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REVIEW 4 major objections 4 minor 108 references

Resolving topological order crossovers on NISQ hardware

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Finite-size Wen–plaquette topological signatures remain measurable and distinguishable from trivial behavior on IBM Quantum hardware across static disorder, amplified circuit noise, and imaginary-field perturbations.

desk verdict A NISQ-era Wen–plaquette study with genuinely new robustness data and an honest scope, but the unreported fidelity of the variational compilation and the missing error bars leave the hardware attribution one step short of fully supported. read the letter →

arxiv 2607.21707 v1 pith:F2KULM24 submitted 2026-07-23 quant-ph cond-mat.str-el

Ruizhe Shen , Yin Zhong , Ching Hua Lee This is my paper · ORCID
classification quant-phcond-mat.str-el
keywords Wen–plaquettemodeltopologicalorderNISQquantumhardwaresimulationvariationalcompilationWilsonloopplaquettestabilizernon-Hermitianquench
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 establishes a practical claim: on today's noisy quantum processors, finite-size signatures of a topologically ordered state can remain measurable and distinguishable from trivial behavior even when hardware is pushed with disorder, lengthened circuits, and effective loss. Working with the exactly solvable Wen–plaquette model, the authors use a nine-qubit system to map how the local plaquette stabilizer ⟨F₁⟩ and a nonlocal Wilson loop respond as the model is driven between stabilizer-dominated, trivial, and disorder-dominated regimes. They find the stabilizer-sector crossover clearly resolved even at disorder strength h = 5, the Wilson-loop crossover still resolved after j = 5 identity layers of added noise, and quench dynamics whose plaquette signal is fragile near the finite-size crossover but substantially more stable deep in the strong-stabilizer regime, both for static disorder and for imaginary-field (non-Hermitian) perturbations. They then prepare a 5×5 stabilizer-sector state on a two-dimensional processor using a projection-inspired representative-qubit construction that needs no variational training, and show its lattice-averaged plaquette response degrades only weakly under intentionally amplified local rotations. The authors are explicit that the result is not a thermodynamic demonstration of topological order, but a hardware-resolved map of where these finite-size diagnostics survive — which matters because it tells a near-term experiment what can and cannot be concluded from such signatures on existing machines.

What carries the argument

The load-bearing objects are the mutually commuting plaquette stabilizers F_{i,j} = σˣσʸσˣσʸ, whose ±1 eigenvalues label the two Z₂ sectors and whose expectation value ⟨F₁⟩ is the local diagnostic; and the boundary Wilson loop W, a long alternating Pauli string tracking the nonlocal loop response across the crossover into the trivial transverse-field regime. State preparation uses two mechanisms: variational compilation, which trains a hardware-efficient circuit V_n(θ) to minimize the fidelity loss C(θ) = 1 − F[V_n(θ)|φ₀⟩, |ψ_tar⟩] for classically computed equilibrium, quench, and normalized non-Hermitian targets, with noise amplified by appending j identity layers, V_extend = (V_id)ᶨ V_prep

What would settle it

Report the optimized fidelity loss C(θ_opt) for representative hardware points in Figs. 3–6, and re-run one crossover curve with an ansatz of markedly different depth or layer structure: if ⟨F₁⟩ or ⟨W⟩ shifts materially while the classical target is unchanged, the hardware 'signatures' track the variational approximation rather than the Wen–plaquette physics. A noiseless simulation of the exact compiled circuits (including the inserted identity layers) that reproduces the measured curves would confirm the signals are intrinsic; divergence would indict the variational layer.

Watch

Extended reading notes

Core claim

The paper's claim: finite-size Wen–plaquette signatures stay measurable and distinguishable from trivial behavior on IBM Quantum hardware. Variationally compiled nine-qubit states reproduce the expected crossovers — ⟨F₁⟩ swapping sectors with the sign of J, the Wilson loop rising at J ≃ g — surviving disorder h = 5 and j = 5 noise-adding layers. Quench dynamics collapse near the crossover but persist deep in the stabilizer regime. A non-variational construction extends this to a 5×5 lattice whose plaquette response stays near unperturbed under up to 15 amplified rotations. The authors frame the result as hardware-resolved, not a thermodynamic proof of topological order.

