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arxiv: 2501.03851 · v1 · submitted 2025-01-07 · ❄️ cond-mat.str-el · cond-mat.stat-mech· quant-ph

Gapless Symmetry-Protected Topological States in Measurement-Only Circuits

Xue-Jia Yu , Sheng Yang , Shuo Liu , Hai-Qing Lin , Shao-Kai Jian This is my paper

Pith reviewed 2026-05-23 06:17 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechquant-ph
keywords measurement-only circuitsgapless SPT statessymmetry-enriched percolationMajorana loop modelClifford circuitstopological edge modessteady-state criticalityZ4 symmetry
0
0 comments X

The pith

Measurement-only circuits realize gapless symmetry-protected topological states in critical steady states.

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

This paper establishes that gapless symmetry-protected topological states can emerge in the steady states of measurement-only quantum circuits. Through extensive Clifford circuit simulations, the authors identify topological edge states persisting at criticality in several circuit families. In Ising cluster circuits, they find a symmetry-enriched non-unitary critical point with both edge states and string operators. A Z4 circuit model hosts a steady-state gSPT phase that survives symmetry-preserving perturbations. These findings are supported by a mapping to the Majorana loop model.

Core claim

The paper claims that gapless symmetry-protected topological (gSPT) states can be realized in the critical steady states of measurement-only circuits. In the Ising cluster circuits, this manifests as a symmetry-enriched percolation critical point featuring topologically nontrivial edge states and string operators. In the Z4 circuit model, a steady-state gSPT phase with topological edge modes is realized and persists under symmetry-preserving perturbations. The system is mapped to the Majorana loop model to explain the underlying mechanisms.

What carries the argument

Mapping to the Majorana loop model, which accounts for the emergence of topological edge states and string operators in the steady-state phases.

Load-bearing premise

Large-scale Clifford circuit simulations faithfully capture the topological properties of the steady-state phase diagram and the mapping to the Majorana loop model correctly explains the edge states and string operators.

What would settle it

A Clifford circuit simulation or equivalent experiment on the Ising cluster model showing the absence of topologically nontrivial edge states and string operators at the identified symmetry-enriched percolation point would falsify the central claim.

Figures

Figures reproduced from arXiv: 2501.03851 by Hai-Qing Lin, Shao-Kai Jian, Sheng Yang, Shuo Liu, Xue-Jia Yu.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Circuit diagram of the Ising cluster circuit model [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The generalized topological entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Majorana representation of the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a1) Steady state phase diagram of the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Majorana representation of the cluster circuit model [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

Measurement-only quantum circuits offer a versatile platform for realizing intriguing quantum phases of matter. However, gapless symmetry-protected topological (gSPT) states remain insufficiently explored in these settings. In this Letter, we generalize the notion of gSPT to the critical steady state by investigating measurement-only circuits. Using large-scale Clifford circuit simulations, we investigate the steady-state phase diagram across several families of measurement-only circuits that exhibit topological nontrivial edge states at criticality. In the Ising cluster circuits, we uncover a symmetry-enriched non-unitary critical point, termed symmetry-enriched percolation, characterized by both topologically nontrivial edge states and string operator. Additionally, we demonstrate the realization of a steady-state gSPT phase in a $\mathbb Z_4$ circuit model. This phase features topological edge modes and persists within steady-state critical phases under symmetry-preserving perturbations. Furthermore, we provide a unified theoretical framework by mapping the system to the Majorana loop model, offering deeper insights into the underlying mechanisms.

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 manuscript studies gapless symmetry-protected topological (gSPT) states realized as steady states of measurement-only circuits. Large-scale Clifford simulations of Ising cluster circuits are used to identify a symmetry-enriched non-unitary critical point (termed symmetry-enriched percolation) that exhibits both topologically nontrivial edge states and a string operator. A Z4 circuit model is shown to host a steady-state gSPT phase with topological edge modes that survives symmetry-preserving perturbations while remaining critical. A mapping of the circuits onto a Majorana loop model is presented as a unified theoretical framework.

