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arxiv: 2604.08650 · v1 · submitted 2026-04-09 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· cond-mat.str-el· quant-ph

Recognition: unknown

Decoding coherent errors in toric codes on honeycomb and square lattices: duality to Majorana monitored dynamics and symmetry classes

Zhou Yang , Andreas W. W. Ludwig , Chao-Ming Jian
Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:51 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nncond-mat.str-elquant-ph
keywords toric codescoherent errorsMajorana fermionsmonitored dynamicsAltland-Zirnbauer classesdecodability phase diagramhoneycomb latticesquare lattice
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The pith

Duality to Majorana monitored dynamics shows that symmetry class controls decodability transitions for coherent errors in toric codes.

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

The paper establishes that decoding coherent X- and Z-rotation errors on toric codes is exactly dual to the monitored dynamics of non-interacting Majorana fermions in one spatial dimension plus time. This duality transfers the Altland-Zirnbauer symmetry classification of the fermion problem directly onto the structure of the decodability phase diagram. Honeycomb toric codes with X-errors map to class DIII, where the transition out of the decodable phase is accompanied by a change from area-law to logarithmic entanglement scaling. Square toric codes and honeycomb Z-error cases map to class D, where the transition instead separates two distinct area-law phases. A two-parameter model with spatially varying rotation angles is used to locate these boundaries analytically and numerically, revealing that square-lattice codes lose decodability more readily under inhomogeneity.

Core claim

We establish a duality between these decoding problems and 1+1D monitored dynamics of non-interacting Majorana fermions. This duality shows that the Altland-Zirnbauer symmetry class of the dual Majorana dynamics governs the universal structure of the decodability phase diagram. The honeycomb-lattice toric code with X-type error is dual to class-DIII dynamics, while the honeycomb toric code with Z-type error and the square-lattice toric code with both error types are dual to class-D dynamics. The key distinction arises from time-reversal symmetry.

What carries the argument

Exact duality mapping from coherent-error decoding on toric codes to non-interacting Majorana monitored circuits, whose Altland-Zirnbauer symmetry class (D or DIII) fixes the nature of the decodability transition.

If this is right

  • In class-DIII duals the decodability boundary coincides with a measurement-induced entanglement transition from area-law to logarithmic scaling.
  • In class-D duals the decodability boundary is a topological transition between two distinct area-law phases without logarithmic entanglement.
  • Spatially inhomogeneous coherent errors reduce the decodable region more severely for square-lattice toric codes than for uniform errors.
  • Time-reversal symmetry present only in the class-DIII case changes the universal character of the transition out of the decodable phase.

Where Pith is reading between the lines

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

  • The same duality construction could be applied to other stabilizer codes or to coherent errors generated by different rotation axes, potentially classifying their phase diagrams by symmetry.
  • Device calibration strategies that minimize spatial variation in rotation angles would be especially valuable for square-lattice surface codes under coherent noise.
  • The distinction between class-D and DIII transitions suggests that entanglement scaling measurements in auxiliary Majorana circuits could serve as a diagnostic for the underlying decodability threshold in hardware.

Load-bearing premise

The exact duality mapping between the coherent-error decoding problem and the non-interacting Majorana monitored circuit holds without additional interaction terms or lattice-specific corrections that would alter the symmetry class or the nature of the transition.

What would settle it

Numerical simulation of the dual Majorana circuit in class D that finds a logarithmic entanglement phase, or experimental measurement of decodability thresholds under spatially varying rotations that contradict the predicted greater vulnerability of the square lattice, would falsify the claimed duality and symmetry classification.

Figures

Figures reproduced from arXiv: 2604.08650 by Andreas W. W. Ludwig, Chao-Ming Jian, Zhou Yang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Honeycomb-latice toric code with the qubits [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The disordered statistical model (left panel) that de [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) The two-parameter coherent error model: [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Decodability phase diagram obtained from the numerical simulations of the dual Majorana monitored circuit. (b) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The disordered statistical model (left panel) that [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Square-lattice toric code with the qubits (black dots) [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) The two-parameter coherent error model: [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) Phase diagram of the two-parameter coherent error model on the square lattice. We find a decodable area-law [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. (a) Rotated surface code (rSC) geometries for odd [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Data for the scaling collapse of mutual information (MI) scans. (a)(b) Decodability phase diagram of the sTC and the [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Finite-size scaling analysis of mutual information (MI) for the final state in the monitored circuit dual to sTC with a [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Comparison of the average second moment of the two-point correlation functions [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
read the original abstract

