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arxiv: 2606.03956 · v1 · pith:YKDE2ZFCnew · submitted 2026-06-02 · 🪐 quant-ph · cond-mat.stat-mech

Operator spreading in random circuits with orthogonal or symplectic symmetry

Zhiyang Tan , Piet W. Brouwer This is my paper

Pith reviewed 2026-06-28 09:20 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords operator spreadingrandom quantum circuitsorthogonal symmetrysymplectic symmetryPauli stringsbutterfly velocitydomain wallHaar measure
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The pith

Random circuits with orthogonal or symplectic symmetry relax Pauli-string weights to a ternary structure with finite-width domain walls.

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

The paper examines operator spreading in quantum circuits built from gates drawn from ensembles that are invariant under orthogonal or symplectic transformations. It shows that the averaged weights assigned to Pauli strings settle into three distinct values instead of the two values found in unitary-invariant circuits. The boundary separating unscrambled from scrambled regions stays spread out even when the gates are fully random according to the Haar measure on those groups. A split within the orthogonal ensemble between its two components produces different lower bounds on the speed of spreading.

Core claim

For circuits whose two-qubit gates are drawn from orthogonal-invariant or symplectic-invariant ensembles, the ensemble-averaged Pauli-string weights relax to a ternary-valued structure. The domain wall separating trivial and scrambled regions has finite width even for Haar-random gates. The negative-determinant sector of the orthogonal group exhibits a non-zero lower bound for butterfly velocity, and for qudit size two this velocity can exceed the Haar value.

What carries the argument

The domain wall in the spatial profile of ensemble-averaged Pauli-string weights.

If this is right

  • Butterfly velocity is bounded below by a positive constant in the negative-determinant orthogonal ensemble for any gate distribution.
  • For qubits the butterfly velocity can exceed the value obtained from Haar-random unitary circuits.
  • Domain walls retain finite width for Haar-distributed gates taken from the orthogonal or symplectic ensembles.
  • Symplectic-invariant circuits exhibit the same ternary relaxation pattern as orthogonal-invariant circuits.

Where Pith is reading between the lines

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

  • Additional discrete symmetries such as time reversal may produce structured relaxation of operator weights beyond the continuous symmetry groups studied here.
  • The ternary pattern could appear in other models whose gates respect only a subgroup of the full unitary group.
  • Small-system simulations could directly test whether the three-valued structure survives beyond the ensemble average.

Load-bearing premise

Two-qubit gates are drawn independently from the orthogonal- or symplectic-invariant ensembles and the ensemble averaging remains well-defined at finite qudit dimension.

What would settle it

Numerical computation of the spatial profile of averaged Pauli weights after many layers of orthogonal gates, checking whether exactly three weight values appear and whether the transition region width stays finite.

Figures

Figures reproduced from arXiv: 2606.03956 by Piet W. Brouwer, Zhiyang Tan.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p032_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
read the original abstract

We investigate operator spreading in random quantum circuits with gates drawn from orthogonal-invariant or symplectic-invariant ensembles, revealing several key distinctions from the well-studied unitary-invariant case. We find that the ensemble-averaged Pauli-string weights relax to a ternary-valued structure, instead of the binary structure of unitary-invariant circuits. For orthogonal- or symplectic-invariant circuits, the domain wall separating trivial and scrambled regions has a finite width even for Haar-random gates, whereas domain walls are sharp for Haar-distributed random unitary circuits. We further find a fundamental dichotomy between random circuits with two-qubit gates from the two disconnected components of the orthogonal group: While the butterfly velocity for the special orthogonal ensemble lies between zero and the Haar value, the negative-determinant sector exhibits a non-zero lower bound for any gate distribution. Moreover, for qudit size $q=2$, the butterfly velocity can exceed that of the Haar-random ensemble.

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

0 major / 2 minor

Summary. The manuscript examines operator spreading in random quantum circuits with two-qubit gates drawn from orthogonal-invariant or symplectic-invariant ensembles (including disconnected components). It claims that ensemble-averaged Pauli-string weights relax to a ternary-valued structure (distinct from the binary structure of unitary-invariant circuits), that the domain wall has finite width even for Haar-random gates (unlike sharp walls in the unitary case), and that butterfly velocity exhibits a dichotomy: the special orthogonal ensemble yields velocities between zero and the Haar value, while the negative-determinant sector has a non-zero lower bound for any gate distribution, with the q=2 case allowing velocities exceeding the Haar value.

Significance. If the central claims hold, the work provides a clear extension of operator-spreading results to symmetry-restricted ensembles, identifying qualitative distinctions (ternary weights, finite-width domain walls) that are absent in the unitary case. The explicit treatment of the two components of the orthogonal group and the resulting bounds on butterfly velocity constitute a substantive contribution to the literature on random circuits and scrambling. No machine-checked proofs or parameter-free derivations are indicated, but the symmetry-based distinctions are presented as direct consequences of the ensemble definitions.

minor comments (2)
  1. [§3] The definition of the domain wall and the precise averaging procedure over the disconnected components should be stated explicitly in the main text (near the discussion of the ternary structure) to allow direct reproduction of the finite-width result.
  2. [§2] Notation for the Pauli-string weights and the butterfly velocity extraction could be unified between the orthogonal and symplectic cases to improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee's summary correctly captures the central claims regarding ternary Pauli weights, finite-width domain walls, and the dichotomy in butterfly velocities between the two components of the orthogonal group. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract presents claims about ternary Pauli weights, finite domain-wall width, and butterfly-velocity bounds as direct consequences of symmetry-restricted ensembles (orthogonal/symplectic vs unitary). No equations, fitted parameters, self-citations, or ansatzes are quoted that reduce predictions to inputs by construction. The reader's assessment of 2.0 aligns with absence of load-bearing self-reference or definitional loops; the skeptic note confirms no derivations available to inspect for hidden reductions. The derivation chain appears self-contained against external ensemble definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition of the orthogonal and symplectic ensembles and on the standard mapping from random circuits to operator growth; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Gates are drawn independently from the orthogonal-invariant or symplectic-invariant measure on the appropriate Lie group.
    Invoked to define the ensemble-averaged weights and domain-wall dynamics.
  • domain assumption The Pauli-string weight and domain-wall observables are well-defined for the chosen qudit dimension.
    Required for the ternary structure and velocity statements to apply at q=2.

