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arxiv: 2606.13470 · v1 · pith:6QTBDAPYnew · submitted 2026-06-11 · 🪐 quant-ph

Invariant Measures and Weak-Magic-Injection Asymptotics in Random Monitored Quantum Circuits

Guocheng Zhen , Xuanrong Yang , Chengkai Zhu , Ranyiliu Chen , Xin Wang This is my paper

Pith reviewed 2026-06-27 06:41 UTC · model grok-4.3

classification 🪐 quant-ph
keywords monitored quantum circuitsmagic resourcesstationary distributionClifford gatesGross-Wigner manastabilizer Rényi entropyweak magic injection
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The pith

Random monitored quantum circuits admit a unique stationary law with dimension-dependent magic asymptotics in the weak injection limit.

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

The paper develops an analytical framework for random monitored quantum circuits driven by Clifford gates with weak non-Clifford perturbations. It proves the existence and uniqueness of the stationary law that ergodically describes the long-time dynamics. The leading asymptotics of steady magic are resolved in the weak-magic-injection limit, revealing a linear scaling for the Gross-Wigner mana in odd-prime local dimensions and a quadratic scaling for the 2-stabilizer Rényi entropy in qubit systems. This contrast arises from the distinct local geometries of the resource measures near the stabilizer layer. The work provides the first rigorous foundation for the stochastic many-body dynamics beyond numerical simulations.

Core claim

We prove the existence and uniqueness of the stationary law for the monitored quantum circuit dynamics, giving an ergodic description of the long-time behavior. In the weak-magic-injection limit, the steady Gross-Wigner mana has a linear leading asymptotic in odd-prime local dimension, while the steady 2-stabilizer Rényi entropy has a quadratic leading asymptotic in qubit systems.

What carries the argument

The stationary law derived from the random Clifford-driven monitored circuits, which supports the ergodic description and enables calculation of the magic asymptotics under weak non-Clifford injection.

If this is right

  • The long-time dynamics become independent of initial conditions due to the unique stationary distribution.
  • Steady-state magic content is determined by the scaling of the injection strength and the choice of resource measure.
  • The power of the leading asymptotic reflects the local geometry of each magic measure around the stabilizer states.
  • This framework allows analytical treatment of the competition between scrambling, measurements, and resource injection.

Where Pith is reading between the lines

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

  • The difference in scaling suggests that qubit systems suppress magic accumulation more strongly near the stabilizer layer compared to higher-dimensional systems.
  • Similar stationary laws might exist for other types of monitored dynamics or different gate sets.
  • The results could be extended to study how magic affects measurement-induced phase transitions in these circuits.
  • Experimental verification could involve preparing small quantum circuits and measuring the relevant resource quantifiers in the steady state.

Load-bearing premise

The monitored circuits are driven exclusively by random Clifford gates with only weak non-Clifford perturbations in the weak-magic-injection scaling regime.

What would settle it

A numerical computation of the steady-state Gross-Wigner mana for small odd-prime dimensional systems or 2-stabilizer Rényi entropy for qubits, in the limit of vanishing magic injection strength, that deviates from the predicted linear or quadratic scaling would falsify the asymptotics.

Figures

Figures reproduced from arXiv: 2606.13470 by Chengkai Zhu, Guocheng Zhen, Ranyiliu Chen, Xin Wang, Xuanrong Yang.

Figure 1
Figure 1. Figure 1: Schematic view of one monitored update step of the model. At time step [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical steady-state diagnostic for the monitored dynamics. The plot tracks exact mana [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Odd-prime weak-magic-injection asymptotics from the computational-basis product initial state. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Qubit weak-magic-injection asymptotics from the computational-basis product initial state. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic road map of the proof. The preliminary conventions feed into the Markov-kernel [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original abstract

Monitored quantum circuits provide a natural setting in which scrambling, measurements, and measurement-conditioned updates compete within a stochastic many-body dynamics. From the viewpoint of nonstabilizer resource theory, this competition is especially relevant because Clifford-compatible operations preserve the stabilizer structure, while weak non-Clifford perturbations inject magic resource. Most of the existing understanding of monitored quantum circuits has been shaped by numerical simulations and phenomenological descriptions, while a rigorous dynamics theory remains less developed. In this paper, we address this gap by developing an analytical framework which lays a rigorous mathematical foundation for the study of random monitored quantum dynamics. Specifically, we study a class of monitored quantum circuits driven by random Clifford. We prove the existence and uniqueness of the stationary law, which gives an ergodic description of the long-time dynamics. We then resolve the leading asymptotics of steady magic in the weak-magic-injection limit. This tangent description makes the contrast between resource measures transparent: in odd-prime local dimension, the steady Gross--Wigner mana has a linear leading asymptotic, whereas in qubit systems the steady 2-stabilizer R\'enyi entropy has a quadratic leading asymptotic. These different powers reflect the distinct local geometries of the two resource measures near the stabilizer layer. In this way, this work develops an analytical framework that first establishes the stationary ergodic dynamics of random monitored quantum circuits.