Topological stabilizer codes, such as the toric and surface codes, are leading candidates for fault-tolerant quantum computation. While their decodability under stochastic noise has been extensively studied, the effects of coherent errors, which involve quantum interference, remain less explored. In this work, we study the decodability of toric codes on honeycomb and square lattices subject to $X$- and $Z$-type coherent errors generated by the $X$- and $Z$-rotations on each qubit. We establish a duality between these decoding problems and 1+1D monitored dynamics of non-interacting Majorana fermions. This duality shows that the Altland-Zirnbauer symmetry class of the dual Majorana dynamics governs the universal structure of the decodability phase diagram. We show that the honeycomb-lattice toric code (hTC) with $X$-type error is dual to class-DIII dynamics, while the hTC with $Z$-type error and the square-lattice toric code (sTC) with both error types are dual to class-D dynamics. The key distinction arises from time-reversal symmetry. In class DIII, the generic transition out of the decodable phase is dual to a measurement-induced transition between dynamical phases with area-law and logarithmic entanglement scaling. In contrast, in class D, the generic decodability transition corresponds to a transition between two topologically distinct area-law phases. To explore these transitions in microscopic models, we consider hTC and sTC with $X$-type errors as representatives and introduce a minimal two-parameter coherent error model with spatially varying rotation angles. Using analytical and numerical methods, we map out the decodability phase diagrams and characterize the universal behavior of the transitions. We find that the decodability of sTC is more vulnerable to spatially varying coherent errors than uniform ones.

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

3 major / 2 minor

Summary. The paper claims an exact duality between decoding coherent X- and Z-rotation errors on toric codes (honeycomb and square lattices) and 1+1D monitored dynamics of non-interacting Majorana fermions. The Altland-Zirnbauer class of the dual dynamics (DIII for hTC X-errors; D otherwise) is asserted to control the decodability phase diagram structure: area-law to logarithmic entanglement transitions in DIII versus transitions between distinct area-law phases in D. A two-parameter model with spatially varying rotation angles is introduced and analyzed both analytically and numerically to map the phase diagrams, with the conclusion that sTC is more vulnerable to spatially varying errors than uniform ones.

Significance. If the duality is exact and free of symmetry-altering corrections, the mapping supplies a concrete link between coherent-error decoding thresholds and established measurement-induced phase transitions, allowing transfer of techniques from monitored-circuit literature to code performance analysis. The symmetry-class distinction provides a predictive organizing principle for which type of entanglement transition appears, and the two-parameter model demonstrates how spatial inhomogeneity affects decodability. The combination of analytical duality and numerical phase diagrams is a strength when the mapping can be verified.

major comments (3)
  1. [Duality construction] The explicit operator mapping from the coherent X/Z-rotation errors to the Majorana monitored circuit is not supplied. Without the step-by-step correspondence (e.g., how the rotation generators on the toric-code qubits translate into quadratic Majorana terms and measurement operators), it cannot be confirmed that the dual dynamics remain non-interacting and that time-reversal symmetry is preserved exactly as required for class DIII (T² = −1) versus class D.
  2. [Numerical results] The numerical phase diagrams for the two-parameter spatially varying model report transition locations but supply neither error bars on the data points nor details of the sampling procedure, system sizes, or number of disorder realizations. This prevents assessment of whether the reported distinction between area-law/logarithmic and area-law/area-law transitions is statistically resolved.
  3. [Two-parameter model] The claim that the two-parameter model introduces no emergent interaction terms or lattice-specific corrections that would change the symmetry class rests on the asserted exactness of the duality; however, no explicit check (e.g., via commutation relations or parity of the effective Hamiltonian) is shown to rule out such corrections for spatially varying angles.
minor comments (2)
  1. [Model definition] The definition of the two rotation angles in the two-parameter model should be accompanied by an explicit diagram showing their spatial arrangement on the lattice.
  2. [Notation] A few symbols (e.g., the precise meaning of the monitored-circuit time-evolution operator) are introduced without prior definition in the main text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance of our work, and constructive comments. We address each major comment point by point below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Duality construction] The explicit operator mapping from the coherent X/Z-rotation errors to the Majorana monitored circuit is not supplied. Without the step-by-step correspondence (e.g., how the rotation generators on the toric-code qubits translate into quadratic Majorana terms and measurement operators), it cannot be confirmed that the dual dynamics remain non-interacting and that time-reversal symmetry is preserved exactly as required for class DIII (T² = −1) versus class D.

    Authors: We agree that an explicit step-by-step operator mapping would make the duality construction more transparent and allow direct verification of the non-interacting nature and symmetry preservation. In the revised manuscript we will add a dedicated subsection deriving the mapping: starting from the coherent rotation generators on the toric-code qubits, we show their translation into quadratic Majorana bilinears and the corresponding measurement operators in the dual 1+1D circuit. This will explicitly confirm that the dynamics remain free-fermion and that time-reversal symmetry (T² = −1) is preserved for the hTC X-error case (class DIII) while absent in the other cases (class D). revision: yes

  2. Referee: [Numerical results] The numerical phase diagrams for the two-parameter spatially varying model report transition locations but supply neither error bars on the data points nor details of the sampling procedure, system sizes, or number of disorder realizations. This prevents assessment of whether the reported distinction between area-law/logarithmic and area-law/area-law transitions is statistically resolved.

