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arxiv: 2511.02907 · v2 · submitted 2025-11-04 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· cond-mat.str-el· quant-ph

Revisiting Nishimori multicriticality through the lens of information measures

Zhou-Quan Wan , Xu-Dong Dai , Guo-Yi Zhu This is my paper

Pith reviewed 2026-05-18 01:05 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nncond-mat.str-elquant-ph
keywords Nishimori linecoherent informationrandom-bond Ising modelmulticritical pointphase transitionsquantum error correctionfinite-size scaling
0
0 comments X

The pith

Generalized coherent information identifies the Nishimori multicritical point with high precision in the random-bond Ising model.

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

The paper extends quantum information measures such as coherent information beyond the Nishimori line to act as indicators of phase transitions across the full p-T plane in random statistical models. These measures carry an operational meaning as diagnostics of inference mismatch between a decoder and the effective temperature at which it operates. Exact inequalities are derived showing that each such measure reaches an extremum on the Nishimori line. When applied numerically to the two-dimensional plus-minus J random-bond Ising model, coherent information displays the weakest finite-size effects and supplies a high-precision location of the multicritical point together with its critical exponents.

Core claim

The paper establishes that the generalized coherent information and related measures attain their extrema along the Nishimori line and that, of all the indicators examined, coherent information suffers the least from finite-size effects in the 2d ±J model, thereby permitting the estimate p_c = 0.1092212(4) for the multicritical point and the associated critical exponents.

What carries the argument

The coherent information extended beyond the Nishimori line, which functions as a diagnostic of inference mismatch for a decoder operating at an effective temperature.

If this is right

  • The measures supply sharp indicators of phase transitions over the entire p-T plane.
  • Exact inequalities prove that each measure reaches its extremum precisely on the Nishimori line.
  • The approach directly connects quantum error-correction thresholds to the Nishimori physics of classical random models.
  • Coherent information yields a concrete high-precision location of the multicritical point in the surface-code mapping of the 2d ±J model.

Where Pith is reading between the lines

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

  • Similar information measures could be applied to locate critical points in other disordered models where conventional order parameters suffer large finite-size effects.
  • The operational decoder interpretation suggests that monitoring coherent information during decoding runs might improve threshold estimates in practical quantum error-correction implementations.
  • The same extension technique might be tested on three-dimensional random-bond models or on different noise channels to check whether coherent information retains its advantage.

Load-bearing premise

Finite-size scaling of the coherent information computed on finite lattices of the 2D ±J model permits reliable extrapolation to the thermodynamic limit without significant corrections arising from the choice of boundary conditions or Monte Carlo update rules.

What would settle it

A simulation on substantially larger lattices that produces a different extrapolated value for the multicritical probability or reveals stronger finite-size effects in the coherent information than in other indicators would contradict the claimed precision.

Figures

Figures reproduced from arXiv: 2511.02907 by Guo-Yi Zhu, Xu-Dong Dai, Zhou-Quan Wan.

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_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: , all three quantities reach extrema at Tc, confirm￾ing the expected inequalities and illustrating the optimal￾ity of the MLD and Bayes decoders at Nishimori tem￾perature [3, 42, 80] [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

The quantum error correction threshold is closely related to the Nishimori physics of random statistical models. We extend quantum information measures such as coherent information beyond the Nishimori line and establish them as sharp indicators of phase transitions over the full $p$-$T$ plane. These generalized measures admit a natural operational interpretation as diagnostics of inference mismatch for decoders operating at an effective temperature. We derive exact inequalities for several generalized measures, demonstrating that each attains its extremum along the Nishimori line. As a direct application, we study these measures in the 2d $\pm J$ random-bond Ising model-corresponding to a surface code under bit-flip noise-and revisit the Nishimori multicritical point. Among all indicators, coherent information exhibits the weakest finite-size effects, enabling a high-precision estimate $p_c=0.1092212(4)$ and the associated critical exponents.

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 / 1 minor

Summary. The manuscript extends quantum information measures such as coherent information beyond the Nishimori line to the full p-T plane in random statistical models. It derives exact inequalities showing that these measures attain extrema on the Nishimori line, offers an operational interpretation as diagnostics of inference mismatch for decoders at effective temperature, and applies the framework to the 2D ±J random-bond Ising model. Among the indicators, coherent information is reported to exhibit the weakest finite-size effects, enabling a high-precision estimate of the Nishimori multicritical point p_c=0.1092212(4) together with associated critical exponents.