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Reference graph

Works this paper leans on

68 extracted references · 3 linked inside Pith

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    , q 2 −1, form a basis for operators acting on a qudit, a quantum sys- tem withqdegrees of freedom|m⟩,m= 0,1,

    orthogonal case The generalized Pauli matricesσ a,a= 0,1, . . . , q 2 −1, form a basis for operators acting on a qudit, a quantum sys- tem withqdegrees of freedom|m⟩,m= 0,1, . . . , q−1. We represent the indexa= (a ′, a′′)as a pair of two integersa ′, a′′ ∈Z q. The Weyl-Heisenberg choice for the complete fam- ily ofq 2 independent generalized Pauli matric...

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    In this case, the qudit dimensionqmust be even

    symplectic case Symplectic invariance applies to qudits of half-integer spin. In this case, the qudit dimensionqmust be even. Without loss of generality, we may choose the qudit basis such that the involution matrixZ x of Eq. (16) takes the form Zx =11q/2 ⊗σ 2 (A15) 13 parityσparityσ ′ countσ: [σ, σ ′] = 0countσ:{σ, σ ′}= 0 e e ′ q(q+ 2)/4q 2/4 e o ′ q2/4...

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    Orthogonal and symplectic ensembles When the two-qudit gate operatorsUsatisfyU † =U T (or- thogonal case) orU † =U R (symplectic case),i.e.,Uis an orthogonal or symplecticq 2×q2 matrix, respectively, the sym- metry constraints on the coefficientsBj simplify significantly. In both cases, we find that additional equalities hold for the coefficientsB j, B1 =...

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    The circular or- thogonal and symplectic ensembles (COE and CSE, respec- tively) from random matrix theory are examples of ensembles with such symmetry [29]

    Symmetric and self-dual ensembles The symmetry contraints on the coefficientsB j also sim- plify significantly if the two-qudit gate operatorsUare sym- metric or self-dual matrices,U=U T orU=U R for the orthogonal or symplectic case, respectively. The circular or- thogonal and symplectic ensembles (COE and CSE, respec- tively) from random matrix theory ar...

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    Specifically, we examine the Symmetric Orthogonal case (U=U ∗ =U T , corresponding toσ= 1) and the Self-dual Symplectic case (U=U R =U †, corresponding toσ=−1)

    Combined symmetries Finally, we consider the most constrained ensembles, where the matrixUpossesses both symmetries. Specifically, we examine the Symmetric Orthogonal case (U=U ∗ =U T , corresponding toσ= 1) and the Self-dual Symplectic case (U=U R =U †, corresponding toσ=−1). The combined symmetries impose constraints on the coef- ficientsB j, B1 =B 2 =B...

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    o” (corresponding to ternary index−1) and “11

    The caseq= 2 m Forq= 2 m, to show that the projected binary-string distri- bution¯ρ¯psatisfies a closed evolution equation, it is sufficient to consider the adjacent quditsxandx+ 1, which are subject to a two-qudit gate. From Eqs. (31) and (35) we then have ¯ρ¯px,¯px+1(t) = X ax,ax+1 ρax,ax+1;ax,ax+1(t−1) × X px→¯px X px+1→¯px+1 ⟨|Wax,ax+1;px,px+1 |2⟩. (C...

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    (36), where the transition matrixT ¯a¯phas a block structure as in Eq

    General qudit sizeq For a general qudit sizeq, which is not a power of2, we can also obtain a closed Markovian evolution equation of the 24 form of Eq. (36), where the transition matrixT ¯a¯phas a block structure as in Eq. (63). This matrix is constructed by summing the transition prob- abilitiesW ab;pq over all Pauli stringspandqwith the weight factorsχ ...

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    and combine¯ρ (n)(x;t)and¯ρ (n)(x+ 1, t)into a two- component spinor, R(n)(∆x;t) = ¯ρ(n)(∆x;t) ¯ρ(n)(∆x+ 1;t) ,(D1) where now∆xis always even. The evolution equations (45)– (46) with the truncation prescription (47), may then be repre- sented as R(n)(∆x;t) =D (n)R(n)(∆x;t−1) +D ′(n)R(n)(∆x+ 2;t−1),(D2) whereD (n) andD ′(n) are(4×3 n)×(4×3 n)matrices. The ...

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    Detailed derivation Before giving the full derivation, we first outline the logical structure. We denote the ensemble of two-qudit gate operators byEand the transition matrix corresponding to the two-qudit operatorUbyW(U)

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    Symplectic invariant ensembles For the symplectic-invariant ensemble, one proceeds almost the same as the orthogonal case, with a few small differences. The constructed groupGis invariant under symplectic trans- formations. In theW(V)representation, up to the trivial ac- tion, there are only two nontrivial possibilities of the image of G:Sp(q 2/2), the pr...

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    Relaxation time One can verify that⟨W⟩commutes withB, by using the Weingarten calculus result in App. B, which also guarantees the validation of the projection method. As an example, consider the termC 16 = B16 tr ΣaΣ† pΣ∗ bΣT q and investigate its action on a density matrixρ pq of ternary form. The nonzero elements of such density matrices havep=qorp=q T...

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