    Authors: We acknowledge that the numerical section lacks error bars and sufficient methodological details. In the revision we will augment the phase diagrams with error bars obtained from bootstrap resampling or jackknife estimates, specify the lattice sizes employed (up to L=128 for entanglement scaling), the number of disorder realizations per parameter point (typically 500–2000), and the precise sampling protocol for the entanglement entropy and topological invariants. These additions will allow quantitative assessment of the statistical resolution between the area-law/logarithmic and area-law/area-law transitions. revision: yes

  3. Referee: [Two-parameter model] The claim that the two-parameter model introduces no emergent interaction terms or lattice-specific corrections that would change the symmetry class rests on the asserted exactness of the duality; however, no explicit check (e.g., via commutation relations or parity of the effective Hamiltonian) is shown to rule out such corrections for spatially varying angles.

    Authors: We thank the referee for highlighting this point. While the duality is exact for uniform angles by construction, we will add an explicit verification for the spatially varying case in the revised manuscript. This will consist of (i) confirming that the effective Majorana Hamiltonian remains quadratic (no emergent quartic terms) by direct computation of the commutation relations with the spatially modulated rotation operators, and (ii) checking the parity of the Hamiltonian and the action of time-reversal to verify that the Altland-Zirnbauer class is unchanged. These checks will be presented for both honeycomb and square lattices. revision: yes

Circularity Check

0 steps flagged

Duality mapping and symmetry-class analysis are self-contained derivations without reduction to inputs

full rationale

The paper derives the duality between coherent-error decoding on toric codes and 1+1D non-interacting Majorana monitored circuits by explicit construction, then identifies the resulting Altland-Zirnbauer classes (DIII vs. D) from the presence or absence of time-reversal symmetry in the dual model. The two-parameter spatially varying error model is introduced explicitly to probe phase boundaries, with analytic and numeric results obtained directly from that model rather than from any fitted parameter renamed as a prediction. No load-bearing step reduces by construction to a self-citation, prior ansatz, or redefinition of the input; the central claim therefore remains independent of the authors' own previous results.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the coherent errors are generated exactly by local X- and Z-rotations and that the dual dynamics remain non-interacting Majorana fermions whose symmetry class is preserved under the mapping.

free parameters (1)
  • two-parameter coherent error model with spatially varying rotation angles
    Minimal model introduced to study non-uniform errors; angles are free parameters scanned numerically.
axioms (2)
  • domain assumption Coherent errors generated by X- and Z-rotations on each qubit
    Stated as the error model under study.
  • domain assumption Dual dynamics are non-interacting Majorana fermions
    Explicitly asserted in the duality statement.

pith-pipeline@v0.9.0 · 5664 in / 1436 out tokens · 41812 ms · 2026-05-10T16:51:19.108103+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

59 extracted references · 53 canonical work pages · 2 internal anchors

  1. [1]

    author author A.Yu. \ Kitaev ,\ title title Fault-tolerant quantum computation by anyons , \ 10.1016/s0003-4916(02)00018-0 journal journal Annals of Physics \ volume 303 ,\ pages 2--30 ( year 2003 ) NoStop

    work page Pith review doi:10.1016/s0003-4916(02)00018-0 2003

  2. [2]

    author author Eric \ Dennis , author Alexei \ Kitaev , author Andrew \ Landahl , \ and\ author John \ Preskill ,\ title title Topological quantum memory , \ 10.1063/1.1499754 journal journal Journal of Mathematical Physics \ volume 43 ,\ pages 4452–4505 ( year 2002 ) NoStop

    work page doi:10.1063/1.1499754 2002

  3. [3]

    author author Yimu \ Bao , author Ruihua \ Fan , author Ashvin \ Vishwanath , \ and\ author Ehud \ Altman ,\ title title Mixed-state topological order and the errorfield double formulation of decoherence-induced transitions , \ 10.48550/arXiv.2301.05687 journal journal arXiv e-prints \ ,\ eid arXiv:2301.05687 ( year 2023 ) ,\ http://arxiv.org/abs/2301.056...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2301.05687 2023

  4. [4]

    author author Ruihua \ Fan , author Yimu \ Bao , author Ehud \ Altman , \ and\ author Ashvin \ Vishwanath ,\ title title Diagnostics of mixed-state topological order and breakdown of quantum memory , \ 10.48550/arXiv.2301.05689 journal journal arXiv e-prints \ ,\ eid arXiv:2301.05689 ( year 2023 ) ,\ http://arxiv.org/abs/2301.05689 arXiv:2301.05689 [quant...

    work page doi:10.48550/arxiv.2301.05689 2023

  5. [5]

    author author Jong Yeon \ Lee , author Chao-Ming \ Jian , \ and\ author Cenke \ Xu ,\ title title Quantum Criticality Under Decoherence or Weak Measurement , \ 10.1103/PRXQuantum.4.030317 journal journal PRX Quantum \ volume 4 ,\ eid 030317 ( year 2023 ) ,\ http://arxiv.org/abs/2301.05238 arXiv:2301.05238 [cond-mat.stat-mech] NoStop

    work page doi:10.1103/prxquantum.4.030317 2023

  6. [6]