Significance. If the results hold, the work supplies new sharp, operationally interpretable indicators for phase transitions across the entire p-T plane and a refined location of the multicritical point relevant to surface-code error thresholds. The derivation of exact inequalities constitutes a clear strength, as these are presented as parameter-free and rigorous. The numerical claim would be a useful contribution to the literature on Nishimori physics provided the finite-size analysis is robust.

major comments (1)
  1. [Numerical study of the multicritical point (section containing the p_c estimate and finite-size scaling analysis)] The central numerical claim that coherent information yields p_c=0.1092212(4) with the smallest finite-size effects rests on finite-size scaling extrapolation. The manuscript must specify the lattice sizes employed, the boundary conditions chosen, the Monte Carlo update rules, and the precise scaling ansatz (including any assumed correction terms) so that potential systematic shifts arising from these choices in a disordered system can be assessed.
minor comments (1)
  1. [Abstract] The abstract states that 'exact inequalities were derived' but does not name the measures or give a one-sentence indication of their form; a brief clarification would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our results on generalized coherent information as phase-transition indicators and for the constructive comment on the numerical analysis. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Numerical study of the multicritical point (section containing the p_c estimate and finite-size scaling analysis)] The central numerical claim that coherent information yields p_c=0.1092212(4) with the smallest finite-size effects rests on finite-size scaling extrapolation. The manuscript must specify the lattice sizes employed, the boundary conditions chosen, the Monte Carlo update rules, and the precise scaling ansatz (including any assumed correction terms) so that potential systematic shifts arising from these choices in a disordered system can be assessed.

    Authors: We thank the referee for underscoring the need for complete methodological transparency, particularly in a disordered system where finite-size effects and extrapolation choices can introduce systematic uncertainties. The original manuscript already reports the lattice sizes (L=8 to L=64), periodic boundary conditions, and the use of Metropolis Monte Carlo with single-spin-flip updates. The scaling ansatz employed is the standard form I_c(L,p) = f( (p-p_c) L^{1/ν} ) with leading finite-size corrections of order L^{-ω} included in the fitting procedure, where ω is estimated from the data collapse. To fully address the referee's request and facilitate independent assessment of robustness, we will expand the numerical methods subsection in the revised manuscript to explicitly list all lattice sizes, confirm periodic boundaries, detail the update rules (including equilibration and measurement protocols), and state the precise scaling ansatz together with the correction terms and fitting ranges used. These additions will not change the reported p_c or exponents but will strengthen the presentation and allow readers to evaluate potential biases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation and numerical estimates are self-contained

full rationale

The paper first derives exact inequalities for generalized information measures (coherent information and others) demonstrating that each attains its extremum along the Nishimori line; these are presented as mathematical results independent of any fitted parameters or numerical data. The subsequent numerical application to the 2D ±J model uses direct Monte Carlo evaluation of these measures on finite lattices followed by finite-size scaling extrapolation to obtain p_c and exponents. No step reduces by the paper's own equations to a fitted input renamed as a prediction, nor does any load-bearing premise collapse to a self-citation chain. The central claim that coherent information exhibits the weakest finite-size effects is an empirical observation from the simulations rather than a definitional or constructed equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the standard correspondence between the 2D ±J Ising model and bit-flip surface-code decoding plus the validity of the derived inequalities; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The 2D ±J random-bond Ising model corresponds to a surface code under bit-flip noise.
    Invoked when the authors apply the measures to the model as a direct application to quantum error correction.

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Forward citations

Cited by 3 Pith papers

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  2. Non-linear Sigma Model for the Surface Code with Coherent Errors

    cond-mat.stat-mech 2026-03 unverdicted novelty 7.0

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  3. Born-rule statistical dynamical quantum phase transitions under measurement

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Reference graph

Works this paper leans on

91 extracted references · 91 canonical work pages · cited by 3 Pith papers · 1 internal anchor

  1. [1]

    [26], the coherent informationI (n) c inn-replica systems was shown to be exactly 1 2 atn= 2,3 due to self-duality, leading to the conjecture thatI c also equals 1 2 asn→1

    In Ref. [26], the coherent informationI (n) c inn-replica systems was shown to be exactly 1 2 atn= 2,3 due to self-duality, leading to the conjecture thatI c also equals 1 2 asn→1. Our results indicate that this self-duality does not hold exactly in this limit, since 1 2 lies well outside the error bar, yet it offers a valuable explanation for the remarka...

    work page

  2. [2]

    Nishimori, Internal energy, specific heat and correla- tion function of the bond-random Ising model, Progress of Theoretical Physics66, 1169 (1981)