Significance. If the central claims hold, the work extends gSPT physics to critical steady states of measurement circuits and introduces the notion of symmetry-enriched percolation as a non-unitary fixed point. The combination of Clifford numerics across multiple circuit families with an explicit loop-model mapping supplies concrete, falsifiable diagnostics (edge correlations and string-operator scaling) that could be tested in other platforms.

major comments (2)
  1. [theoretical framework (Majorana loop model mapping)] Theoretical framework paragraph (mapping to Majorana loop model): the mapping is described as obtained by rewriting the measurement operators, yet the abstract invokes the same loop connectivity and symmetry action to diagnose the edge states and string operators observed numerically. An explicit derivation is required that extracts the edge-mode wavefunction or the string-operator scaling dimension directly from the loop-ensemble statistics without feeding back the circuit-simulated correlation functions.
  2. [Ising cluster circuits results] Ising cluster circuits section: the claim that the critical point is simultaneously characterized by topologically nontrivial edge states and a string operator rests on the numerical data, but the manuscript does not report how the string-operator expectation value is extracted in the steady state or whether its scaling is shown to be independent of the edge-state diagnostic used to identify the phase.
minor comments (1)
  1. [Abstract] The term 'symmetry-enriched percolation' is introduced in the abstract without a one-sentence definition or reference to prior percolation literature; a brief clarifying clause would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments. We address each major comment below and will incorporate the requested clarifications in a revised version.

read point-by-point responses

  1. Referee: Theoretical framework paragraph (mapping to Majorana loop model): the mapping is described as obtained by rewriting the measurement operators, yet the abstract invokes the same loop connectivity and symmetry action to diagnose the edge states and string operators observed numerically. An explicit derivation is required that extracts the edge-mode wavefunction or the string-operator scaling dimension directly from the loop-ensemble statistics without feeding back the circuit-simulated correlation functions.

    Authors: We agree that an explicit, self-contained derivation is needed. The current manuscript sketches the mapping via operator rewriting but does not fully derive the edge-mode wavefunction or string-operator dimension from loop statistics alone. In the revision we will add an appendix that starts from the loop-ensemble measure, incorporates the symmetry action on loops, and directly computes the relevant scaling dimensions and edge correlations without reference to the circuit numerics. revision: yes

  2. Referee: Ising cluster circuits section: the claim that the critical point is simultaneously characterized by topologically nontrivial edge states and a string operator rests on the numerical data, but the manuscript does not report how the string-operator expectation value is extracted in the steady state or whether its scaling is shown to be independent of the edge-state diagnostic used to identify the phase.

    Authors: We thank the referee for this observation. The string operator is evaluated by measuring the appropriate multi-qubit Pauli string on the final steady-state stabilizer state after many circuit layers; its expectation value is obtained by averaging over the ensemble of measurement outcomes. In the revision we will add an explicit description of this procedure together with a supplementary figure demonstrating that the extracted scaling dimension remains unchanged when the edge-state diagnostic is varied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on independent simulations and mapping

full rationale

The provided abstract and context present large-scale Clifford circuit simulations and a mapping to the Majorana loop model as separate lines of evidence for the symmetry-enriched percolation point and gSPT phase. No quoted equations or sections demonstrate that any prediction reduces by construction to fitted parameters, self-definitions, or load-bearing self-citations. The mapping is described as offering 'deeper insights' rather than being derived from the same observables it explains. Per the hard rules, absent explicit paper text exhibiting a reduction (e.g., Eq. X = Eq. Y by construction), the finding is no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no details provided on free parameters, axioms, or invented entities.

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Forward citations

Cited by 5 Pith papers

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  1. Anomalous Dynamical Scaling at Topological Quantum Criticality

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  2. PT symmetry-enriched non-unitary criticality

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    PT symmetry enriches non-Hermitian critical points with topological nontriviality, robust edge modes, and a quantized imaginary subleading term in entanglement entropy scaling.

  3. Generalized Li-Haldane Correspondence in Critical Dirac-Fermion Systems

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  4. Generalized Li-Haldane Correspondence in Critical Dirac-Fermion Systems

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