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arxiv: 2501.04679 · v1 · submitted 2025-01-08 · 🪐 quant-ph · cond-mat.str-el

Exploring nontrivial topology at quantum criticality in a superconducting processor

Ziqi Tan , Ke Wang , Sheng Yang , Fanhao Shen , Feitong Jin , Xuhao Zhu , Yujie Ji , Shibo Xu
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Jiachen Chen Yaozu Wu Chuanyu Zhang Yu Gao Ning Wang Yiren Zou Aosai Zhang Tingting Li Zehang Bao Zitian Zhu Jiarun Zhong Zhengyi Cui Yihang Han Yiyang He Han Wang Jianan Yang Yanzhe Wang Jiayuan Shen Gongyu Liu Zixuan Song Jinfeng Deng Hang Dong Pengfei Zhang Shao-Kai Jian Hekang Li Zhen Wang Qiujiang Guo Chao Song Xue-Jia Yu H. Wang Hai-Qing Lin Fei Wu
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Pith reviewed 2026-05-23 05:36 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-el
keywords topological criticalitycluster Ising modelboundary g-functionentanglement spectrumbulk-boundary correspondencesuperconducting processorquantum simulation
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The pith

Preparing low-energy states of the cluster Ising model on a 100-qubit processor reveals nontrivial topology at criticality through boundary measurements.

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

The paper prepares low-lying critical states of the cluster Ising model on a superconducting processor with up to 100 qubits. It measures the boundary g-function from these states to identify nontrivial topology and uses entanglement spectrum analysis to detect two-fold degeneracy under periodic boundaries. This setup tests the bulk-boundary correspondence that is expected to hold even when the bulk has no energy gap. A sympathetic reader would care because the result shows how quantum processors can access topological features in gapless critical regimes that are hard to reach by other means.

Core claim

By preparing low-lying critical states of the cluster Ising model on a superconducting processor with up to 100 qubits and probing the boundary g-function together with two-fold topological degeneracy in the entanglement spectrum under periodic boundary conditions, the work experimentally verifies the universal bulk-boundary correspondence in topological critical systems.

What carries the argument

The boundary g-function extracted from prepared low-energy states, together with entanglement-spectrum degeneracy under periodic boundaries, which together identify the topological content of the critical cluster Ising model.

If this is right

  • Low-lying critical states can serve as resources for studying topology without requiring a bulk gap.
  • Programmable processors enable direct access to the interplay of topology and criticality in models with many qubits.
  • The bulk-boundary correspondence holds for the tested critical cluster Ising model, extending the correspondence beyond gapped topological phases.

Where Pith is reading between the lines

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

  • The same preparation and measurement approach could be applied to other critical Hamiltonians to test whether additional models host topological criticality.
  • If the method scales, it opens a route to classifying quantum phase transitions by their topological signatures rather than by symmetry alone.
  • Critical states prepared this way might be usable in hybrid quantum protocols that combine criticality with boundary topological protection.

Load-bearing premise

The states prepared on the processor remain close enough to the ideal low-energy eigenstates of the critical cluster Ising Hamiltonian that the measured g-function and entanglement degeneracy still reflect the intended topological properties.

What would settle it

A measurement on a well-calibrated processor that yields a g-function value matching the trivial case or an entanglement spectrum lacking the expected two-fold degeneracy at the critical point would falsify the verification of bulk-boundary correspondence.

Figures

Figures reproduced from arXiv: 2501.04679 by Aosai Zhang, Chao Song, Chuanyu Zhang, Fanhao Shen, Feitong Jin, Fei Wu, Gongyu Liu, Hai-Qing Lin, Hang Dong, Han Wang, Hekang Li, H. Wang, Jiachen Chen, Jianan Yang, Jiarun Zhong, Jiayuan Shen, Jinfeng Deng, Ke Wang, Ning Wang, Pengfei Zhang, Qiujiang Guo, Shao-Kai Jian, Sheng Yang, Shibo Xu, Tingting Li, Xue-Jia Yu, Xuhao Zhu, Yanzhe Wang, Yaozu Wu, Yihang Han, Yiren Zou, Yiyang He, Yu Gao, Yujie Ji, Zehang Bao, Zhengyi Cui, Zhen Wang, Ziqi Tan, Zitian Zhu, Zixuan Song.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
read the original abstract

The discovery of nontrivial topology in quantum critical states has introduced a new paradigm for classifying quantum phase transitions and challenges the conventional belief that topological phases are typically associated with a bulk energy gap. However, realizing and characterizing such topologically nontrivial quantum critical states with large particle numbers remains an outstanding experimental challenge in statistical and condensed matter physics. Programmable quantum processors can directly prepare and manipulate exotic quantum many-body states, offering a powerful path for exploring the physics behind these states. Here, we present an experimental exploration of the critical cluster Ising model by preparing its low-lying critical states on a superconducting processor with up to $100$ qubits. We develop an efficient method to probe the boundary $g$-function based on prepared low-energy states, which allows us to uniquely identify the nontrivial topology of the critical systems under study. Furthermore, by adapting the entanglement Hamiltonian tomography technique, we recognize two-fold topological degeneracy in the entanglement spectrum under periodic boundary condition, experimentally verifying the universal bulk-boundary correspondence in topological critical systems. Our results demonstrate the low-lying critical states as useful quantum resources for investigating the interplay between topology and quantum criticality.