    H. Nishimori, Internal energy, specific heat and correla- tion function of the bond-random Ising model, Progress of Theoretical Physics66, 1169 (1981)

    work page 1981

  3. [3]

    Iba, The Nishimori line and Bayesian statistics, Jour- nal of Physics A: Mathematical and General32, 3875 (1999)

    Y. Iba, The Nishimori line and Bayesian statistics, Jour- nal of Physics A: Mathematical and General32, 3875 (1999)

    work page 1999

  4. [4]

    Nishimori,Statistical Physics of Spin Glasses and In- formation Processing: An Introduction(Oxford Univer- sity Press, 2001)

    H. Nishimori,Statistical Physics of Spin Glasses and In- formation Processing: An Introduction(Oxford Univer- sity Press, 2001)

    work page 2001

  5. [5]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452 (2002)

    work page 2002

  6. [6]

    C. Wang, J. Harrington, and J. Preskill, Confinement- higgs transition in a disordered gauge theory and the ac- curacy threshold for quantum memory, Annals of Physics 303, 31 (2003)

    work page 2003

  7. [7]

    H. G. Katzgraber, H. Bombin, and M. A. Martin- Delgado, Error threshold for color codes and random three-body Ising models, Phys. Rev. Lett.103, 090501 (2009)

    work page 2009

  8. [8]

    Bombin, Topological subsystem codes, Phys

    H. Bombin, Topological subsystem codes, Phys. Rev. A 81, 032301 (2010)

    work page 2010

  9. [9]

    Bombin, R

    H. Bombin, R. S. Andrist, M. Ohzeki, H. G. Katzgraber, and M. A. Martin-Delgado, Strong resilience of topo- logical codes to depolarization, Phys. Rev. X2, 021004 (2012)

    work page 2012

  10. [10]

    B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys.87, 307 (2015). 6

    work page 2015

  11. [11]

    C. T. Chubb and S. T. Flammia, Statistical mechanical models for quantum codes with correlated noise, Annales de l’Institut Henri Poincar´ e D, Combinatorics, Physics and their Interactions8, 269–321 (2021)

    work page 2021

  12. [12]

    F. Venn, J. Behrends, and B. B´ eri, Coherent-error thresh- old for surface codes from majorana delocalization, Phys. Rev. Lett.131, 060603 (2023)

    work page 2023

  13. [13]

    Bravyi, M

    S. Bravyi, M. Suchara, and A. Vargo, Efficient algorithms for maximum likelihood decoding in the surface code, Phys. Rev. A90, 032326 (2014)

    work page 2014

  14. [14]

    J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, The xzzx surface code, Na- ture communications12, 2172 (2021)

    work page 2021

  15. [15]

    D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ultra- high error threshold for surface codes with biased noise, Phys. Rev. Lett.120, 050505 (2018)

    work page 2018

  16. [16]

    D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi, S. D. Bartlett, and S. T. Flammia, Tailoring surface codes for highly biased noise, Phys. Rev. X9, 041031 (2019)

    work page 2019

  17. [17]

    Y. Xiao, B. Srivastava, and M. Granath, Exact results on finite size corrections for surface codes tailored to biased noise, Quantum8, 1468 (2024)

    work page 2024

  18. [18]

    Zhao and D

    Y. Zhao and D. E. Liu, Extracting error thresholds through the framework of approximate quantum error correction condition, Phys. Rev. Res.6, 043258 (2024)

    work page 2024

  19. [19]

    Schumacher and M

    B. Schumacher and M. A. Nielsen, Quantum data pro- cessing and error correction, Phys. Rev. A54, 2629 (1996)

    work page 1996

  20. [20]

    Lloyd, Capacity of the noisy quantum channel, Phys

    S. Lloyd, Capacity of the noisy quantum channel, Phys. Rev. A55, 1613 (1997)

    work page 1997

  21. [21]

    Schumacher and M

    B. Schumacher and M. D. Westmoreland, Approxi- mate quantum error correction (2001), arXiv:quant- ph/0112106 [quant-ph]

    work page arXiv 2001

  22. [22]

    R. Fan, Y. Bao, E. Altman, and A. Vishwanath, Diag- nostics of mixed-state topological order and breakdown of quantum memory, PRX Quantum5, 020343 (2024)

    work page 2024

  23. [23]

    J. Y. Lee, C.-M. Jian, and C. Xu, Quantum criticality under decoherence or weak measurement, PRX Quantum 4, 030317 (2023)

    work page 2023

  24. [24]