Load-bearing premise

Every hardware measurement in Figs. 3–6 is made on a state V_n(θ_opt)|φ₀⟩ that only approximates the classically computed target via the fidelity loss C(θ) = 1 − F (Eq. 9); the paper reports neither the achieved fidelity, the ansatz depth n, nor the optimizer convergence. If the ansatz is insufficiently expressive or trapped in a poor local minimum, the measured 'signatures' reflect the approximate circuit rather than the Wen–plaquette model; the same reliance applies to the

Editorial extensions

If this is right

  • The local plaquette stabilizer crossover between the two Z₂ sectors remains clearly resolved on hardware at disorder strength h = 5, so static disorder alone does not destroy the finite-size stabilizer diagnostic.
  • The nonlocal Wilson-loop crossover remains resolved after j = 5 inserted identity layers that amplify gate and decoherence errors, meaning depth-induced noise up to this level does not wash out the loop signal.
  • The dynamical robustness of the stabilizer signature is controlled by the ratio of disorder (or imaginary-field strength) to the plaquette defect energy scale 2J: fragile at J ≈ 1 near the crossover, stable at J = 4–5 deep in the stabilizer regime.
  • A non-variational, projection-inspired preparation protocol builds a 5×5 stabilizer-sector state on a two-dimensional processor without variational training, and its lattice-averaged plaquette response degrades only weakly under up to 15 amplified local single-qubit perturbations.
  • The paper's stated scope limits these claims to finite-size, hardware-resolved signatures: they are diagnostics for NISQ devices, not a thermodynamic demonstration of topological order.

Reading between the lines

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

  • If the achieved variational fidelities were reported (C(θ_opt) per data point), one could check whether the hardware curves track the Wen–plaquette physics or merely the ansatz; the paper's silence on achieved fidelity leaves this the single most direct target for replication.
  • The appendix's noise estimate — a ≈0.78 ratio of measured to ideal 5×5 plaquette response from 64 CZ gates at ≈2.5×10⁻³ infidelity — implies a concrete scaling budget: pushing the representative-qubit protocol to larger lattices requires lower two-qubit error rates or shallower transpiled sweeps, since attenuation compounds with each CZ gate in the causal cone.
  • The qualitative similarity between the disorder and imaginary-field response maps suggests the two perturbations degrade the Wilson-loop signal through a common mechanism at the level of these finite-size diagnostics; a testable extension is to see whether matching a single scale (e.g., comparing h to γ) collapses both data sets onto one curve.
  • Because the layered sweep structure scales as roughly L_x + L_y − 3 stages rather than with the number of plaquettes, the representative-qubit construction is a natural template for preparing other commuting-stabilizer sectors (e.g., toric-code-type states) on 2D processors, provided the hardware connectivity supports the local controlled Pauli strings.
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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 / 4 minor

Summary. The paper reports a hardware study of finite-size topological crossover signatures in the Wen–plaquette model on IBM Quantum processors. For a 3×3 nine-qubit system, the authors variationally compile classically computed target states (ground states, quench states, and normalized non-Hermitian evolved states) and measure a plaquette stabilizer and a boundary Wilson loop, studying robustness against static disorder, amplified circuit noise, and an imaginary field. They find that local plaquette signatures remain resolvable in the strong-stabilizer regime, while nonlocal Wilson-loop response is more fragile, and they extend the protocol to a 5×5 stabilizer-sector state on a 2D processor using a coherent representative-qubit construction. The central claim is explicitly scoped as a finite-size, hardware-resolved study rather than a thermodynamic demonstration of topological order.

