pith. sign in

arxiv: 2605.28162 · v2 · pith:6GLL4HYQnew · submitted 2026-05-27 · 🪐 quant-ph · cs.LG

Learning Logical Operations for Arbitrary Quantum Error Correction Codes

Nico Meyer , Christopher Mutschler , Dominik Seu{\ss} , Andreas Maier , Daniel D. Scherer This is my paper

Pith reviewed 2026-06-29 12:17 UTC · model grok-4.3

classification 🪐 quant-ph cs.LG
keywords quantum error correctionlogical operationsvariational optimizationnon-additive codestransversal gatesfault-tolerant quantum computingcircuit co-design
0
0 comments X

The pith

A learning framework finds physical circuits realizing logical operations on arbitrary quantum error-correcting codes from the encoding alone.

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

The paper develops a variational method that takes only an encoding circuit and learns physical gate implementations enforcing constraints such as transversality or limited depth. It first recovers known logical operations on stabilizer codes, then introduces a co-design procedure that simultaneously adjusts non-additive encodings to a target noise model while requiring specific logical gate families. The method supplies concrete loss functions and ansatz families that guide the optimizer toward valid solutions. A reader cares because this removes the need for manual stabilizer-based design when exploring hardware-adapted codes for early fault-tolerant regimes.

Core claim

Given solely an encoding circuit, the framework constructs physical realizations of logical operations while enforcing structural properties such as transversality or shallow depth; the same pipeline extends to a co-design loop that tailors non-additive encodings to a noise model and enforces chosen logical gate sets, including transversal IQP-type families or low-depth universal sets.

What carries the argument

The variational optimization procedure that tunes circuit parameters to match a target logical action subject to structural constraints, extended to joint optimization of the encoding itself.

If this is right

  • Logical operations become discoverable for non-additive codes that have no stabilizer description.
  • Encodings can be co-optimized with a noise model to support chosen logical gate sets.
  • The approach directly supplies hardware-adapted gadgets for early fault-tolerant computation.
  • A software pipeline automates the full loop of loss design, ansatz selection, and optimization.

Where Pith is reading between the lines

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

  • Custom non-additive codes could be generated on demand for specific device noise profiles rather than selected from a pre-existing catalog.
  • The same optimization loop might be reused to discover fault-tolerant gadgets that go beyond the transversal or low-depth families considered here.
  • If the method scales, it could shorten the design cycle between hardware characterization and code selection.

Load-bearing premise

The chosen loss functions and circuit families are sufficient for the optimizer to converge on implementations that actually perform the desired logical operations without requiring separate verification.

What would settle it

Apply the framework to the Steane code with a transversality constraint and test whether the learned circuit matches the known transversal Hadamard or CNOT up to local corrections.

Figures

Figures reproduced from arXiv: 2605.28162 by Andreas Maier, Christopher Mutschler, Daniel D. Scherer, Dominik Seu{\ss}, Nico Meyer.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic overview of the framework for learning [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Variational learning pipeline for a logical operation [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Circuit-level view of the learning setup for a two-qubit [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of different loss variants on the target two [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic ansatz families used for learning logical op [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Numerical success rates for learning transversal real [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Training behaviour of VarEFTQC for a ((5 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Illustration of training procedure on the two-design [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Geometric interpretation of loss function variants for [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Geometric interpretation of loss function variants for [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Illustration of convergence to local minima on the [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 10
Figure 10. Figure 10: Strictly speaking, this omits some orthogonal [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
read the original abstract

Logical operations are essential for quantum computation within quantum error-correcting codes. However, discovering their physical realizations is challenging, especially for non-additive codes that lack a stabilizer description. We present a general learning-based framework that, given only an encoding circuit, constructs physical implementations of logical operations while enforcing structural properties such as transversality or shallow depth. Our approach is validated by rediscovering known logical operations of standard stabilizer codes. We then extend it to a co-design procedure, dubbed variational early fault-tolerant quantum computing (VarEFTQC), which tailors non-additive encodings to a given noise model and enforces desired logical gate sets, such as transversal IQP-type families or low-depth universal sets. A software library implements the complete learning pipeline, including loss-function variants, ansatz families, and optimization routines. Together, these results position VarEFTQC as a practical tool for discovering hardware-adapted logical gadgets for early fault-tolerant quantum computing.

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

2 major / 2 minor

Summary. The manuscript presents a learning-based framework that, given only an encoding circuit for a quantum error-correcting code, discovers physical implementations of logical operations while enforcing constraints such as transversality or shallow depth. It validates the approach by rediscovering known operations on stabilizer codes and extends it to a variational co-design method (VarEFTQC) that tailors non-additive encodings to noise models and desired logical gate sets (e.g., transversal IQP or low-depth universal families). A software library implementing the pipeline (loss variants, ansatze, optimizers) is provided.

