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arxiv: 2603.29695 · v3 · pith:QBJZRFNInew · submitted 2026-03-31 · 🪐 quant-ph

Probes of chaos over the Clifford group and approach to Haar values

Stefano Cusumano , Gianluca Esposito , Alioscia Hamma This is my paper

Pith reviewed 2026-05-13 23:33 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum chaosisospectral twirlingHaar measureGaussian Unitary EnsembleT-doped circuitsClifford groupToric Codeprobes of chaos
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The pith

Expectation values of chaos probes match Haar distribution moments for GUE and T-doped circuits but deviate for GDE.

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

The paper applies isospectral twirling, which fixes a Hamiltonian spectrum while selecting eigenvectors randomly, to track how probes of chaos behave as bases transition from stabilizer to Haar-random. It models the transition explicitly with T-doped random quantum circuits and computes probe averages over spectra drawn from the Gaussian Diagonal Ensemble for non-chaotic cases and the Gaussian Unitary Ensemble for chaotic cases. The same probes are evaluated on the Toric Code Hamiltonian to obtain concrete reference values. The central finding is that the averages adhere to Haar moments precisely in the chaotic ensembles and T-doped circuits while deviating in the non-chaotic ensemble.

Core claim

Using isospectral twirling on fixed spectra with random eigenvectors, the expectation values of the probes adhere to moments of the Haar distribution for chaotic behavior modeled by the Gaussian Unitary Ensemble and T-doped circuits, while deviating for the non-chaotic Gaussian Diagonal Ensemble and showing specific behavior on the Toric Code Hamiltonian.

What carries the argument

Isospectral twirling, which fixes the Hamiltonian spectrum and selects eigenvectors randomly, applied across T-doped circuits and random-matrix ensembles.

If this is right

  • The probes distinguish chaotic from non-chaotic regimes by whether their averages match Haar moments.
  • T-doped circuits provide a tunable family of states interpolating between Clifford and fully random bases.
  • Gaussian Diagonal Ensemble spectra yield systematically different probe values from Gaussian Unitary Ensemble spectra.
  • The Toric Code supplies a concrete, non-random benchmark for probe behavior outside the two ensembles.

Where Pith is reading between the lines

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

  • The same twirling procedure could be applied to experimental Hamiltonians to test whether observed probe values indicate chaos without requiring full eigenstate tomography.
  • Varying the doping level in T-doped circuits might map a quantitative threshold where probes cross from non-Haar to Haar statistics.
  • If the modeling is accurate, spectral statistics alone with randomized eigenvectors suffice to reproduce the probe signatures of chaos without simulating full time evolution.

Load-bearing premise

Randomly chosen eigenvectors from a fixed spectrum accurately capture the statistical signatures of chaotic dynamics in actual quantum systems.

What would settle it

Numerical or experimental computation of the same probe expectation values in a many-body system whose eigenstates are known to be fully chaotic and comparison against the reported GUE and Haar averages.

Figures

Figures reproduced from arXiv: 2603.29695 by Alioscia Hamma, Gianluca Esposito, Stefano Cusumano.