Significance. If fully supported, the paper would provide a practical benchmark for which finite-size topological diagnostics remain measurable on current NISQ hardware, with useful guidance on noise resilience and scalable stabilizer-sector preparation. The authors are appropriately cautious in the Discussion, and the inclusion of exact-diagonalization reference curves for the equilibrium diagnostics in Fig. 4 is a strength, as is the availability of data and code upon request. However, the central evidence chain depends on an unreported variational compilation fidelity and on comparisons that in Figs. 5–6 lack an exact-target reference. The reader's stress-test concern is legitimate and lands on a load-bearing gap: without knowing the achieved fidelity and ansatz depth for Eqs. (8)–(9), the hardware data cannot be unambiguously attributed to the Wen–plaquette target states.

major comments (4)
  1. [§II.B–II.C, Eqs. (8)–(9), App. S2] The paper never reports the achieved variational fidelity C(θ_opt), the number of layers n, or the optimizer convergence for any of the hardware data in Figs. 3–6. Since the noiseless reference curves are simulations of the same optimized variational circuits, agreement between hardware and noiseless data can be saturated even when the ansatz poorly approximates the target state; both share the same ansatz bias. The exact-diagonalization curves in Fig. 4 provide an independent benchmark for the equilibrium cases, but Figs. 5 and 6 compare only noiseless-circuit versus hardware. The variationally compressed readout in Eq. (S6) adds a second unquantified approximation. I request per-figure reporting of n, the optimized fidelity, and the optimization procedure, plus exact-target (statevector/ED) reference curves for the quench and non-Hermitian data in Figs. 5 and 6.
  2. [§II.C, Figs. 3–6] Several load-bearing qualitative claims — 'clearly resolved', 'close agreement', 'substantially more stable' — are made without any per-point statistics. Appendix S2 states that each data point is estimated from 20,000 shots, which is enough to compute binomial confidence intervals, but no error bars, standard deviations, or confidence intervals appear in any figure. Without uncertainty quantification, the apparent separation between curves cannot be quantitatively distinguished from shot noise or device drift. Please add error bars or, at minimum, representative confidence intervals for all hardware data points.
  3. [§II.C.2, Eq. (6), Fig. 4(b,d)] The disordered Hamiltonian in Eq. (6) uses disorder fields h_i drawn uniformly from [−h/2, h/2], but the manuscript does not state whether the h=5 curves in Fig. 4 correspond to a single disorder realization or to an average over many realizations. This is not a technicality: the claim that 'strong disorder strongly suppresses' the Wilson-loop response and that the plaquette stabilizer remains 'clearly resolved' depends on the statistical representativeness of the disorder sample. Please specify the number of realizations and how the plotted curves are aggregated, and provide the per-realization spread for the disorder cases.
  4. [§II.D, Fig. 7(b), Eq. (24)] The scalability benchmark in Fig. 7(b) claims that the lattice-averaged plaquette response degrades only weakly under up to 15 inserted rotations, but the figure shows no error bars or spread over the stated five random realizations. In addition, the analytic estimate in Eq. (S20) predicts F_hw/F_ideal ≈ 0.78, yet no quantitative comparison between Eq. (S20) and the measured values is shown. Please provide the measured mean and spread, and indicate whether Eq. (S20) is intended as a fitted estimate or a first-principles prediction. This is needed to support the claim that the prepared stabilizer sector is robust under the deliberately amplified perturbations.
minor comments (4)
  1. [Appendix S5] Typo: 'ibm miamiNightHawk device' should likely be 'ibm_miami (NightHawk) device' or similar.
  2. [Eq. (12) and App. S3] The ancilla-assisted interferometric readout is described in detail for the plaquette F1, but the controlled-W operation used for the Wilson loop is only mentioned in the main text. Please provide the corresponding circuit decomposition for controlled-W, or state explicitly that the same compressed-readout workflow of Eq. (S6) is used for W.
  3. [§II.C.4, Fig. 6] The caption of Fig. 6 says 'close agreement between noiseless and experimental curves', but the dashed and solid curves in panels (c,d) appear to differ systematically in some regimes. Please clarify whether the plotted values are raw or corrected for the systematic offset discussed in §II.C.3, and state this explicitly in the caption.
  4. [References] Several references to the authors' own work (e.g., Refs. [37,53,80,88]) are used for methodological context rather than for the specific results; this is acceptable, but the manuscript would benefit from a sentence in the Methods identifying which prior protocols are being reused from those papers.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: hardware data benchmark classically computed Wen–plaquette target states; self-citations are precedential, not load-bearing.