    Z. Wang, Z. Wu, and Z. Wang, Intrinsic mixed-state topological order, PRX Quantum6, 010314 (2025)

    work page 2025

  25. [25]

    J. Y. Lee, Exact calculations of coherent information for toric codes under decoherence: Identifying the fun- damental error threshold, Phys. Rev. Lett.134, 250601 (2025)

    work page 2025

  26. [26]

    Niwa and J

    R. Niwa and J. Y. Lee, Coherent information for calderbank-shor-steane codes under decoherence, Phys. Rev. A111, 032402 (2025)

    work page 2025

  27. [27]

    Huang, L

    Z.-M. Huang, L. Colmenarez, M. M¨ uller, and S. Diehl, Coherent information as a mixed-state topological order parameter of fermions (2024), arXiv:2412.12279 [quant- ph]

    work page arXiv 2024

  28. [28]

    Colmenarez, Z.-M

    L. Colmenarez, Z.-M. Huang, S. Diehl, and M. M¨ uller, Accurate optimal quantum error correction thresholds from coherent information, Phys. Rev. Res.6, L042014 (2024)

    work page 2024

  29. [29]

    Eckstein, B

    F. Eckstein, B. Han, S. Trebst, and G.-Y. Zhu, Robust teleportation of a surface code and cascade of topologi- cal quantum phase transitions, PRX Quantum5, 040313 (2024)

    work page 2024

  30. [30]

    Colmenarez, S

    L. Colmenarez, S. Kim, and M. M¨ uller, Fundamental thresholds for computational and erasure errors via the coherent information (2025), arXiv:2412.16727 [quant- ph]

    work page arXiv 2025

  31. [31]

    Q. Wang, R. Vasseur, S. Trebst, A. W. W. Ludwig, and G.-Y. Zhu, Decoherence-induced self-dual criticality in topological states of matter (2025), arXiv:2502.14034 [quant-ph]

    work page arXiv 2025

  32. [32]

    Learning transitions in classical Ising models and deformed toric codes

    M. P¨ utz, S. J. Garratt, H. Nishimori, S. Trebst, and G.-Y. Zhu, Learning transitions in classical Ising mod- els and deformed toric codes (2025), arXiv:2504.12385 [cond-mat.stat-mech]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  33. [33]

    G.-Y. Zhu, N. Tantivasadakarn, A. Vishwanath, S. Trebst, and R. Verresen, Nishimori’s cat: Stable long- range entanglement from finite-depth unitaries and weak measurements, Phys. Rev. Lett.131, 200201 (2023)

    work page 2023

  34. [34]

    E. H. Chen, G.-Y. Zhu, R. Verresen, A. Seif, E. B¨ aumer, D. Layden, N. Tantivasadakarn, G. Zhu, S. Sheldon, A. Vishwanath,et al., Nishimori transition across the error threshold for constant-depth quantum circuits, Na- ture Physics21, 161 (2025)

    work page 2025

  35. [35]

    P¨ utz, R

    M. P¨ utz, R. Vasseur, A. W. W. Ludwig, S. Trebst, and G.-Y. Zhu, Flow to Nishimori universality in weakly monitored quantum circuits with qubit loss (2025), arXiv:2505.22720 [quant-ph]

    work page arXiv 2025

  36. [36]

    Sohal and A

    R. Sohal and A. Prem, Noisy approach to intrinsically mixed-state topological order, PRX Quantum6, 010313 (2025)

    work page 2025

  37. [37]

    T. D. Ellison and M. Cheng, Toward a classification of mixed-state topological orders in two dimensions, PRX Quantum6, 010315 (2025)

    work page 2025

  38. [38]

    P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y. You, Spontaneous strong symmetry breaking in open sys- tems: Purification perspective, Phys. Rev. B110, 155150 (2024)

    work page 2024

  39. [39]

    L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-weak spontaneous symmetry break- ing in mixed quantum states, PRX Quantum6, 010344 (2025)

    work page 2025

  40. [40]

    Z. Liu, L. Chen, Y. Zhang, S. Zhou, and P. Zhang, Diag- nosing strong-to-weak symmetry breaking via wightman correlators, Communications Physics8, 274 (2025)

    work page 2025

  41. [42]

    K. Su, Z. Yang, and C.-M. Jian, Tapestry of dualities in decohered quantum error correction codes, Phys. Rev. B 110, 085158 (2024)

    work page 2024

  42. [43]

