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REVIEW 4 major objections 6 minor 61 references

Measurement-only circuit of perturbed toric code on triangular lattice: Topological entanglement, 1-form symmetry and logical qubits

T0 review · 4 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper claims that in a measurement-only toric code on a triangular lattice, the topological-order transition (as measured by topological entanglement entropy) and the spontaneous breaking of the 1-form symmetries occur at different meas

desk verdict A genuinely new triangular-lattice measurement-only-circuit study with a plausible but under-controlled claim that TEE and 1-form-symmetry critical points split; fitting problems (one critical point outside the physical range, a non-growing variance peak) must be fixed before the quantitative phase diagram is trusted. read the letter →

arxiv 2510.23162 v3 pith:5PRG2ZV2 submitted 2025-10-27 quant-ph cond-mat.othercond-mat.stat-mechcond-mat.str-el

Keisuke Kataoka , Yoshihito Kuno , Takahiro Orito , Ikuo Ichinose This is my paper
classification quant-phcond-mat.othercond-mat.stat-mechcond-mat.str-el
keywords measurement-onlycircuittoriccodetriangularlatticetopologicalentanglemententropy1-formsymmetrystabilizerformalismlogicalqubitspercolationtransition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies a measurement-only quantum circuit that repeatedly measures the stabilizers of a toric code on a triangular lattice together with single-qubit X and Z measurements. It claims that the steady-state phase transition seen in the topological entanglement entropy (TEE) does not happen at the same parameter value as the transition in the disorder parameters that diagnose spontaneous breaking of the 1-form symmetries. The authors find the symmetry-breaking signal appears first, at measurement strength p_g around 0.72–0.79, while the TEE only switches on around p_g = 0.85–0.88, with a correlation-length exponent near 1.7 that differs from the percolation-like exponent (about 4/3) of the string and loop observables. Because the triangular lattice lacks the self-duality and bipartiteness of the square lattice, the two transitions are not forced to coincide by symmetry, so the setup provides a clean test of whether 1-form symmetry breaking is the origin of topological order. The result challenges a widely held belief and gives a concrete playground for separating the signatures of topological order.

What carries the argument

The measurement-only circuit itself — projective measurements of toric-code stabilizers and competing local Pauli measurements — is the central object. The key identity is the six-region formula for the topological entanglement entropy, computed from boundary stabilizer counts; the key diagnostic operators are the zigzag string operators, which serve as disorder parameters of the electric and magnetic 1-form symmetries, and the non-contractible loops, which act as logical operators. The triangular lattice is the load-bearing geometric choice: without self-duality or bipartiteness, the electric and magnetic transitions are not forced to coincide, so the different observables can be compared a

What would settle it

A concrete check: run the same measurement-only circuit at p_g = 0.80 for system size L = 24 and measure both the open-string disorder parameter and the six-region TEE. The paper predicts the string has essentially vanished while the TEE is still zero; if the string remains substantially nonzero there, or if the two crossing points shift together with increasing L, the claimed separation is an artifact.

Watch

Extended reading notes

Core claim

The paper's central claim is that in the steady state of a measurement-only circuit built from triangular-lattice toric-code stabilizers (vertex X-checks, plaquette Z-checks, plus local X and Z measurements), the phase transition in the topological entanglement entropy does not coincide with the spontaneous breaking of the electric and magnetic 1-form symmetries. Using exact stabilizer simulations and finite-size scaling, the authors find TEE critical points p_g^c = 0.846±0.016 (confinement side) and 0.876±0.048 (Higgs side) with correlation-length exponent ν ≈ 1.7, whereas the string disorder parameters (diagnosing 1-form symmetry breaking) and the non-contractible loop order parameters (lo

Load-bearing premise

The paper assumes that a vanishing expectation value of the string disorder parameter — the quantity that diagnoses whether the 1-form symmetry is spontaneously broken — is an exact diagnostic, just as an ordinary order parameter diagnoses 0-form symmetry breaking; if the string can vanish for other reasons, the claimed ordering does not follow.