full rationale

The paper's derivation chain is not circular. Target states for the hardware experiments are defined independently by exact diagonalization of the Wen–plaquette Hamiltonians in Eqs. 4, 6, and 7, and Eq. 8 states explicitly that the variational circuit only approximates these classically computed targets: V_n(θ_opt)|φ0⟩ ≈ |ψ_tar⟩. The measured observables, ⟨F1⟩ and ⟨W⟩, are not fitted parameters; the variational parameters in Eq. 9 are optimized to match the target state, not to reproduce the hardware data. The crossover locations (J≈g, onset of the complex-spectrum fraction R) come from the exactly solvable model and ED response maps in Appendix S4, so the hardware curves are used in the correct evidential direction: they test whether known target-state signatures survive noise. The representative-qubit construction is derived from the stabilizer projectors in Eqs. 21–23 and Appendix S6.2, with the external reference [45] serving as a methodological precedent rather than as load-bearing support. The self-citations [37,53,80,88] appear in methodological or contextual clusters, but no central claim reduces to them. The main limitation—that the achieved variational fidelity, ansatz depth, and optimizer convergence for Eqs. 8–9 and Eq. S6 are not reported—is a genuine evidence gap about how well V_n represents |ψ_tar⟩, but it is not a circular reduction: the target states are independently specified, and the observables are not defined in terms of the fitted parameters. The paper's own stated limitations (e.g., the uncorrected systematic offset in Fig. 5 and the noise estimates in Appendix S6.3) affect precision and scalability, not the logical independence of the derivation.

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

The central claim rests on the standard algebra of the Wen–plaquette stabilizers plus several unquantified engineering assumptions. The dominant load-bearing assumption is that the variational ansatz V_n achieves high fidelity to the classically computed targets (Eq. 8) — the achieved fidelity is never reported. Secondary assumptions: identity-layer noise amplification isolates hardware noise (Eq. 11); the depolarizing-plus-T_eff error model (Eq. S19) describes the 5×5 device well enough for the 'fits our measured data' claim; normalized right-state expectation values are the relevant non-Hermitian observables. Study parameters (h=5, γ=0.1/0.5, perturbation angle range ±0.1, j=1,5 identity layers) are chosen by hand and bound the robustness claims. No new physical entities are postulated.