Significance. If the central claim holds, the work would offer a practical, general tool for logical-gate discovery on non-stabilizer codes and for hardware-adapted early fault-tolerant gadgets, with the open-source library as a concrete contribution that enables reproducibility.

major comments (2)
  1. [Abstract / validation paragraph] The validation claim (rediscovering known operations) only demonstrates reachability of correct solutions; it does not establish that every low-loss minimum realizes the intended logical action on the code subspace. No explicit post-optimization verification (e.g., computation of the logical channel or commutation with the code projector) is described, leaving open the possibility of spurious minima that satisfy the numerical loss but fail to act correctly as logical unitaries.
  2. [Framework description] The loss-function construction that is asserted to enforce both logical action and structural constraints (transversality, depth) is not specified in sufficient detail to assess whether it is free of the circularity risk noted in the stress-test: minima could satisfy the loss by construction without guaranteeing the logical map. Concrete forms of the loss, the ansatz families, and any regularization terms are required to evaluate this.
minor comments (2)
  1. Notation for the encoding circuit and the logical operators should be introduced with explicit definitions before the optimization procedure is described.
  2. The manuscript should include at least one concrete example (e.g., a small non-additive code) showing the explicit circuit found, the loss value, and an independent check that the logical action is correct.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The two major comments identify areas where additional verification and explicit detail are needed to strengthen the claims. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / validation paragraph] The validation claim (rediscovering known operations) only demonstrates reachability of correct solutions; it does not establish that every low-loss minimum realizes the intended logical action on the code subspace. No explicit post-optimization verification (e.g., computation of the logical channel or commutation with the code projector) is described, leaving open the possibility of spurious minima that satisfy the numerical loss but fail to act correctly as logical unitaries.

    Authors: We agree that reachability alone does not rule out spurious minima. Although the loss is constructed to penalize deviations from the target logical action on the code subspace, the manuscript does not report explicit post-optimization checks. In the revision we will add a verification procedure that extracts the effective logical channel via projection onto the code subspace and verifies commutation with the code projector for all rediscovered operations. These checks will be included in the validation section and the supplementary material. revision: yes

  2. Referee: [Framework description] The loss-function construction that is asserted to enforce both logical action and structural constraints (transversality, depth) is not specified in sufficient detail to assess whether it is free of the circularity risk noted in the stress-test: minima could satisfy the loss by construction without guaranteeing the logical map. Concrete forms of the loss, the ansatz families, and any regularization terms are required to evaluate this.

    Authors: The referee is correct that the current manuscript provides only a high-level description of the loss components. We will expand the framework section with the explicit mathematical expressions for each loss term (logical-action fidelity, transversality penalty, depth penalty), the precise parameterization of the variational ansatze, and the regularization coefficients. These additions will make it possible to evaluate whether the loss enforces the logical map independently of the structural constraints and will reference the corresponding implementations in the released software library. revision: yes

Circularity Check

0 steps flagged

No circularity: optimization framework with external validation

full rationale

The paper describes a variational learning pipeline that takes an encoding circuit as input and optimizes ansatzes under explicitly constructed loss functions to enforce logical action plus structural constraints. Rediscovery of known stabilizer-code gates serves as external validation rather than definitional closure. No equations reduce the output to the input by construction, no load-bearing self-citations are invoked for uniqueness or ansatz choice, and the central claim (that the framework produces correct gadgets) rests on the design of the loss rather than on renaming fitted quantities. This is a standard non-circular variational search setup.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Only abstract available, so ledger is inferred from described components. The method relies on optimization of ansatz parameters and assumptions that loss functions enforce logical correctness and structural properties.

free parameters (1)
  • ansatz parameters
    Variational optimization tunes parameters in the ansatz families to discover implementations.
axioms (2)
  • domain assumption The given encoding circuit is correct and sufficient as input.
    Framework takes encoding circuit as sole input.
  • ad hoc to paper Loss functions can be constructed to enforce both logical action and constraints like transversality.
    Central to the learning pipeline and VarEFTQC.

pith-pipeline@v0.9.1-grok · 5697 in / 1308 out tokens · 34557 ms · 2026-06-29T12:17:37.378328+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