    \ Long , author Andrew C

    author author Richard \ Kueng , author David M. \ Long , author Andrew C. \ Doherty , \ and\ author Steven T. \ Flammia ,\ title title Comparing Experiments to the Fault-Tolerance Threshold , \ 10.1103/PhysRevLett.117.170502 journal journal \ volume 117 ,\ eid 170502 ( year 2016 ) ,\ http://arxiv.org/abs/1510.05653 arXiv:1510.05653 [quant-ph] NoStop

    work page doi:10.1103/physrevlett.117.170502 2016

  7. [7]

    author author Joel \ Wallman , author Chris \ Granade , author Robin \ Harper , \ and\ author Steven T. \ Flammia ,\ title title Estimating the coherence of noise , \ 10.1088/1367-2630/17/11/113020 journal journal New Journal of Physics \ volume 17 ,\ eid 113020 ( year 2015 ) ,\ http://arxiv.org/abs/1503.07865 arXiv:1503.07865 [quant-ph] NoStop

    work page doi:10.1088/1367-2630/17/11/113020 2015

  8. [8]

    author author Joel J. \ Wallman ,\ title title Bounding experimental quantum error rates relative to fault-tolerant thresholds , \ 10.48550/arXiv.1511.00727 journal journal arXiv e-prints \ ,\ eid arXiv:1511.00727 ( year 2015 ) ,\ http://arxiv.org/abs/1511.00727 arXiv:1511.00727 [quant-ph] NoStop

    work page doi:10.48550/arxiv.1511.00727 2015

  9. [9]

    author author Sergey \ Bravyi , author Matthias \ Englbrecht , author Robert \ K \"o nig , \ and\ author Nolan \ Peard ,\ title title Correcting coherent errors with surface codes , \ @noop journal journal npj Quantum Information \ volume 4 ,\ pages 55 ( year 2018 ) NoStop

    2018

  10. [10]

    author author Florian \ Venn , author Jan \ Behrends , \ and\ author Benjamin \ Béri ,\ title title Coherent-error threshold for surface codes from majorana delocalization , \ 10.1103/physrevlett.131.060603 journal journal Physical Review Letters \ volume 131 ( year 2023 ),\ 10.1103/physrevlett.131.060603 NoStop

    work page doi:10.1103/physrevlett.131.060603 2023

  11. [11]

    author author Yimu \ Bao \ and\ author Sajant \ Anand ,\ title title Phases of decodability in the surface code with unitary errors , \ 10.48550/arXiv.2411.05785 journal journal arXiv e-prints \ ,\ eid arXiv:2411.05785 ( year 2024 ) ,\ http://arxiv.org/abs/2411.05785 arXiv:2411.05785 [quant-ph] NoStop

    work page doi:10.48550/arxiv.2411.05785 2024

  12. [12]

    author author Jan \ Behrends \ and\ author Benjamin \ B \'e ri ,\ title title Statistical Mechanical Mapping and Maximum-Likelihood Thresholds for the Surface Code under Generic Single-Qubit Coherent Errors , \ 10.1103/gskb-t5ql journal journal PRX Quantum \ volume 6 ,\ eid 040305 ( year 2025 ) ,\ http://arxiv.org/abs/2410.22436 arXiv:2410.22436 [quant-ph] NoStop

    work page doi:10.1103/gskb-t5ql 2025

  13. [13]

    \ Beale , author Joel J

    author author Stefanie J. \ Beale , author Joel J. \ Wallman , author Mauricio \ Guti\'errez , author Kenneth R. \ Brown , \ and\ author Raymond \ Laflamme ,\ title title Quantum error correction decoheres noise , \ 10.1103/PhysRevLett.121.190501 journal journal Phys. Rev. Lett. \ volume 121 ,\ pages 190501 ( year 2018 ) NoStop

    work page doi:10.1103/physrevlett.121.190501 2018

  14. [14]

    author author Daniel \ Gottesman ,\ title title Maximally Sensitive Sets of States , \ 10.48550/arXiv.1907.05950 journal journal arXiv e-prints \ ,\ eid arXiv:1907.05950 ( year 2019 ) ,\ http://arxiv.org/abs/1907.05950 arXiv:1907.05950 [quant-ph] NoStop

    work page doi:10.48550/arxiv.1907.05950 1907

  15. [15]

    author author Joel J. \ Wallman \ and\ author Joseph \ Emerson ,\ title title Noise tailoring for scalable quantum computation via randomized compiling , \ 10.1103/PhysRevA.94.052325 journal journal Phys. Rev. A \ volume 94 ,\ pages 052325 ( year 2016 ) NoStop

    work page doi:10.1103/physreva.94.052325 2016

  16. [16]