free parameters (5)
  • variational ansatz depth n and optimization tolerance = unreported
    The n-layer hardware-efficient circuit of App. S2 is used for every hardware target state, but the depth per figure and the achieved fidelity loss C(θ_opt) are never stated; this is the main uncontrolled quantity in the paper.
  • identity-layer count j = 0, 1, 5
    Noise-amplification protocol Eq. 11; the 'remains clearly resolved' robustness claim extends only to j=5 at an unspecified magnitude of introduced error.
  • disorder strength h and imaginary field γ = h = 5 (also 1, 10); γ = 0.1, 0.5
    The robustness conclusions are demonstrated at these hand-chosen strengths; the claims are relative to these values.
  • device calibration values in Eq. S20 = ε̄_CZ = 2.5×10⁻³, τ_CZ = 68 ns, T_eff = 100 μs
    The attenuation estimate is stated to 'fit our measured data' for the 5×5 plaquette response; if these values were selected to match the data, the agreement is fitted rather than predicted.
  • perturbation rotation range = [−0.1, 0.1]
    Random Rx angles for the Fig. 7(b) perturbation protocol; the 'weak degradation' claim is measured relative to this deliberately narrow range.
assumptions (6)
  • standard math Wen–plaquette stabilizers commute and square to identity: [F_i,F_j]=0, F_i²=I (Eqs. 1–3, App. S1)
    Underlies sector labeling, exact solvability, and the projector identity Eq. 10 / Eq. S2 used by the scalable preparation.
  • domain assumption Thermodynamic-limit Z2 topological order and |J|=|g| transition of the transverse Wen–plaquette model (Refs. [59,60])
    The finite-size 'crossover signatures' are interpreted as precursors of this transition; the physics is imported from prior literature, not derived.
  • ad hoc to paper Hardware-efficient variational ansatz can approximate every classically computed target state to high fidelity (Eq. 8, App. S2)
    Load-bearing for all of Figs. 3–6; the paper reports no achieved fidelity, so the approximation quality is assumed.
  • ad hoc to paper Identity layers V_id leave the ideal state unchanged, so degradation is attributable to hardware noise (Eq. 11)
    If V_id deviates from identity, the noise-amplification protocol measures compilation error plus noise, conflating the two.
  • domain assumption Two-qubit depolarizing error plus exponential T_eff decoherence models the device (Eq. S19)
    The 5×5 attenuation estimate that 'fits' Fig. 7(b) assumes this error model.
  • domain assumption Normalized right-eigenstate expectation values define the non-Hermitian observables (Eqs. 7, 18)
    A modeling convention for loss-like effects; hardware never runs nonunitary evolution, so the compiled states are classical constructs.

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Pith. "Pith review of Resolving topological order crossovers on NISQ hardware." pith.science (2026). https://pith.science/paper/F2KULM24

@misc{pith2026260721707,
  author       = {Pith},
  title        = {Pith review of: Resolving topological order crossovers on NISQ hardware},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F2KULM24}},
  note         = {Machine review of arXiv:2607.21707}
}
read the original abstract

Topological phases of matter provide a promising route toward robust quantum information processing, but on present-day noisy intermediate-scale quantum devices the experimentally relevant question is whether signatures of topological crossovers remain resolvable under realistic imperfections. Here, we address this question in the Wen--plaquette model through a two-stage strategy on the IBM Quantum hardware. We first use a tractable system to systematically characterize crossover signatures. Using variationally compiled equilibrium and quench-generated states, we resolve crossovers between stabilizer-dominated and trivial or disorder-dominated regimes through local plaquette stabilizers and a Wilson loop, and quantify their robustness against static disorder, deliberately amplified circuit noise, and effective non-Hermitian fields. The quench dynamics further reveal that plaquette-sector signatures remain substantially more stable deep in the strong-stabilizer regime than near the finite-size crossover. Building on the properties established in the small system, we extend the implementation to a physical two-dimensional IBM processor using a layered representative-qubit construction. The resulting lattice-averaged plaquette response exhibits only weak degradation under intentionally amplified local coherent perturbations. Together, these results connect controlled finite-size characterization with a scalable hardware implementation, providing a practical route for preparing and probing topological signatures on near-term quantum processors.

Figures

Figures reproduced from arXiv: 2607.21707 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. (b) presents a hardware demonstration of the robustness of the Wilson-loop signal under controllably amplified circuit noise. We probe the finite-size crossover between the Wen–plaquette and trivial polarized regimes by sweeping J/g at fixed g = 1 and measuring the bound￾ary Wilson loop ⟨W⟩ [Eq. 5]. To read out this long Pauli string, we use an ancilla-assisted interferometric protocol: the ancilla is prepared in |+… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Square-lattice non-variational stabilizer-sector preparation and robustness benchmark. (a) Schematic of the coherent [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

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Reference graph

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