61 extracted references · 10 canonical work pages · 5 internal anchors

  1. [1]

    andp Z = 0.086

    (c) Same VarEFTQC setup as in (b), but without warm-start initialization of the encoding. andp Z = 0.086. Moreover, bitflip noiseN bit of strength p= 0.1. Both noise models are applied independently to all physical qubits. The quality of an encoding is mea- sured by the suppression of information loss compared to a bare physical execution. In particular, ...

    work page doi:10.5281/zenodo.20280560

  2. [2]

    P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A52, R2493 (1995)

    1995

  3. [3]

    A. M. Steane, Error Correcting Codes in Quantum The- ory, Phys. Rev. Lett.77, 793 (1996)

    1996

  4. [4]

    Gottesman,Stabilizer codes and quantum error cor- rection, Ph.D

    D. Gottesman,Stabilizer codes and quantum error cor- rection, Ph.D. thesis, California Institute of Technology (1997)

    1997

  5. [5]

    Learning Encodings by Maximizing State Distinguishability: Variational Quantum Error Correction

    N. Meyer, C. Mutschler, A. Maier, and D. D. Scherer, Learning Encodings by Maximizing State Dis- tinguishability: Variational Quantum Error Correction, arXiv:2506.11552 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  6. [6]

    Meyer, C

    N. Meyer, C. Mutschler, A. Maier, and D. D. Scherer, Variational Quantum Error Correction, inIEEE Inter- national Conference on Quantum Computing and Engi- neering (QCE), Vol. 2 (2025) pp. 456–457

    2025

  7. [7]

    P. W. Shor, Fault-tolerant quantum computation, inSymposium on Foundations of Computer Science (FOCS)(1996) pp. 56–65

    1996

  8. [8]

    Gottesman, An Introduction to Quantum Error Cor- rection and Fault-Tolerant Quantum Computation, in Proc

    D. Gottesman, An Introduction to Quantum Error Cor- rection and Fault-Tolerant Quantum Computation, in Proc. Sympos. Appl. Math., Vol. 68 (2010) pp. 13–58

    2010

  9. [9]

    Aliferis, D

    P. Aliferis, D. Gottesman, and J. Preskill, Quantum accu- racy threshold for concatenated distance-3 codes, Quan- tum Inf. Comput.6, 97 (2005)

    2005

  10. [10]

    D. A. Lidar and T. A. Brun,Quantum Error Correction (Cambridge University Press, 2013)

    2013

  11. [11]

    Eastin and E

    B. Eastin and E. Knill, Restrictions on Transversal En- coded Quantum Gate Sets, Phys. Rev. Lett.102, 110502 (2009)

    2009

  12. [12]

    Horsman, A

    D. Horsman, A. G. Fowler, S. Devitt, and R. Van Meter, Surface code quantum computing by lattice surgery, New J. Phys.14, 123011 (2012)

    2012

  13. [13]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal Clifford gates and noisy ancillas, Phys. Rev. A71, 022316 (2005)

    2005

  14. [14]

    Paetznick and B

    A. Paetznick and B. W. Reichardt, Universal Fault- Tolerant Quantum Computation with Only Transver- sal Gates and Error Correction, Phys. Rev. Lett.111, 090505 (2013)

    2013

  15. [15]

    Katabarwa, K

    A. Katabarwa, K. Gratsea, A. Caesura, and P. D. John- son, Early Fault-Tolerant Quantum Computing, PRX Quantum5, 020101 (2024)

    2024

  16. [16]

    M. A. Webster, A. O. Quintavalle, and S. D. Bartlett, Transversal diagonal logical operators for stabiliser codes, New J. Phys.25, 103018 (2023)

    2023

  17. [17]

    H. Chen, M. Vasmer, N. P. Breuckmann, and E. Grant, Automated discovery of logical gates for quantum error correction, Quantum Inf. Comput.22, 947 (2022)

    2022

  18. [18]

    Kraus, General state changes in quantum theory, Ann

    K. Kraus, General state changes in quantum theory, Ann. Phys.64, 311 (1971)

    1971

  19. [19]

    J. Olle, R. Zen, M. Puviani, and F. Marquardt, Si- multaneous discovery of quantum error correction codes and encoders with a noise-aware reinforcement learning agent, npj Quantum Inf.10, 1 (2024)

    2024

  20. [20]

    D. W. Leung, M. A. Nielsen, I. L. Chuang, and Y. Ya- mamoto, Approximate quantum error correction can lead to better codes, Phys. Rev. A56, 2567 (1997)

    1997

  21. [21]

    Laflamme, C

    R. Laflamme, C. Miquel, J. P. Paz, and W. H. Zurek, Per- fect Quantum Error Correcting Code, Phys. Rev. Lett. 77, 198 (1996)