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 paper develops an analytical framework for random monitored quantum circuits driven exclusively by random Clifford gates with weak non-Clifford perturbations. It proves existence and uniqueness of the stationary law of the induced Markov process, yielding an ergodic description of the long-time dynamics. It then derives the leading asymptotics of steady-state magic in the weak-magic-injection limit, establishing a linear leading term for the Gross-Wigner mana in odd-prime local dimension and a quadratic leading term for the 2-stabilizer Rényi entropy in qubit systems; these scalings are attributed to the distinct local geometries of the two resource measures near the stabilizer layer.

Significance. If the claimed proofs and expansions hold, the work supplies a rigorous ergodic-theory foundation for magic-resource dynamics in monitored circuits, replacing purely numerical or phenomenological descriptions with an explicit stationary measure and perturbative asymptotics. The explicit contrast between linear and quadratic vanishing orders, tied to local geometry, is a clear conceptual advance in resource theory.

major comments (2)
  1. [Abstract] Abstract (and the model definition): the existence/uniqueness proof for the stationary law is asserted for the Markov process generated by random Clifford gates plus weak non-Clifford injection, yet no explicit construction of the transition kernel, compactness argument, or application of standard ergodic theorems (e.g., on the space of density operators) is visible; without these steps the claim that the stationary law is unique and ergodic cannot be verified.
  2. [Abstract] Abstract (weak-magic-injection asymptotics): the linear leading term for Gross-Wigner mana (odd-prime dimension) and quadratic term for 2-stabilizer Rényi entropy (qubits) are stated as following from a perturbative expansion around the stabilizer layer, but no explicit Taylor expansion, error bound, or scaling assumption on the injection strength is provided; the claimed powers therefore rest on an unshown local-geometry calculation.
minor comments (2)
  1. The double-dash notation “Gross--Wigner mana” should be standardized to the conventional single hyphen or the full name with citation.
  2. [Abstract] The phrase “tangent description” in the final sentence of the abstract is unclear; a more precise statement of the perturbative regime would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and constructive suggestions. We address each major comment below. The full manuscript contains the detailed proofs referenced in the abstract; we will revise the abstract and model section to improve visibility of the key steps without altering the results.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the model definition): the existence/uniqueness proof for the stationary law is asserted for the Markov process generated by random Clifford gates plus weak non-Clifford injection, yet no explicit construction of the transition kernel, compactness argument, or application of standard ergodic theorems (e.g., on the space of density operators) is visible; without these steps the claim that the stationary law is unique and ergodic cannot be verified.

    Authors: The transition kernel is constructed explicitly in Section II as the composition of random Clifford unitaries (drawn uniformly from the Clifford group) followed by weak non-Clifford injections at each site. The state space is the compact convex set of density operators on a finite-dimensional Hilbert space, which is compact in the trace norm. Irreducibility follows from the fact that the Clifford group generates a transitive action on stabilizer states while the weak injections ensure positive probability of reaching any magic direction; aperiodicity is immediate from the continuous-time embedding. These properties allow direct application of the standard ergodic theorem for Markov processes on compact metric spaces, yielding uniqueness of the stationary measure. We will add a one-sentence pointer to Section II in the abstract and expand the model definition paragraph to name the compactness and ergodicity arguments. revision: partial

  2. Referee: [Abstract] Abstract (weak-magic-injection asymptotics): the linear leading term for Gross-Wigner mana (odd-prime dimension) and quadratic term for 2-stabilizer Rényi entropy (qubits) are stated as following from a perturbative expansion around the stabilizer layer, but no explicit Taylor expansion, error bound, or scaling assumption on the injection strength is provided; the claimed powers therefore rest on an unshown local-geometry calculation.

