate-averaged contraction rules or the precise S4-adapted tensor representation. In the revised version we will add an appendix that (i) lists the explicit contraction rules for the four-replica U(1)-symmetric gates, (ii) shows the reduced tensor network after S4 projection, and (iii) specifies the iTEBD bond-dimension and truncation parameters used in the thermodynamic-limit calculations. These additions will enable independent reproduction of the 1/t scaling. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper computes the stabilizer Rényi entropy gap via iTEBD on a four-replica tensor network in the thermodynamic limit and verifies the 1/t scaling in an independent energy-conserving Ising chain. The hydrodynamic argument is invoked only to interpret the observed scaling as belonging to a diffusive class; no quoted equation or self-citation reduces the scaling itself to a fitted input, self-definition, or load-bearing prior result by the same authors. The numerical evidence is therefore independent of the interpretive step, satisfying the criteria for a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that disorder-averaged dynamics of the stabilizer Rényi entropy are captured by a four-replica tensor network and that a hydrodynamic argument correctly predicts the 1/t gap closure.

axioms (2)
  • domain assumption Disorder-averaged dynamics of the stabilizer Rényi entropy can be represented by a four-replica tensor network
    Invoked to evaluate the dynamics in the thermodynamic limit via iTEBD.
  • domain assumption A hydrodynamic argument yields the 1/t scaling of the stabilizer Rényi entropy gap
    Used to classify the late-time approach to the random-state value.

pith-pipeline@v0.9.1-grok · 5699 in / 1327 out tokens · 25586 ms · 2026-06-27T06:23:14.625577+00:00 · methodology

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Forward citations

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