Editorial extensions

If this is right

  • If the discrepancy is real, topological order (as diagnosed by TEE) and 1-form symmetry breaking are not interchangeable diagnostics in monitored circuits; the conventional view that symmetry breaking is the origin of topological order needs qualification.
  • The TEE critical exponent ν ≈ 1.7 differs from the 2D percolation value 4/3, suggesting the entanglement transition belongs to a different universality class from the string and loop transitions.
  • The string and loop observables, including the logical operators, transition near the percolation threshold, indicating that emergence of a quantum memory in this circuit is tied to percolation of stabilizer voids rather than to the TEE transition.
  • The two 1-form symmetry transitions (electric and magnetic) occur at slightly different p_g values on the triangular lattice, a direct manifestation of broken self-duality that could be used to tune the two symmetries independently.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: the same protocol on a square lattice, where self-duality forces coincident transitions, should show TEE and symmetry-breaking signals crossing at the same point; observing the triangular-lattice discrepancy there would confirm that the separation is due to lattice asymmetry.
  • The paper's footnote assumption — that a vanishing disorder parameter is an exact diagnosis of 1-form symmetry breaking — could be checked by comparing the string correlator with a differently normalized disorder parameter in the same simulations; a disagreement in the region p_g ≈ 0.72–0.85 would indicate the ordering is a property of the specific diagnostic rather than of the phase.
  • If the discrepancy persists in larger systems and in Hamiltonian (non-measurement) versions of the triangular toric code, it would suggest that TEE and 1-form symmetry breaking are generically independent signatures in non-self-dual topological models.
  • The reported ν ≈ 1.7 for the TEE transition invites comparison with other monitored-circuit critical points; one could look for the same exponent in measurement-only circuits with different stabilizer codes to test whether it is universal across this class.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies a measurement-only circuit (MoC) for the toric code on the triangular lattice, where projective measurements of stabilizers (A_s, B_p) compete with local Pauli measurements, parameterized by probabilities p_g, p_x, p_z. Using stabilizer simulations, the authors compute three independent non-local diagnostics: the topological entanglement entropy (TEE), the Rényi-2 expectation values of open string operators (disorder parameters of the 1-form symmetries), and non-contractible loop operators (logical operators). The main quantitative claim is that on the lines p_x+p_g=1 and p_z+p_g=1, the TEE transitions at p_g ≈ 0.85 and 0.88 with correlation-length exponent ν ≈ 1.7, while the string and loop observables transition at p_g ≈ 0.72–0.79 with ν closer to the 2D percolation value 4/3. From this, the authors conclude in Sec. IV that the TEE phase transition does not coincide with the emergence of spontaneous symmetry breaking (SSB) of the 1-form symmetries; rather, SSB occurs first and the TEE becomes finite later. The triangular lattice is chosen because it is neither self-dual nor bipartite, so the degeneracies of the square-lattice model are absent.

Significance. If the central claim is correct, the paper provides a concrete counterexample to the common assumption that the topological-order transition and the 1-form-SSB transition necessarily occur at the same point. This would be of substantial interest for monitored quantum circuits and for the broader gauge-theory understanding of topological order. The numerical method is well-suited to the problem: stabilizer circuits allow exact trajectory simulations, the TEE is extracted from boundary stabilizer counts on hexagon complexes, and the paper reports bootstrap error estimates and multiple observables. These are genuine strengths. However, the central conclusion rests on finite-size scaling (FSS) analyses whose reliability is not fully established, and on an interpretive premise about 1-form SSB that is stated but not justified. The paper is therefore a promising contribution whose main claim currently outruns the numerical evidence.