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

1 major / 0 minor

Summary. The manuscript reports an experimental study of the critical cluster Ising model on a superconducting quantum processor with up to 100 qubits. Low-lying critical states are prepared, a method is developed to probe the boundary g-function for identifying nontrivial topology, and entanglement Hamiltonian tomography is used to observe two-fold degeneracy in the entanglement spectrum under periodic boundary conditions, thereby claiming experimental verification of the universal bulk-boundary correspondence in topological critical systems.

Significance. If the measured signatures are shown to be robust against hardware imperfections, the work would be significant as one of the first large-scale experimental realizations (N=100) of topology at quantum criticality, using programmable quantum hardware to access states that are difficult to prepare classically and providing direct tests of bulk-boundary correspondence beyond small-system numerics.

major comments (1)
  1. [State preparation and tomography sections] State-preparation and tomography sections: The central claim that the prepared states are faithful low-energy eigenstates of the ideal critical cluster Ising Hamiltonian (required for the g-function and entanglement degeneracy to reflect topology) lacks quantitative bounds on energy variance or overlap with the true critical states for N=100. Typical superconducting-processor error rates make this load-bearing; without such bounds or scaling analysis, the observed two-fold degeneracy and g-function could arise from non-critical or topologically trivial states mixed by noise.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point regarding the characterization of the prepared states. We address the concern below.

read point-by-point responses

  1. Referee: [State preparation and tomography sections] State-preparation and tomography sections: The central claim that the prepared states are faithful low-energy eigenstates of the ideal critical cluster Ising Hamiltonian (required for the g-function and entanglement degeneracy to reflect topology) lacks quantitative bounds on energy variance or overlap with the true critical states for N=100. Typical superconducting-processor error rates make this load-bearing; without such bounds or scaling analysis, the observed two-fold degeneracy and g-function could arise from non-critical or topologically trivial states mixed by noise.

    Authors: We agree that quantitative bounds on preparation fidelity would strengthen the central claim. Direct computation of energy variance or overlap with the exact critical eigenstate is intractable for N=100. However, the manuscript demonstrates that the measured boundary g-function and the two-fold entanglement-spectrum degeneracy match the universal predictions for the topologically nontrivial critical point of the cluster Ising model. These signatures are highly sensitive to deviations from criticality or loss of topology and are unlikely to appear in generic noisy or trivial states. We have performed supporting numerical checks on smaller systems (where exact diagonalization is feasible) that confirm the preparation protocol yields states with the expected topological features. We will revise the manuscript to include an expanded discussion of these smaller-system benchmarks together with a qualitative error analysis based on the observed robustness of the signatures. revision: partial

Circularity Check

0 steps flagged

No circularity: experimental measurements on hardware

full rationale

The paper's central claims rest on direct state preparation and tomography on a superconducting processor (up to 100 qubits), followed by measurement of the boundary g-function and entanglement spectrum degeneracy. These are empirical observations, not mathematical derivations. No equations reduce a prediction to a fitted input by construction, no self-citation chain carries the load-bearing premise, and no ansatz or uniqueness theorem is smuggled in. The bulk-boundary correspondence verification is data-driven and externally falsifiable via hardware results, placing the work in the self-contained experimental category.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the work relies on standard assumptions of quantum simulation fidelity that are not quantified here.

pith-pipeline@v0.9.0 · 5872 in / 1174 out tokens · 29456 ms · 2026-05-23T05:36:38.376536+00:00 · methodology

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Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Anomalous Dynamical Scaling at Topological Quantum Criticality

    cond-mat.str-el 2025-12 unverdicted novelty 7.0

    Topological quantum critical points exhibit anomalous dynamical scaling in boundary dynamics and defect production due to edge modes, beyond conventional Kibble-Zurek scaling.

  2. PT symmetry-enriched non-unitary criticality

    quant-ph 2025-09 unverdicted novelty 7.0

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

  3. Topology and edge modes surviving criticality in non-Hermitian Floquet systems

    cond-mat.mes-hall 2026-02 unverdicted novelty 6.0

    Non-Hermitian Floquet systems host gapless symmetry-protected topological phases with unified winding numbers and robust edge modes surviving at criticality.

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

    cond-mat.str-el 2025-09 unverdicted novelty 6.0

    An exact relation is derived between bulk entanglement spectrum and boundary energy spectrum at topological criticality in free-fermion systems, allowing edge-mode degeneracy to be read from bulk data in arbitrary dimensions.

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

    cond-mat.str-el 2025-09 unverdicted novelty 6.0

    Derives exact bulk-boundary correspondence allowing extraction of edge-mode degeneracy from bulk entanglement spectrum in critical free-fermion systems of arbitrary dimensions.

  6. Deconfined criticality as intrinsically gapless topological state in one dimension

    cond-mat.str-el 2025-03 unverdicted novelty 6.0

    Deconfined criticality in a 1D lattice model is shown to be an intrinsically gapless topological state whose mixed anomaly enforces robust edge modes without gapped counterparts.

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