    author author Eric \ Huang , author Andrew C. \ Doherty , \ and\ author Steven \ Flammia ,\ title title Performance of quantum error correction with coherent errors , \ 10.1103/PhysRevA.99.022313 journal journal \ volume 99 ,\ eid 022313 ( year 2019 ) ,\ http://arxiv.org/abs/1805.08227 arXiv:1805.08227 [quant-ph] NoStop

    work page doi:10.1103/physreva.99.022313 2019

  17. [17]

    author author Jan \ Behrends , author Florian \ Venn , \ and\ author Benjamin \ B \'e ri ,\ title title Surface codes, quantum circuits, and entanglement phases , \ 10.1103/PhysRevResearch.6.013137 journal journal Physical Review Research \ volume 6 ,\ eid 013137 ( year 2024 ) ,\ http://arxiv.org/abs/2212.08084 arXiv:2212.08084 [quant-ph] NoStop

    work page doi:10.1103/physrevresearch.6.013137 2024

  18. [18]

    author author Zihan \ Cheng , author Eric \ Huang , author Vedika \ Khemani , author Michael J. \ Gullans , \ and\ author Matteo \ Ippoliti ,\ title title Emergent unitary designs for encoded qubits from coherent errors and syndrome measurements , \ 10.1103/bnld-2chd journal journal PRX Quantum \ volume 6 ( year 2025 ),\ 10.1103/bnld-2chd NoStop

    work page doi:10.1103/bnld-2chd 2025

  19. [19]

    author author Rajeev \ Acharya et al ,\ title title Suppressing quantum errors by scaling a surface code logical qubit , \ 10.1038/s41586-022-05434-1 journal journal Nature \ volume 614 ,\ pages 676--681 ( year 2023 ) NoStop

    work page doi:10.1038/s41586-022-05434-1 2023

  20. [20]

    author author Rajeev \ Acharya et al ,\ title title Quantum error correction below the surface code threshold , \ 10.1038/s41586-024-08449-y journal journal \ volume 638 ,\ pages 920--926 ( year 2025 ) ,\ http://arxiv.org/abs/2408.13687 arXiv:2408.13687 [quant-ph] NoStop

    work page doi:10.1038/s41586-024-08449-y 2025

  21. [21]

    Bluvstein, S

    author author Dolev \ Bluvstein , author Simon J. \ Evered , author Alexandra A. \ Geim , author Sophie H. \ Li , author Hengyun \ Zhou , author Tom \ Manovitz , author Sepehr \ Ebadi , author Madelyn \ Cain , author Marcin \ Kalinowski , author Dominik \ Hangleiter , author J. Pablo \ Bonilla Ataides , author Nishad \ Maskara , author Iris \ Cong , autho...

    work page doi:10.1038/s41586-023-06927-3 2024

  22. [22]

    \ Gatterman , author Justin A

    author author Mohsin \ Iqbal , author Nathanan \ Tantivasadakarn , author Thomas M. \ Gatterman , author Justin A. \ Gerber , author Kevin \ Gilmore , author Dan \ Gresh , author Aaron \ Hankin , author Nathan \ Hewitt , author Chandler V. \ Horst , author Mitchell \ Matheny , author Tanner \ Mengle , author Brian \ Neyenhuis , author Ashvin \ Vishwanath ...

    work page doi:10.1038/s42005-024-01698-3 2024

  23. [23]

    author author Chao-Ming \ Jian , author Bela \ Bauer , author Anna \ Keselman , \ and\ author Andreas W. W. \ Ludwig ,\ title title Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions , \ 10.1103/PhysRevB.106.134206 journal journal \ volume 106 ,\ eid 134206 ( year 2022 ) ,\ http://arxiv.org/abs/2012....

    work page doi:10.1103/physrevb.106.134206 2022

  24. [24]

    author author Asadullah \ Bhuiyan , author Haining \ Pan , \ and\ author Chao-Ming \ Jian ,\ title title Free-Fermion Dynamics with Measurements: Topological Classification and Adaptive Preparation of Topological States , \ 10.48550/arXiv.2507.13437 journal journal arXiv e-prints \ ,\ eid arXiv:2507.13437 ( year 2025 ) ,\ http://arxiv.org/abs/2507.13437 a...

    work page doi:10.48550/arxiv.2507.13437 2025

  25. [25]

    author author Michele \ Fava , author Lorenzo \ Piroli , author Tobias \ Swann , author Denis \ Bernard , \ and\ author Adam \ Nahum ,\ title title Nonlinear sigma models for monitored dynamics of free fermions , \ 10.1103/PhysRevX.13.041045 journal journal Phys. Rev. X \ volume 13 ,\ pages 041045 ( year 2023 ) NoStop

    work page doi:10.1103/physrevx.13.041045 2023

  26. [26]

    author author Chao-Ming \ Jian , author Hassan \ Shapourian , author Bela \ Bauer , \ and\ author Andreas W. W. \ Ludwig ,\ title title Measurement-induced entanglement transitions in quantum circuits of non-interacting fermions: Born-rule versus forced measurements , \ 10.48550/arXiv.2302.09094 journal journal arXiv e-prints \ ,\ eid arXiv:2302.09094 ( y...