major comments (4)
  1. [Table 1A / Sec. III.D] The Z-loop FSS row reports p_g = 1.009 ± 0.017, outside the physical range p_g ∈ [0,1] on the line p_x+p_g=1. A critical point cannot lie outside the domain of the tuning parameter; this result indicates that the FSS ansatz is being applied to data that do not contain a genuine transition in the scanned interval. This entry should be removed or refit with a model that respects the domain. More importantly, this unphysical fit undermines confidence in the other FSS results in the same table, since the same procedure is used for all observables.
  2. [Sec. III.D, Fig. 7 (center and right panels)] The text states that the X-loop variance peaks "do not develop for increasing system size" while the peak becomes sharper. For a conventional continuous transition, the variance (susceptibility) peak height grows as L^{γ/ν}; a peak that saturates in height cannot be collapsed by the standard FSS ansatz. Therefore the quoted values p_x^c,loop = 0.719 ± 0.003 and ν = 1.27 ± 0.06 are not reliable. The authors should show the peak-height scaling, or replace this estimator with a crossing-based quantity such as a Binder cumulant or a ratio of string correlators.
  3. [Secs. III.B–III.D and Table I] The central separation between the TEE transition (from variance peaks at p_g = 0.846/0.876) and the string/loop transitions (from means of order parameters at p_g ≈ 0.72–0.79) is extracted with different FSS estimators. Finite-size corrections to variance peaks and order-parameter means are generally different, and the unphysical Z-loop fit shows that not all fits in the family are well-controlled. Without a common estimator or a demonstration that both observables are controlled by the same divergent length scale, the apparent ~0.1 separation does not establish distinct transitions. Please provide, e.g., Binder-cumulant crossings for both TEE and string observables, or a simultaneous collapse using the same scaling variable.
  4. [Footnote [52] and Sec. IV] The ordering claim "SSB takes place first, and then the TEE gets a finite value" relies on the assumption stated in footnote [52] that a vanishing expectation value of the disorder parameter is an exact diagnosis of SSB of the 1-form symmetry, in the same way as for ordinary 0-form symmetry. This assumption is neither proved nor standard in general: a string disorder parameter can vanish for other reasons (e.g., boundary or finite-size effects specific to the monitored trajectory ensemble) without implying true 1-form SSB. The conclusion should be reformulated as a statement about the behavior of the string correlators, or the diagnostic assumption must be justified explicitly.
minor comments (6)
  1. [Abstract and Introduction] There are grammatical slips in the abstract and introduction (e.g., "gives possibility", "takes place"). These should be corrected for a journal submission.
  2. [Sec. III.B] The notation p_x^gc and p_z^gc is confusing: the text says "for the MoC with p_x+p_g=1 ... p_x^gc = 0.846". Since the line is p_x+p_g=1, p_x^gc is actually the critical value of p_g on that line, not the probability p_x. Please rename to avoid ambiguity.
  3. [Appendix D.1] The text says "we fixed the value of p_g with keeping p_x+p_z = p_g", but the figure captions and the phase diagram use p_x+p_z = 1-p_g. Please reconcile this inconsistency.
  4. [Figure captions (Fig. A5, A6)] The captions contain "collapse of the date" which should be "collapse of the data". Similar typos appear in the main text (e.g., "orders" for "operators").
  5. [Eq. (5) and Table II] In the FSS ansatz χ_TE(p_g) = L^ζ F((p_g-p_gc)L^{1/ν}), the exponent ζ is not defined or reported. Please clarify what ζ is and whether it is consistent with a dimensionless TEE variance. Also, Table II should explain explicitly what "TEE(1~6)" means relative to "2~6 hexagons".
  6. [Appendix C, Eq. (A2)] The definition of the Fredenhagen-Marcu operator uses L_{1/2} and L, but the relationship between these lengths is only stated in words. A more precise definition would make the connection to the string order parameters easier to follow.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the central comparison uses independent observables and the only self-citations are methodological, not load-bearing.

full rationale

The paper's central claim—that the TEE transition (p_g ≈ 0.846/0.876) does not coincide with the 1-form-symmetry-breaking transition inferred from string/loop observables (p_g ≈ 0.72–0.79)—is obtained by separately finite-size-scaling three independent families of observables: the TEE combination in Eq. (2), the Rényi-2 string correlation functions in Eqs. (7)–(8), and the non-contractible loop operators in Eq. (9). None of these quantities is defined in terms of a fitted p_gc or ν, and no fitted value from one observable is reused as an input to another. The FSS ansatz in Eq. (5) is applied separately to each observable, so the reported discrepancy is an empirical comparison rather than a construction. Self-citations [27,30,31,45,51] provide numerical and diagnostic techniques (stabilizer simulation, Rényi-2 correlators, the percolation picture of string expectation values), but the triangular-lattice result is not imported from them. The interpretive assumption in footnote [52] is explicitly stated as an assumption, not derived from a self-citation. Concerns such as the unphysical Z-loop fit p_g = 1.009 ± 0.017, the non-growing X-loop variance peak, and the use of different FSS estimators for TEE versus strings are potential finite-size or methodological risks, but they are not circularity: the observables remain independent diagnostics, and a flawed comparison is not a definitional reduction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new particles or entities are introduced. The free parameters are FSS outputs rather than model inputs; the central claim depends on the FSS ansatz, the footnote-52 SSB diagnosis, and numerical convergence assumptions. The measurement probabilities p_x, p_z, p_g are circuit controls, not hidden fitted parameters.