Figure 1
Figure 1. Figure 1: Comparison of g2(t), g˜3(t) and g4(t) averaged over the GDE (panel a) and the GUE (panel b) for d = 216. One can observe in both cases how the Clifford spectral form factor g˜3(t) shares the same equilibrium value of g2(t). Moreover, one can observe the suppression of oscillations in the case of GUE. Furthermore, we find that, in order to meaningfully average over the Clifford group, one has to impose the … view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of Clifford 2a and T-doped 2b circuits. As the addition of even a single T-gate to the circuit is sufficient to bring one out of the Clifford orbit, one is interested in what happens to the moment of an operator O. It turns out [85] that one can write an explicit expression for the 4-th moment of an operator averaged over the T-doped Clifford circuits. This expression depends explicitly on bot… view at source ↗
Figure 3
Figure 3. Figure 3: Plot of the normalized versions of g2(t) GDE ( 3a) and g2(t) GUE ( 3b) for d = 2N . As shown in [1, 135], in order to compute the envelope curves one has to first compute the large d limit, and then suppress all the oscillating terms. One can consider for instance the two point-spectral form factor g2(t) averaged over the GDE and GUE. One has: g2(t) GUE = d + (dr1(t))2 − dr2(t) (57) Taking the large d limi… view at source ↗
Figure 4
Figure 4. Figure 4: The Toric code. A qubit lives on each of the [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the functions g Tor 2 (t) (panel a), g˜3(t) (panel 5b) and g Tor 4 (t) (panel c) for different lattice size N. One can once again observe how the behavior of g˜3(t) is in between the ones of g2(t) and g4(t). IV. PROBES OF CHAOS Let us finally turn the the object of this paper, probes of chaos. In this section we are going to illustrate the results for the two main probes for which one can observe a… view at source ↗
Figure 6
Figure 6. Figure 6: Plot of the Loschmidt echo of the second kind [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plot of the Loschnmidt echo for the Toric Code for different lattice size. One can observe a [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plot of the OTOC4 averaged over the GDE ( 8a)and the GUE( 8b) for d = 212. One can observe how in both cases averaging over the Clifford or Unitary group only influences the asymptotic value, but not the equilibration time. For both spectral families the asymptotic value of the Haar average is O(d −2 ), while for the Clifford average it goes as O(d) −1 . 10 3 10 2 10 1 10 0 10 1 t 10 2 10 1 10 0 O T O C To… view at source ↗
Figure 9
Figure 9. Figure 9: Plot of the OTOC4 for the Toric code for different lattice size. Similarly as for the Loschmidt echo, one can once again observe a different minimum whern the average is taken over the Clifford (O(d −1 )) or the Unitary (O(d −2 )) group. Thus after averaging one has: ⟨OTOC4⟩G = d−1Tr h T(1423)(A⊗1,1 ⊗ B †⊗1,1 )R (4) G (V)i (89) Assuming the operators A, B to be non overlapping Pauli strings, one can comput… view at source ↗
Figure 10
Figure 10. Figure 10: Plot of the bound on the tripartite info [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Tripartite Mutual Information for the Toric Code over different lattice size. [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Plot of the normalized versions of g3(t) GDE (a) and g3(t) GUE (b) for d = 2N . Notice how in both cases the curve essentially does not depend on the dimension and the asymptotic value is reached extremely quickly. The main difference between g3(t) GDE and g3(t) GUE is that the latter shortly becomes negative before reaching its asymptotic value. 45 [PITH_FULL_IMAGE:figures/full_fig_p045_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Plot of the normalized four points spectral form factor [PITH_FULL_IMAGE:figures/full_fig_p047_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Plot of the function g˜3(t) averaged over the GDE (panel a) and the GUE(panel b) for d = 2N . For the permutation of the form T(ij) we get: ⃗qT(ij) = Trh T(ij)QV ⊗2,2 stab i = d −2X P Trh P VstabP V † stabi Tr[P Vstab] Trh P V † stabi = d −2 X P ∈Pab Trh P VstabP V † stabi Tr[P Vstab] Trh P V † stabi = d −2 X P ∈Pab Trh VstabV † stabi Tr[P Vstab] Trh P V † stabi = d −1 X P ∈Pab Tr[P Vstab] Trh P V † stabi… view at source ↗
Figure 15
Figure 15. Figure 15: Comparison between g2(t), g˜3(t), g4(t) for the Toric Code for lattice size N = 2 and N = 4 Appendix G: Probes of chaos 1. Loschmidt echo We can write the expressions for the Loschmidt echo of the second kind in matrix notation, which are more useful for practical calculations. We have: ⟨L2(t)⟩U = ⃗cW · L⃗ H 2 (G1) ⟨L2(t)⟩C = ⃗qW+L⃗ Q 2 + ⃗q⊥W−L⃗ Q⊥ 2 (G2) ⟨L2(t)⟩Ck = ⃗t Ξ kL⃗ Q 2 + ⃗t Γ (k)L⃗ H 2 +⃗b · L… view at source ↗
Figure 16
Figure 16. Figure 16: Plot of the bound on the 2-Rényi entropy averaged over the GDE( [PITH_FULL_IMAGE:figures/full_fig_p060_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Plot of the entanglement entropy S2 for the Toric code for different lattice size. Let us set PurAB = Tr h T(13)(24)(V⊗2,2 )(ψ ⊗2 ⊗ T (A) (12)) i . For the average value under isospectral twirling, we can use the usual matrix notation and obtain: ⟨PurAB⟩U = ⃗c W ⃗pH (G63) ⟨PurAB⟩C = ⃗q W+⃗pQ + ⃗q⊥ W−⃗pQ⊥ (G64) ⟨PurAB⟩Ck = ⃗t Ξ k ⃗pQ + ⃗t Γ (k) ⃗pH +⃗b · ⃗pH (G65) where we have defined: (⃗pH)π = Tr h T(12)… view at source ↗
Figure 18
Figure 18. Figure 18: Plot of the 2 norm of coherence averaged over the GDE( [PITH_FULL_IMAGE:figures/full_fig_p068_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Coherence CB for the Toric Code for different lattice size. Once again we can cast the averages in matrix form for all three scenarios. We have: ⟨CB(ψV )⟩U = 1 − ⃗t WD⃗ H B (G140) ⟨CB(ψV )⟩C = 1 −  ⃗qW+D⃗ Q B + ⃗q⊥W−(D⃗ Q⊥ B )  (G141) ⟨CB⟩CT = 1 −  ⃗t Ξ kD⃗ Q B + ⃗t Γ (k)D⃗ H B +⃗b · D⃗ H B  (G142) 68 [PITH_FULL_IMAGE:figures/full_fig_p068_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Plot of the WYD skew information for the case [PITH_FULL_IMAGE:figures/full_fig_p071_20.png] view at source ↗
read the original abstract

Chaotic behavior of quantum systems can be characterized by the adherence of the expectation values of given probes to moments of the Haar distribution. In this work, we analyze the behavior of several probes of chaos using a technique known as Isospectral Twirling [1]. This consists in fixing the spectrum of the Hamiltonian and picking its eigenvectors at random. Here, we study the transition from stabilizer bases to random bases according to the Haar measure by T-doped random quantum circuits. We then compute the average value of the probes over ensembles of random spectra from Random Matrix Theory, the Gaussian Diagonal Ensemble and the Gaussian Unitary Ensemble, associated with non-chaotic and chaotic behavior respectively. We also study the behavior of such probes over the Toric Code Hamiltonian.