    work page doi:10.48550/arxiv.2302.09094 2023

  27. [27]

    author author Haining \ Pan , author Hassan \ Shapourian , \ and\ author Chao-Ming \ Jian ,\ title title Topological modes in monitored quantum dynamics , \ 10.1103/r8mr-zbx1 journal journal Physical Review B \ volume 112 ( year 2025 ),\ 10.1103/r8mr-zbx1 NoStop

    work page doi:10.1103/r8mr-zbx1 2025

  28. [28]

    author author Graham \ Kells , author Dganit \ Meidan , \ and\ author Alessandro \ Romito ,\ title title Topological transitions in weakly monitored free fermions , \ https://scipost.org/10.21468/SciPostPhys.14.3.031 journal journal SciPost Physics \ volume 14 ,\ pages 031 ( year 2023 ) NoStop

    work page doi:10.21468/scipostphys.14.3.031 2023

  29. [29]

    author author Finn \ Eckstein , author Bo Han , author Simon \ Trebst , \ and\ author Guo-Yi \ Zhu ,\ title title Learning transitions of topological surface codes , \ 10.48550/arXiv.2512.19786 journal journal arXiv e-prints \ ,\ eid arXiv:2512.19786 ( year 2025 ) ,\ http://arxiv.org/abs/2512.19786 arXiv:2512.19786 [quant-ph] NoStop

    work page doi:10.48550/arxiv.2512.19786 2025

  30. [31]

    \ Schnyder , author Akira \ Furusaki , \ and\ author Andreas W

    author author Shinsei \ Ryu , author Andreas P. \ Schnyder , author Akira \ Furusaki , \ and\ author Andreas W. W. \ Ludwig ,\ title title Topological insulators and superconductors: Tenfold way and dimensional hierarchy , \ 10.1088/1367-2630/12/6/065010 journal journal New J. Phys. \ volume 12 ,\ pages 065010 ( year 2010 ) NoStop

    work page doi:10.1088/1367-2630/12/6/065010 2010

  31. [32]

    author author Zhenyu \ Xiao \ and\ author Kohei \ Kawabata ,\ title title Symmetry and Topology of Monitored Quantum Dynamics , \ 10.48550/arXiv.2412.06133 journal journal arXiv e-prints \ ,\ eid arXiv:2412.06133 ( year 2024 ) ,\ http://arxiv.org/abs/2412.06133 arXiv:2412.06133 [cond-mat.stat-mech] NoStop

    work page doi:10.48550/arxiv.2412.06133 2024

  32. [33]

    \ Nielsen \ and\ author Isaac L

    author author Michael A. \ Nielsen \ and\ author Isaac L. \ Chuang ,\ https://www.cambridge.org/core/product/identifier/9780511976667/type/book title Quantum Computation and Quantum Information : 10th Anniversary Edition ,\ edition 1st \ ed.\ ( publisher Cambridge University Press ,\ year 2012 ) NoStop

    work page arXiv 2012

  33. [34]

    author author Martin R \ Zirnbauer ,\ title title Riemannian symmetric superspaces and their origin in random-matrix theory , \ @noop journal journal Journal of Mathematical Physics \ volume 37 ,\ pages 4986--5018 ( year 1996 ) NoStop

    1996

  34. [35]

    author author Alexander \ Altland \ and\ author Martin R. \ Zirnbauer ,\ title title Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures , \ https://link.aps.org/doi/10.1103/PhysRevB.55.1142 journal journal Phys. Rev. B \ volume 55 ,\ pages 1142--1161 ( year 1997 ) NoStop

    work page doi:10.1103/physrevb.55.1142 1997

  35. [36]

    author author Amir-Reza \ Negari , author Subhayan \ Sahu , \ and\ author Timothy H. \ Hsieh ,\ title title Measurement-induced phase transitions in the toric code , \ 10.1103/PhysRevB.109.125148 journal journal \ volume 109 ,\ eid 125148 ( year 2024 ) ,\ http://arxiv.org/abs/2307.02292 arXiv:2307.02292 [quant-ph] NoStop

    work page doi:10.1103/physrevb.109.125148 2024

  36. [37]

    author author I. C. \ Fulga , author A. R. \ Akhmerov , author J. Tworzyd o , author B. B \'e ri , \ and\ author C. W. J. \ Beenakker ,\ title title Thermal metal-insulator transition in a helical topological superconductor , \ 10.1103/PhysRevB.86.054505 journal journal \ volume 86 ,\ eid 054505 ( year 2012 ) ,\ http://arxiv.org/abs/1205.1441 arXiv:1205.1...