    Authors: The local-geometry calculation appears in Section IV (and Appendix B). For odd-prime dimension the mana is expanded to first order in the injection strength ε; the linear term is the directional derivative of the mana functional at the stabilizer boundary, which is nonzero. For qubits the 2-stabilizer Rényi entropy has vanishing first derivative at the stabilizer layer, so the leading term is quadratic and given by the Hessian; an explicit error bound O(ε^3) is derived from the C^3 smoothness of both functionals on the compact set of states. The scaling assumption is ε → 0 with system size fixed. We will insert a brief clause in the abstract referencing this expansion and the resulting powers. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation establishes existence and uniqueness of the stationary measure for the Markov process generated by random Clifford gates plus weak non-Clifford perturbations via standard ergodic-theory arguments on compact state spaces, followed by local Taylor expansions of the resource measures (linear vanishing of mana in odd-prime dimension, quadratic for qubit 2-Rényi entropy). These steps rely on the stated circuit assumptions and local geometry of the quantifiers rather than any fitted parameters, self-citations, or ansatzes that reduce the claims to their inputs by construction. The framework is self-contained against external benchmarks of Markov-chain ergodicity and perturbative analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated beyond the modeling choice of random Clifford gates plus weak magic injection.

pith-pipeline@v0.9.1-grok · 5785 in / 1143 out tokens · 20780 ms · 2026-06-27T06:41:48.888028+00:00 · methodology

discussion (0)

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    for everyn≥0,h n ∈ H2 andh n ≥0

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    there existCh <∞andρ h ∈(0,1)such that, for everyn≥0, ∥hn∥H2 ≤C hρn h.(291) Consequently, for everyn≥0and every(s, v)∈ bX2, 0≤h n(s, v)≤C hρn h∥v∥2.(292) 118

  57. [57]

    Since q∈ H 2 and zn ∈ H 2 for all n≥ 0, where H2 is the class of Definition D.25, the recursion(290) and the boundedness ofeP2 on H2 imply inductively thathn ∈ H2 for everyn≥0

    for everyn≥0and everyR <∞, sup s∈S(2) stab,∥v∥≤R (P n 0 S2)((κ(2) s )−1(θv)) θ2 −h n(s, v) − − → θ↓0 0.(293) Proof.We first prove (i) and (ii). Since q∈ H 2 and zn ∈ H 2 for all n≥ 0, where H2 is the class of Definition D.25, the recursion(290) and the boundedness ofeP2 on H2 imply inductively thathn ∈ H2 for everyn≥0. Moreover,q≥0,z n ≥0, and eP2 is posi...

  58. [58]

    for everyn∈N, everyθ M ∈[0,1], and everyf∈ B 1, ∥P n θM f∥0 ≤C 1M n∥f∥0;(379)

  59. [59]

    for everyn∈N, everyθ M ∈[0,1], and everyf∈ B 1, ∥P n θM f∥ B1 ≤C 2βn∥f∥ B1 +C 3M n∥f∥0;(380)

  60. [60]

    for everyθ M ∈[0,1], the part ofσ(P θM )lying in{|z|> β}contains no residual spectrum

  61. [61]

    (i) Fixδ >0andr∈(β, M), and set η := ln(r/β) ln(M/β) >0, V δ,r :={z∈C : |z| ≤rordist(z, σ(P 0))≤δ}

    for everyθ M ∈[0,1], |||PθM −P 0||| ≤τ(θ M).(381) Then the following conclusions hold. (i) Fixδ >0andr∈(β, M), and set η := ln(r/β) ln(M/β) >0, V δ,r :={z∈C : |z| ≤rordist(z, σ(P 0))≤δ}. Then there exist constants θ0 = θ0(δ, r) > 0, AKL(r) > 0, BKL(δ, r) > 0, CKL(δ, r) > 0and DKL(δ, r)>0such that for0≤θ M ≤θ 0 andz∈C\V δ,r ∥(z−P θM )−1f∥ B1 ≤A KL∥f∥ B1 +B...

  62. [62]

    For any fixedδ > 0and r∈ (γ, 1), there exist constantsθ0 = θ0(δ, r) > 0, AKL(r) > 0, BKL(δ, r) > 0, CKL(δ, r)>0andD KL(δ, r)>0such that for0≤θ M ≤θ 0 andz∈C\V δ,r ∥(z−P θM )−1f∥ B1 ≤A KL∥f∥ B1 +B KL∥f∥0,∀f∈ B 1,(382) and |||(z−P θM )−1 −(z−P 0)−1||| ≤τ(θ M)η CKL∥(z−P 0)−1∥B1→B1 +D KL∥(z−P 0)−1∥2 B1→B1 (383) where τ(θM) = C∗θM for some constantC∗ > 0indepe...

  63. [63]

    For each fixedθM ∈[0,1]there existsr ∗(θM)∈(0,1)such that σ(PθM )∩ {|z|> r ∗(θM)}={1}.(385) In particular,1is the unique peripheral eigenvalue ofPθM; its eigenspace is one-dimensional, and in fact1is algebraically simple. Proof. Step 1: verification of (379).Since PθM is a Markov operator, for everyn∈N , every θM ∈[0,1], and everyf∈ B 1 = Lip(X), we have ...