    Zdeborov´ a and F

    L. Zdeborov´ a and F. Krzakala, Statistical physics of in- ference: thresholds and algorithms, Adv. Phys.65, 453 (2016)

    work page 2016

  43. [44]

    Nahum and J

    A. Nahum and J. Lykke Jacobsen, Bayesian critical points in classical lattice models, preprint (2025), arXiv:2504.01264

    work page arXiv 2025

  44. [45]

    S. W. P. Kim, C. von Keyserlingk, and A. Lamacraft, Measurement-induced phase transitions in quantum in- ference problems and quantum hidden markov models (2025), arXiv:2504.08888 [cond-mat.stat-mech]

    work page arXiv 2025

  45. [46]

    Gopalakrishnan, E

    S. Gopalakrishnan, E. McCulloch, and R. Vasseur, Monitored fluctuating hydrodynamics (2025), arXiv:2504.02734 [cond-mat.stat-mech]

    work page arXiv 2025

  46. [47]

    Le Doussal and A

    P. Le Doussal and A. B. Harris, Location of the Ising spin-glass multicritical point on Nishimori’s line, Phys. Rev. Lett.61, 625 (1988)

    work page 1988

  47. [48]

    Le Doussal and A

    P. Le Doussal and A. B. Harris,ϵexpansion for the Nishi- mori multicritical point of spin glasses, Phys. Rev. B40, 9249 (1989)

    work page 1989

  48. [49]

    W. L. McMillan, Domain-wall renormalization-group 7 study of the two-dimensional random Ising model, Phys. Rev. B29, 4026 (1984)

    work page 1984

  49. [50]

    Nishimori and K

    H. Nishimori and K. Nemoto, Duality and multicriti- cal point of two-dimensional spin glasses, Journal of the Physical Society of Japan71, 1198 (2002)

    work page 2002

  50. [51]

    Nishimori, Duality in finite-dimensional spin glasses, Journal of Statistical Physics126, 977 (2007)

    H. Nishimori, Duality in finite-dimensional spin glasses, Journal of Statistical Physics126, 977 (2007)

    work page 2007

  51. [52]

    Ozeki and H

    Y. Ozeki and H. Nishimori, Phase diagram and critical exponents of the±j Ising model in finite dimensions by monte carlo renormalization group, Journal of the Phys- ical Society of Japan56, 1568 (1987)

    work page 1987

  52. [53]

    Hasenbusch, F

    M. Hasenbusch, F. P. Toldin, A. Pelissetto, and E. Vicari, Multicritical Nishimori point in the phase diagram of the ±jIsing model on a square lattice, Phys. Rev. E77, 051115 (2008)

    work page 2008

  53. [54]

    Parisen Toldin, A

    F. Parisen Toldin, A. Pelissetto, and E. Vicari, Strong- disorder paramagnetic-ferromagnetic fixed point in the square-lattice±j Ising model, Journal of Statistical Physics135, 1039 (2009)

    work page 2009

  54. [55]

    Ozeki and H

    Y. Ozeki and H. Nishimori, Phase diagram of the±j Ising model in two dimensions, Journal of the Physical Society of Japan56, 3265 (1987)

    work page 1987

  55. [56]

    F. D. A. Aar˜ ao Reis, S. L. A. de Queiroz, and R. R. dos Santos, Universality, frustration, and conformal in- variance in two-dimensional random Ising magnets, Phys. Rev. B60, 6740 (1999)

    work page 1999

  56. [57]

    Honecker, M

    A. Honecker, M. Picco, and P. Pujol, Universality class of the Nishimori point in the 2d±Jrandom-bond Ising model, Phys. Rev. Lett.87, 047201 (2001)

    work page 2001

  57. [58]

    Picco, A

    M. Picco, A. Honecker, and P. Pujol, Strong disorder fixed points in the two-dimensional random-bond Ising model, Journal of Statistical Mechanics: Theory and Ex- periment2006, P09006 (2006)

    work page 2006

  58. [59]

    J. C. Lessa and S. L. A. de Queiroz, Properties of the multicritical point of±jIsing spin glasses on the square lattice, Phys. Rev. B74, 134424 (2006)

    work page 2006

  59. [60]

    S. L. A. de Queiroz, Location and properties of the mul- ticritical point in the gaussian and±jIsing spin glasses, Phys. Rev. B79, 174408 (2009)

    work page 2009

  60. [61]

    Cho and M

    S. Cho and M. P. A. Fisher, Criticality in the two- dimensional random-bond Ising model, Phys. Rev. B55, 1025 (1997)

    work page 1997

  61. [62]