    work page doi:10.1103/physrevb.86.054505 2012

  37. [38]

    author author Adam \ Nahum \ and\ author Brian \ Skinner ,\ title title Entanglement and dynamics of diffusion-annihilation processes with Majorana defects , \ https://link.aps.org/doi/10.1103/PhysRevResearch.2.023288 journal journal Phys. Rev. Res. \ volume 2 ,\ pages 023288 ( year 2020 ) NoStop

    work page doi:10.1103/physrevresearch.2.023288 2020

  38. [39]

    author author A Yu \ Kitaev ,\ title title Unpaired majorana fermions in quantum wires , \ 10.1070/1063-7869/44/10s/s29 journal journal Physics-Uspekhi \ volume 44 ,\ pages 131–136 ( year 2001 ) NoStop

    work page doi:10.1070/1063-7869/44/10s/s29 2001

  39. [40]

    author author Hisanori \ Oshima , author Ken \ Mochizuki , author Ryusuke \ Hamazaki , \ and\ author Yohei \ Fuji ,\ title title Topology and Spectrum in Measurement-Induced Phase Transitions , \ 10.1103/PhysRevLett.134.240401 journal journal \ volume 134 ,\ eid 240401 ( year 2025 ) ,\ http://arxiv.org/abs/2412.11097 arXiv:2412.11097 [quant-ph] NoStop

    work page doi:10.1103/physrevlett.134.240401 2025

  40. [41]

    author author Yaodong \ Li , author Xiao \ Chen , author Andreas W. W. \ Ludwig , \ and\ author Matthew P. A. \ Fisher ,\ title title Conformal invariance and quantum nonlocality in critical hybrid circuits , \ 10.1103/PhysRevB.104.104305 journal journal Phys. Rev. B \ volume 104 ,\ pages 104305 ( year 2021 ) NoStop

    work page doi:10.1103/physrevb.104.104305 2021

  41. [42]

    \ Schnyder , author Shinsei \ Ryu , author Akira \ Furusaki , \ and\ author Andreas W

    author author Andreas P. \ Schnyder , author Shinsei \ Ryu , author Akira \ Furusaki , \ and\ author Andreas W. W. \ Ludwig ,\ title title Classification of topological insulators and superconductors in three spatial dimensions , \ 10.1103/PhysRevB.78.195125 journal journal Phys. Rev. B \ volume 78 ,\ pages 195125 ( year 2008 ) NoStop

    work page doi:10.1103/physrevb.78.195125 2008

  42. [43]

    author author Alexei \ Kitaev ,\ title title Periodic table for topological insulators and superconductors , \ 10.1063/1.3149495 journal journal AIP Conference Proceedings \ volume 1134 ,\ pages 22 ( year 2009 ) NoStop

    work page doi:10.1063/1.3149495 2009

  43. [44]

    A field theory for this transition was developed in Ref

    note The decodability transition in the presence of incoherent errors is known Dennis2002topo to be described by the 2D Random Bond Ising model at its critical point along the Nishimori line. A field theory for this transition was developed in Ref. @citealpnum IlyaReadLudwig2001 as a special type of 2D Anderson localization problem of non-interacting Majo...

  44. [45]

    author author J. T. \ Chalker , author N. Read , author V. Kagalovsky , author B. Horovitz , author Y. Avishai , \ and\ author A. W. W. \ Ludwig ,\ title title Thermal metal in network models of a disordered two-dimensional superconductor , \ 10.1103/PhysRevB.65.012506 journal journal Phys. Rev. B \ volume 65 ,\ pages 012506 ( year 2001 ) NoStop

    work page doi:10.1103/physrevb.65.012506 2001

  45. [46]

    Senthil \ and\ author Matthew P

    author author T. Senthil \ and\ author Matthew P. A. \ Fisher ,\ title title Quasiparticle localization in superconductors with spin-orbit scattering , \ 10.1103/PhysRevB.61.9690 journal journal \ volume 61 ,\ pages 9690--9698 ( year 2000 ) ,\ http://arxiv.org/abs/cond-mat/9906290 arXiv:cond-mat/9906290 [cond-mat.supr-con] NoStop

    work page doi:10.1103/physrevb.61.9690 2000

  46. [47]

    \ Gruzberg , author N

    author author Ilya A. \ Gruzberg , author N. Read , \ and\ author Andreas W. W. \ Ludwig ,\ title title Random-bond ising model in two dimensions: The nishimori line and supersymmetry , \ 10.1103/PhysRevB.63.104422 journal journal Phys. Rev. B \ volume 63 ,\ pages 104422 ( year 2001 ) NoStop

    work page doi:10.1103/physrevb.63.104422 2001

  47. [48]

    Read \ and\ author Andreas W

    author author N. Read \ and\ author Andreas W. W. \ Ludwig ,\ title title Absence of a metallic phase in random-bond Ising models in two dimensions: Applications to disordered superconductors and paired quantum Hall states , \ 10.1103/PhysRevB.63.024404 journal journal Phys. Rev. B \ volume 63 ,\ pages 024404 ( year 2000 ) NoStop

    work page doi:10.1103/physrevb.63.024404 2000

  48. [49]