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

1 major / 1 minor

Summary. The manuscript claims that probes of quantum chaos adhere to the moments of the Haar distribution when applied to T-doped Clifford circuits approaching Haar-random bases and to isospectral twirling over GUE spectra (modeling chaotic behavior), while deviating for GDE spectra (modeling non-chaotic behavior) and exhibiting specific behavior for the Toric Code Hamiltonian.

Significance. If the results hold, the work provides a controlled numerical approach to benchmarking chaos indicators via isospectral twirling and T-doped circuits, which could help distinguish ergodic from non-ergodic regimes in quantum many-body systems and circuits. The use of standard RMT ensembles as external benchmarks is a clear strength, as is the concrete model for the stabilizer-to-Haar transition.

major comments (1)
  1. [Sections defining the ensembles and isospectral twirling procedure] The central claim that GDE and GUE under isospectral twirling distinguish non-chaotic from chaotic regimes (and that deviations from Haar moments signal non-chaos) is load-bearing. This modeling choice fixes the spectrum while drawing eigenvectors from the Haar measure, decoupling them; however, physical non-chaotic Hamiltonians typically exhibit correlated, non-random eigenvectors (e.g., due to integrability or localization), so the observed GDE deviations may be an artifact of the artificial construction rather than a robust signature. The manuscript should include direct comparisons with actual integrable or localized Hamiltonians to test this.
minor comments (1)
  1. [Abstract] The abstract refers to 'several probes of chaos' and 'the average value of the probes' without defining the probes or reporting any numerical values, error bars, or specific adherence metrics, which hinders immediate assessment of the results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our work and for the constructive major comment. We address the point in detail below and outline the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Sections defining the ensembles and isospectral twirling procedure] The central claim that GDE and GUE under isospectral twirling distinguish non-chaotic from chaotic regimes (and that deviations from Haar moments signal non-chaos) is load-bearing. This modeling choice fixes the spectrum while drawing eigenvectors from the Haar measure, decoupling them; however, physical non-chaotic Hamiltonians typically exhibit correlated, non-random eigenvectors (e.g., due to integrability or localization), so the observed GDE deviations may be an artifact of the artificial construction rather than a robust signature. The manuscript should include direct comparisons with actual integrable or localized Hamiltonians to test this.

    Authors: We appreciate the referee's careful analysis of the isospectral twirling construction and its implications for distinguishing chaotic and non-chaotic regimes. The procedure is designed precisely to isolate the role of the spectrum by sampling eigenvectors from the Haar measure, allowing a controlled comparison between ensembles whose only difference is the level statistics (Poissonian for GDE versus Wigner-Dyson for GUE). The fact that probes reach Haar moments under GUE but deviate under GDE indicates that spectral statistics alone are sufficient to drive the distinction in this controlled setting. We acknowledge that physical non-chaotic Hamiltonians generally possess correlated eigenvectors, which the GDE does not capture. To address this directly, we will revise the manuscript to include an explicit comparison of the probe values obtained for the Toric Code Hamiltonian (a concrete integrable model with structured, non-random eigenvectors) against the corresponding GDE results. This addition will clarify the extent to which the GDE deviations are representative of physical non-chaotic behavior and will be presented in a new subsection discussing the limitations and strengths of the ensemble modeling. We believe these changes will reinforce rather than weaken the central claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external RMT benchmarks and standard approximations used

full rationale

The paper computes probe expectation values directly on isospectral-twirled Hamiltonians whose spectra are drawn from the standard GDE and GUE ensembles of random-matrix theory (treated as independent external models for non-chaotic vs. chaotic regimes) and whose eigenvectors are chosen Haar-randomly. The T-doped circuit analysis invokes the known approximation of the Clifford+T group to the Haar measure, without any fitting of parameters to the probes themselves or redefinition of the target Haar moments. The Toric Code is examined as a concrete Hamiltonian instance. No equation reduces a claimed prediction to a fitted input by construction, and the single citation to the isospectral-twirling technique supplies a computational method rather than a load-bearing uniqueness theorem or ansatz that would force the central result. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard assumptions of random matrix theory for modeling chaotic and non-chaotic spectra plus the validity of isospectral twirling as a proxy for chaotic dynamics; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Random matrix ensembles (GDE, GUE) correctly capture non-chaotic and chaotic spectral statistics of quantum Hamiltonians
    Invoked when averaging probes over these ensembles to represent non-chaotic versus chaotic behavior
  • domain assumption Isospectral twirling with Haar-random eigenvectors models the approach to chaotic dynamics
    Core technique used to study the transition from stabilizer to random bases

pith-pipeline@v0.9.0 · 5421 in / 1468 out tokens · 51299 ms · 2026-05-13T23:33:12.407823+00:00 · methodology

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