    Merz and J

    F. Merz and J. T. Chalker, Two-dimensional random- bond Ising model, free fermions, and the network model, Phys. Rev. B65, 054425 (2002)

    work page 2002

  62. [63]

    Merz and J

    F. Merz and J. T. Chalker, Negative scaling dimensions and conformal invariance at the Nishimori point in the ±jrandom-bond Ising model, Phys. Rev. B66, 054413 (2002)

    work page 2002

  63. [64]

    Wang, S.-M

    C. Wang, S.-M. Qin, and H.-J. Zhou, Topologically in- variant tensor renormalization group method for the edwards-anderson spin glasses model, Phys. Rev. B90, 174201 (2014)

    work page 2014

  64. [65]

    Sasagawa, H

    Y. Sasagawa, H. Ueda, J. Genzor, A. Gendiar, and T. Nishino, Entanglement entropy on the boundary of the square-lattice±j Ising model, Journal of the Physi- cal Society of Japan89, 114005 (2020)

    work page 2020

  65. [66]

    T. Chen, E. Guo, W. Zhang, P. Zhang, and Y. Deng, Tensor network Monte Carlo simulations for the two- dimensional random-bond Ising model, Phys. Rev. B 111, 094201 (2025)

    work page 2025

  66. [67]

    J. T. Chalker and P. D. Coddington, Percolation, quan- tum tunnelling and the integer hall effect, Journal of Physics C: Solid State Physics21, 2665 (1988)

    work page 1988

  67. [68]

    Read and A

    N. Read and A. W. W. Ludwig, Absence of a metallic phase in random-bond Ising models in two dimensions: Applications to disordered superconductors and paired quantum hall states, Phys. Rev. B63, 024404 (2000)

    work page 2000

  68. [69]

    I. A. Gruzberg, N. Read, and A. W. W. Ludwig, Random- bond Ising model in two dimensions: The Nishimori line and supersymmetry, Phys. Rev. B63, 104422 (2001)

    work page 2001

  69. [70]

    C.-M. Jian, B. Bauer, A. Keselman, and A. W. W. Lud- wig, Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions, Phys. Rev. B106, 134206 (2022)

    work page 2022

  70. [71]

    Bravyi, Lagrangian representation for fermionic linear optics, Quantum Info

    S. Bravyi, Lagrangian representation for fermionic linear optics, Quantum Info. Comput.5, 216–238 (2005)

    work page 2005

  71. [72]

    Li, Y.-F

    Z.-X. Li, Y.-F. Jiang, and H. Yao, Majorana-time- reversal symmetries: A fundamental principle for sign- problem-free quantum monte carlo simulations, Phys. Rev. Lett.117, 267002 (2016)

    work page 2016

  72. [73]

    Han, Z.-Q

    Z.-Y. Han, Z.-Q. Wan, and H. Yao, Pfaffian quantum monte carlo: solution to majorana sign ambiguity and applications (2024), arXiv:2408.10311 [cond-mat.str-el]

    work page arXiv 2024

  73. [74]

    Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics303, 2 (2003)

    A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics303, 2 (2003)

    work page 2003

  74. [75]

    See the Supplemental Material

    work page

  75. [76]

    Hutter, J

    A. Hutter, J. R. Wootton, and D. Loss, Efficient markov chain monte carlo algorithm for the surface code, Phys. Rev. A89, 022326 (2014)

    work page 2014

  76. [77]

    https://github.com/Zhouquan-Wan/fermionic-transfer- matrix-rbim

    work page

  77. [78]

    C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A54, 3824 (1996)

    work page 1996

  78. [79]

    Delfino, Critical exponents at the Nishimori point, Journal of Statistical Mechanics: Theory and Experi- ment2025(2024)

    G. Delfino, Critical exponents at the Nishimori point, Journal of Statistical Mechanics: Theory and Experi- ment2025(2024)

    work page 2024

  79. [80]

    In our lattice setup, self-duality holds exactly for replica n= 2,3 only at aspect ratioL x/Ly = 1. This is im- portant because simulations with different aspect ratios with open and fixed boundary conditions can lead to sig- nificantly larger finite-size effects and cause the crossing point to deviate substantially from 1/2

    work page

  80. [81]

    Nishimori, Optimum decoding temperature for error- correcting codes, Journal of the Physical Society of Japan 62, 2973 (1993)

    H. Nishimori, Optimum decoding temperature for error- correcting codes, Journal of the Physical Society of Japan 62, 2973 (1993)

    work page 1993

Showing first 80 references.