free parameters (6)
  • p_gc (TEE) = 0.846±0.016 (p_x line); 0.876±0.048 (p_z line)
    Critical probability extracted by FSS of the TEE variance; the paper's claim of distinct transition locations depends on these fits.
  • ν (TEE) = 1.69±0.26; 1.74±0.29
    Correlation-length exponent from FSS, compared against percolation ν=4/3 to support the claim that TEE is in a different universality class.
  • p_gc (strings) = 0.723±0.013 (X-string); 0.787±0.012 (Z-string)
    FSS estimates for the 1-form-symmetry disorder parameters.
  • ν (strings) = 1.63±0.35; 1.51±0.31
    FSS estimates for string-operator correlation-length exponents.
  • p_gc (loops) = 0.719±0.003 (X-loop); 0.786±0.002 (Z-loop); 1.009±0.017 (Z-loop in Table 1A, outside physical range)
    FSS estimates for logical-loop operators; the 1.009 value is a red flag because p_g is a probability.
  • ν (loops) = 1.27±0.06; 1.45±0.14; 1.37±0.17; 1.39±0.27
    FSS estimates for loop-operator correlation-length exponents.
assumptions (5)
  • standard math The stabilizer formalism and Gottesman-Knill simulation correctly describe the state after each projective Pauli measurement.
    Relied on throughout Sec. II.B and Appendix A; not proved in this paper.
  • standard math The entanglement entropy of a subsystem equals the number of independent stabilizers crossing its boundary, S_A = -L_A + γ (Eq. A1).
    Used in Appendix A to justify the TEE extraction from stabilizer counts.
  • domain assumption The variance of the TEE obeys the finite-size-scaling ansatz χ_TE(p_g) = L^ζ F((p_g - p_gc)L^{1/ν}) with no significant corrections for L ≤ 24 (Eq. 5).
    Used to extract p_gc and ν; its validity for these small system sizes is not independently established.
  • ad hoc to paper Vanishing expectation value of the string disorder parameter is an exact diagnosis of SSB of the 1-form symmetry, as for ordinary 0-form symmetry.
    Explicitly assumed in footnote [52]; it underpins the central interpretive claim that 1-form SSB occurs before TEE onset.
  • domain assumption After more than 10N measurement steps and with 500 trajectory samples, the sampled observables represent the steady state.
    Appendix A verifies steady behavior for a few parameter points, but not for all phase-diagram regions.

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Cite this review

Pith. "Pith review of Measurement-only circuit of perturbed toric code on triangular lattice: Topological entanglement, 1-form symmetry and logical qubits." pith.science (2026). https://pith.science/paper/5PRG2ZV2

@misc{pith2026251023162,
  author       = {Pith},
  title        = {Pith review of: Measurement-only circuit of perturbed toric code on triangular lattice: Topological entanglement, 1-form symmetry and logical qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5PRG2ZV2}},
  note         = {Machine review of arXiv:2510.23162}
}
read the original abstract

Measurement-only (quantum) circuit (MoC) gives possibility to realize the states with rich entanglements, topological orders and quantum memories. This work studies the MoC, in which the projective-measurement operators consist of stabilizers of the toric code and competitive local Pauli operators. The former correspond to terms of the toric code on a triangular lattice and the later to external magnetic and electric fields. We employ efficient numerical stabilizer algorithm to trace evolving states undergoing phase transitions. We elucidate the phase diagram of the MoC system with the observables such as, topological entanglement entropy (TEE), disorder parameters of 1-form symmetries and emergent logical operators. We clarify the locations of the phase transitions through the observation of the above quantities and obtain precise critical exponents to examine if the observables exhibit the critical behavior simultaneously under the MoC and transitions belong to the same universality class. In contrast to the TC Hamiltonian system and toric code MoC on a square lattice, the system on the triangular lattice is not self-dual nor bipartite, and then, coincidence by symmetries, such as critical behaviors across the TC and Higgs/confined phase, does not takes place. Then, the toric code MoC on the triangular lattice provides us a suitable playground to clarify the mutual relationship between the TEE, spontaneous symmetry breaking of the 1-form symmetries, and emergence of logical operators. Obtained results indicate that toric code MoC on the triangular lattice exhibits a few distinct phase transitions with different location and critical exponents, and some of them are closely related with the two-dimensional percolation transition.

Figures

Figures reproduced from arXiv: 2510.23162 by the authors.

Figure 1
Figure 1. FIG. 1. Toric code on triangular lattice. Left panel: Stabilizers [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Left: Subsystems [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Overview of observables from Higgs-confinement phase to TC. There are three kinds of observables, which we investigate in detail [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Topological entanglement entropy for various system sizes. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Calculation of the expectation value of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Variance of topological entanglement entropy for various [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Order parameters as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

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