    1996

  22. [22]

    A. M. Steane, Quantum Reed-Muller Codes, IEEE Trans. Inf. Theory45, 1701 (2002)

    2002

  23. [23]

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A86, 032324 (2012)

    2012

  24. [24]

    C. Cao, C. Zhang, Z. Wu, M. Grassl, and B. Zeng, Quan- tum variational learning for quantum error-correcting codes, Quantum6, 828 (2022)

    2022

  25. [25]

    Watrous,The Theory of Quantum Information(Cam- bridge University Press, 2018)

    J. Watrous,The Theory of Quantum Information(Cam- bridge University Press, 2018)

    2018

  26. [26]

    P. D. Johnson, J. Romero, J. Olson, Y. Cao, and A. Aspuru-Guzik, QVECTOR: an algorithm for device- tailored quantum error correction, arXiv:1711.02249 (2017)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  27. [27]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, 2010)

    2010

  28. [28]

    Camps-Moreno, H

    E. Camps-Moreno, H. H. L´ opez, G. L. Matthews, N. Ren- gaswamy, and R. San-Jos´ e, Transversal gates for quan- tum CSS codes, arXiv:2601.21514 (2026)

    work page arXiv 2026

  29. [29]

    Chen and N

    Z. Chen and N. Rengaswamy, Tailoring Fault-Tolerance to Quantum Algorithms, IEEE J. Sel. Areas Inf. Theory 6, 311 (2025)

    2025

  30. [30]

    Optimized Compilation of Logical Clifford Circuits

    A. Popov, N. Meyer, D. D. Scherer, and G. Di- etl, Optimized Compilation of Logical Clifford Circuits, arXiv:2602.12831 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  31. [31]

    Z. Chen, J. O. Weinberg, and N. Rengaswamy, Fault Tol- erant Quantum Simulation via Symplectic Transvections, inIEEE International Conference on Quantum Comput- ing and Engineering (QCE), Vol. 1 (2025) p. 158

    2025

  32. [32]

    Hastie,The Elements of Statistical Learning: Data Mining, Inference, and Prediction(Springer, 2009)

    T. Hastie,The Elements of Statistical Learning: Data Mining, Inference, and Prediction(Springer, 2009)

    2009

  33. [33]

    Bassily, V

    R. Bassily, V. Feldman, C. Guzm´ an, and K. Talwar, Sta- bility of Stochastic Gradient Descent on Nonsmooth Con- vex Losses, inAdvances in Neural Information Processing Systems (NeurIPS), Vol. 33 (2020) pp. 4381–4391

    2020

  34. [34]

    Sommers and K

    H.-J. Sommers and K. ˙Zyczkowski, Statistical properties of random density matrices, J. Phys. A37, 8457 (2004)

    2004

  35. [35]

    Ambainis and J

    A. Ambainis and J. Emerson, Quantum t-designs: t-wise independence in the quantum world, inIEEE Conference on Computational Complexity (CCC)(2007) pp. 129– 140

    2007

  36. [36]

    Dankert, R

    C. Dankert, R. Cleve, J. Emerson, and E. Livine, Exact and approximate unitary 2-designs and their application to fidelity estimation, Phys. Rev. A80, 012304 (2009)

    2009

  37. [37]

    Knill, Quantum computing with realistically noisy de- vices, Nature434, 39 (2005)

    E. Knill, Quantum computing with realistically noisy de- vices, Nature434, 39 (2005)

    2005

  38. [38]

    D. C. Liu and J. Nocedal, On the limited memory BFGS 23 method for large scale optimization, Math. Program.45, 503 (1989)

    1989

  39. [39]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, PyTorch: An Imperative Style, High- Performance Deep Learning Library, inAdvances in Neu- ral Information Processing S...

    2019

  40. [40]

    PennyLane: Automatic differentiation of hybrid quantum-classical computations

    V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, S. Ahmed, V. Ajith, M. S. Alam, G. Alonso-Linaje, B. Akash- Narayanan, A. Asadi,et al., PennyLane: Automatic dif- ferentiation of hybrid quantum-classical computations, arXiv:1811.04968 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  41. [41]

    Meyer, C

    N. Meyer, C. Ufrecht, M. Periyasamy, A. Plinge, C. Mutschler, D. D. Scherer, and A. Maier, Qiskit-Torch- Module: Fast Prototyping of Quantum Neural Networks, inIEEE International Conference on Quantum Comput- ing and Engineering (QCE), Vol. 1 (2024) pp. 817–823

    2024

  42. [42]

