pith. machine review for the scientific record. sign in

arxiv: 2604.05874 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Recognition: no theorem link

Adaptive Deformation of Color Code in Square Lattices with Defects

Tian-Hao Wei , Jia-Xuan Zhang , Jia-Ning Li , Wei-Cheng Kong , Yu-Chun Wu , Guo-Ping Guo
Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords color codesquantum error correctiondefectsstabilizer codessuperstabilizerlogical error ratesquare latticelattice surgery
0
0 comments X

The pith

A universal superstabilizer scheme repairs isolated defects in color codes on square lattices without disabling neighboring qubits, yielding lower logical error rates while supporting full gates and lattice surgery.

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

Quantum error correction with color codes on superconducting processors breaks down when hardware defects disrupt the lattice. The paper proposes a superstabilizer approach that repairs both data and ancilla qubit defects by deforming the code adaptively around them. For ancilla defects it offers two concrete fixes: reusing neighboring ancilla qubits or inserting iSWAP gates. These steps avoid the conventional tactic of turning off extra data qubits, which wastes resources and raises error rates. The resulting architecture handles defect clusters and preserves the operations needed for computation.

Core claim

We propose a universal superstabilizer scheme applicable to data qubit defects in arbitrary stabilizer codes. Based on this scheme, we develop concrete repair methods for isolated defects of both internal data qubits and ancilla qubits in color codes defined on square lattices. Furthermore, for ancilla qubit defects, we present two optimization schemes. One scheme reuses neighboring ancilla qubits, and the other employs iSWAP gates. Unlike conventional approaches that directly disable neighboring data qubits and thus cause resource waste, both of our schemes avoid such waste and consequently achieve a lower logical error rate. Integrating the above techniques, we construct a comprehensive 1-

What carries the argument

The universal superstabilizer scheme, which deforms the code lattice around defects to restore error-correction capability without extra qubit loss.

If this is right

  • Isolated defects in both data and ancilla qubits can be repaired without disabling extra data qubits.
  • Two specific optimizations for ancilla defects (neighbor reuse and iSWAP insertion) reduce resource waste.
  • The full architecture handles clusters of defects while preserving transversal Clifford gates and lattice surgery.
  • The approach applies to any stabilizer code, not just color codes.

Where Pith is reading between the lines

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

  • Color codes could become competitive with surface codes on real hardware that contains a few percent defects.
  • The scheme might allow smaller code distances to reach the same reliability target, lowering overall qubit overhead.
  • Experimental teams could test the iSWAP-based repair on existing superconducting arrays by deliberately inducing isolated ancilla defects.

Load-bearing premise

Repair operations introduced by the scheme do not create new error sources that would cancel the claimed reduction in logical error rate.

What would settle it

A threshold simulation or hardware experiment on a defective color-code patch that measures the logical error rate after applying the superstabilizer repairs and finds it equal to or higher than the rate obtained by simply disabling neighboring qubits.

Figures

Figures reproduced from arXiv: 2604.05874 by Guo-Ping Guo, Jia-Ning Li, Jia-Xuan Zhang, Tian-Hao Wei, Wei-Cheng Kong, Yu-Chun Wu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Simulation results for isolated ancilla qubit defects, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: (c), a measurement of the logical operator Z¯ 1X¯ 2 can be implemented as in [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. (a) Left: Schematic of the Neighbor-Assisted scheme [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p020_19.png] view at source ↗
read the original abstract

Quantum error correction is a crucial technology for fault tolerant quantum computing. On superconducting platforms, hardware defects in large scale quantum processors can disrupt the regular lattice structure of topological codes and impair their error correction capabilities. Although defect adaptive methods for surface codes have been extensively studied, other topological codes such as color codes still lack a systematic framework for handling defects. To address this issue, we propose a universal superstabilizer scheme applicable to data qubit defects in arbitrary stabilizer codes. Based on this scheme, we develop concrete repair methods for isolated defects of both internal data qubits and ancilla qubits in color codes defined on square lattices. Furthermore, for ancilla qubit defects, we present two optimization schemes. One scheme reuses neighboring ancilla qubits, and the other employs iSWAP gates. Unlike conventional approaches that directly disable neighboring data qubits and thus cause resource waste, both of our schemes avoid such waste and consequently achieve a lower logical error rate.Integrating the above techniques, we construct a comprehensive defect adaptive architecture for color codes to handle various defect clusters. We also show that our scheme supports a full transversal Clifford gate set and lattice surgery operations. These results provide a systematic theoretical pathway for deploying robust and low overhead color codes on defective quantum hardware.

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 / 1 minor

Summary. The manuscript proposes a universal superstabilizer scheme applicable to data qubit defects in arbitrary stabilizer codes. It develops concrete repair methods for isolated defects of internal data qubits and ancilla qubits in square-lattice color codes, presents two optimization schemes for ancilla defects (neighbor reuse and iSWAP gates), constructs a comprehensive defect-adaptive architecture for various defect clusters, and asserts that these approaches avoid resource waste, achieve lower logical error rates than disabling neighbors, support a full transversal Clifford gate set, and enable lattice surgery operations.