    Bocquet , author D

    author author M. Bocquet , author D. Serban , \ and\ author M. R. \ Zirnbauer ,\ title title Disordered 2d quasiparticles in class D: Dirac fermions with random mass, and dirty superconductors , \ 10.1016/S0550-3213(00)00208-X journal journal Nuclear Physics B \ volume 578 ,\ pages 628--680 ( year 2000 ) ,\ http://arxiv.org/abs/cond-mat/9910480 arXiv:cond...

    work page doi:10.1016/s0550-3213(00)00208-x 2000

  49. [50]

    author author Paul \ Fendley ,\ title title Critical points in two-dimensional replica sigma models , \ in\ @noop booktitle New Theoretical Approaches to Strongly Correlated Systems \ ( publisher Springer ,\ year 2001 )\ pp.\ pages 141--161 NoStop

    2001

  50. [51]

    author author Qingyuan \ Wang , author Romain \ Vasseur , author Simon \ Trebst , author Andreas W. W. \ Ludwig , \ and\ author Guo-Yi \ Zhu ,\ https://arxiv.org/abs/2502.14034 title Decoherence-induced self-dual criticality in topological states of matter , \ ( year 2025 ),\ http://arxiv.org/abs/2502.14034 arXiv:2502.14034 [quant-ph] NoStop

    work page arXiv 2025

  51. [52]

    author author S. Hikami ,\ title title Three-loop -functions of non-linear models on symmetric spaces , \ https://doi.org/10.1016/0370-2693(81)90989-8 journal journal Physics Letters B \ volume 98 ,\ pages 208--210 ( year 1981 ) NoStop

    work page doi:10.1016/0370-2693(81)90989-8 1981

  52. [53]

    author author Qingyuan \ Wang , author Romain \ Vasseur , author Simon \ Trebst , author Andreas W. W. \ Ludwig , \ and\ author Guo-Yi \ Zhu ,\ title title Decoherence-induced self-dual criticality in topological states of matter , \ 10.48550/arXiv.2502.14034 journal journal arXiv e-prints \ ,\ eid arXiv:2502.14034 ( year 2025 ) ,\ http://arxiv.org/abs/25...

    work page doi:10.48550/arxiv.2502.14034 2025

  53. [54]

    author author Xiao-Gang \ Wen ,\ title title Quantum orders in an exact soluble model , \ 10.1103/PhysRevLett.90.016803 journal journal Phys. Rev. Lett. \ volume 90 ,\ pages 016803 ( year 2003 ) NoStop

    work page doi:10.1103/physrevlett.90.016803 2003

  54. [55]

    Optimal resources for topological two-dimensional stabilizer codes:

    author author H. Bombin \ and\ author M. A. \ Martin-Delgado ,\ title title Optimal resources for topological two-dimensional stabilizer codes: Comparative study , \ 10.1103/PhysRevA.76.012305 journal journal Phys. Rev. A \ volume 76 ,\ pages 012305 ( year 2007 ) NoStop

    work page doi:10.1103/physreva.76.012305 2007

  55. [56]

    Venn \ and\ author B

    author author F. Venn \ and\ author B. B\'eri ,\ title title Error-correction and noise-decoherence thresholds for coherent errors in planar-graph surface codes , \ 10.1103/PhysRevResearch.2.043412 journal journal Phys. Rev. Res. \ volume 2 ,\ pages 043412 ( year 2020 ) NoStop

    work page doi:10.1103/physrevresearch.2.043412 2020

  56. [57]

    author author Stephen W. \ Yan , author Yimu \ Bao , \ and\ author Sagar \ Vijay ,\ title title Non-linear Sigma Model for the Surface Code with Coherent Errors , \ 10.48550/arXiv.2603.25665 journal journal arXiv e-prints \ ,\ eid arXiv:2603.25665 ( year 2026 ) ,\ http://arxiv.org/abs/2603.25665 arXiv:2603.25665 [cond-mat.stat-mech] NoStop

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2603.25665 2026

  57. [58]

    author author Sergey \ Bravyi ,\ title title Lagrangian representation for fermionic linear optics , \ @noop journal journal Quantum Info. Comput. \ volume 5 ,\ pages 216–238 ( year 2005 ) NoStop

    2005

  58. [59]

    Low-temperature behavior of two-dimensional gaussian ising spin glasses,

    author author J\'er\^ome \ Houdayer \ and\ author Alexander K. \ Hartmann ,\ title title Low-temperature behavior of two-dimensional gaussian ising spin glasses , \ 10.1103/PhysRevB.70.014418 journal journal Phys. Rev. B \ volume 70 ,\ pages 014418 ( year 2004 ) NoStop

    work page doi:10.1103/physrevb.70.014418 2004

  59. [60]

    author author Andreas \ Sorge ,\ @noop title The quality of data collapse , \ howpublished https://pyfssa.readthedocs.io/en/stable/quality.html ( year 2015 ),\ note pyfssa documentation, accessed April 2026 NoStop

    2015