    S. Du, J. Lee, H. Li, L. Wang, and X. Zhai, Gradient De- scent Finds Global Minima of Deep Neural Networks, in International Conference on Machine Learning (ICML), Vol. 95 (2019) pp. 1675–1685

    2019

  43. [43]

    Allen-Zhu, Y

    Z. Allen-Zhu, Y. Li, and Z. Song, A Convergence Theory for Deep Learning via Over-Parameterization, inInterna- tional Conference on Machine Learning (ICML), Vol. 97 (2019) pp. 242–252

    2019

  44. [44]

    Hastie, A

    T. Hastie, A. Montanari, S. Rosset, and R. J. Tibshirani, Surprises in High-Dimensional Ridgeless Least Squares Interpolation, Ann. Stat.50, 949 (2022)

    2022

  45. [45]

    Meyer, J

    N. Meyer, J. Murauer, A. Popov, C. Ufrecht, A. Plinge, C. Mutschler, and D. D. Scherer, Warm-Start Variational Quantum Policy Iteration, inIEEE International Confer- ence on Quantum Computing and Engineering (QCE), Vol. 1 (2024) pp. 1458–1466

    2024

  46. [46]

    Nest, Simulating quantum computers with probabilis- tic methods, Quantum Inf

    M. Nest, Simulating quantum computers with probabilis- tic methods, Quantum Inf. Comput.11, 0784 (2009)

    2009

  47. [47]

    M. J. Bremner, R. Jozsa, and D. J. Shepherd, Classi- cal simulation of commuting quantum computations im- plies collapse of the polynomial hierarchy, Proc. Roy. Soc. Lond.467, 459 (2011)

    2011

  48. [48]

    M. J. Bremner, A. Montanaro, and D. J. Shep- herd, Achieving quantum supremacy with sparse and noisy commuting quantum computations, Quantum1, 8 (2017)

    2017

  49. [49]

    Coyle, D

    B. Coyle, D. Mills, V. Danos, and E. Kashefi, The Born supremacy: quantum advantage and training of an Ising Born machine, npj Quantum Inf.6, 60 (2020)

    2020

  50. [50]

    Recio-Armengol, S

    E. Recio-Armengol, S. Ahmed, and J. Bowles, Train on classical, deploy on quantum: scaling genera- tive quantum machine learning to a thousand qubits, arXiv:2503.02934 (2025)

    work page arXiv 2025

  51. [51]

    Paletta, A

    L. Paletta, A. Leverrier, A. Sarlette, M. Mirrahimi, and C. Vuillot, Robust sparse IQP sampling in constant depth, Quantum8, 1337 (2024)

    2024

  52. [52]

    Hangleiter, M

    D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hyper- cubes, PRX Quantum6, 020338 (2025)

    2025

  53. [53]

    Koutsioumpas, D

    S. Koutsioumpas, D. Banfield, and A. Kay, The Smallest Code with Transversal T, arXiv:2210.14066 (2022)

    work page arXiv 2022

  54. [54]

    Faist, S

    P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, Continuous Symmetries and Approximate Quantum Error Correc- tion, Phys. Rev. X10, 041018 (2020)

    2020

  55. [55]

    Aliferis and J

    P. Aliferis and J. Preskill, Fault-tolerant quantum com- putation against biased noise, Phys. Rev. A78, 052331 (2008)

    2008

  56. [56]

    D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Ul- trahigh Error Threshold for Surface Codes with Biased Noise, Phys. Rev. Lett.120, 050505 (2018)

    2018

  57. [57]

    Learning to Concatenate Quantum Codes

    N. Meyer, C. Mutschler, D. Seuß, A. Maier, and D. D. Scherer, Learning to Concatenate Quantum Codes, arXiv:2604.14931 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  58. [58]

    Knill, R

    E. Knill, R. Laflamme, and W. H. Zurek, Resilient Quan- tum Computation, Science279, 342 (1998)

    1998

  59. [59]

    M. R. Geller and Z. Zhou, Efficient error models for fault- tolerant architectures and the pauli twirling approxima- tion, Phys. Rev. A88, 012314 (2013)

    2013

  60. [60]

    A. S. Darmawan and D. Poulin, Tensor-network simula- tions of the surface code under realistic noise, Phys. Rev. Lett.119, 040502 (2017)

    2017

  61. [61]

    Huang, P.-G

    E. Huang, P.-G. Rozon, A. Dua, S. Gopalakrishnan, and M. J. Gullans, A robust phase of continuous transver- sal gates in quantum stabilizer codes, arXiv:2510.01319 (2025)

    work page arXiv 2025