Significance. If the proposed repairs preserve code distance and deliver a net reduction in logical error rate under circuit-level noise, the work would supply a systematic framework for defect handling in color codes, extending surface-code techniques to another topological code family and supporting practical deployment on defective superconducting hardware with reduced overhead.

major comments (2)
  1. [Abstract] Abstract: the assertion that the schemes 'consequently achieve a lower logical error rate' is unsupported by any threshold analysis, Monte Carlo data, or explicit comparison of effective error rates. This is load-bearing because the net benefit over conventional disabling of neighbors hinges on the added gates and measurements in the superstabilizer repairs not introducing offsetting errors.
  2. [Universal scheme description] Universal superstabilizer scheme: the claim of applicability to arbitrary stabilizer codes and preservation of code distance for all defect cases is stated without an explicit general construction or proof. The manuscript must supply this to substantiate the universality assertion.
minor comments (1)
  1. The abstract and introduction would benefit from a brief statement of the assumed noise model (e.g., depolarizing or circuit-level) under which the lower error-rate claim is expected to hold.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review. The comments highlight important points regarding the strength of our claims, and we address each major comment below with specific revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the schemes 'consequently achieve a lower logical error rate' is unsupported by any threshold analysis, Monte Carlo data, or explicit comparison of effective error rates. This is load-bearing because the net benefit over conventional disabling of neighbors hinges on the added gates and measurements in the superstabilizer repairs not introducing offsetting errors.

    Authors: We agree that the original phrasing in the abstract overstates the result without supporting numerical evidence. The claim was intended to follow qualitatively from resource reuse (avoiding the disabling of neighboring qubits, which would otherwise reduce the effective code distance or stabilizer count). However, as the referee correctly notes, the added gates and measurements in the superstabilizer construction could offset this benefit under circuit-level noise. We will revise the abstract to state that the schemes 'avoid resource waste and are expected to yield a lower logical error rate compared to disabling neighbors, subject to verification via numerical simulation.' We will also add a dedicated paragraph in the discussion section explaining the qualitative advantage and explicitly noting that full threshold analysis and Monte Carlo simulations under circuit-level noise are left for future work. revision: partial

  2. Referee: [Universal scheme description] Universal superstabilizer scheme: the claim of applicability to arbitrary stabilizer codes and preservation of code distance for all defect cases is stated without an explicit general construction or proof. The manuscript must supply this to substantiate the universality assertion.

    Authors: The manuscript introduces the superstabilizer idea in Section II as a general method: for an isolated data-qubit defect in any stabilizer code, one replaces the affected stabilizers with a new set of 'superstabilizers' whose support is deformed onto neighboring qubits while keeping the logical operators invariant. This construction is illustrated for color codes but is presented as applicable to any CSS or non-CSS stabilizer code. We acknowledge that an explicit general algorithm and a formal proof of distance preservation were not provided. We will add a new subsection (II.B) that gives the general construction: (1) identify all stabilizers touching the defect qubit, (2) augment them by including an auxiliary qubit or by linear combination to cancel the defect, and (3) verify that the minimum weight of nontrivial logical operators remains unchanged for isolated defects. A short proof sketch will be included showing that, for any isolated defect, the new stabilizer group still commutes, has the same rank, and the code distance is at least the original distance minus a constant that depends only on the defect size (which is 1 in our case). This will substantiate the universality claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity: constructive proposal for superstabilizer defect repairs without fitted inputs or self-referential reductions.

full rationale

The manuscript presents a new universal superstabilizer scheme for data qubit defects in arbitrary stabilizer codes, followed by concrete repair constructions for isolated defects in square-lattice color codes, two ancilla optimizations (neighbor reuse and iSWAP), and an integrated architecture supporting transversal Cliffords and lattice surgery. No equations, fitted parameters, or parameter-renaming steps appear in the provided text. Claims of lower logical error rate are asserted as a direct consequence of avoiding the resource waste of disabling neighbors, but this is a design-level consequence rather than a reduction of any derived quantity to its own inputs by construction. No self-citation is invoked as a uniqueness theorem or load-bearing premise that would force the result. The work is therefore a self-contained constructive proposal whose central results do not collapse to tautological re-labeling of prior quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a workable superstabilizer construction for arbitrary stabilizer codes and on the assumption that the proposed ancilla repairs preserve the code distance and logical operators without new error channels.

axioms (2)
  • domain assumption Defects can be treated as isolated or in small clusters without global lattice reconfiguration.
    Invoked when describing repair methods for isolated defects and defect clusters.
  • domain assumption The underlying quantum hardware supports the required iSWAP gates and measurements with sufficient fidelity.
    Implicit in the claim that the schemes achieve lower logical error rates.

pith-pipeline@v0.9.0 · 5533 in / 1324 out tokens · 29543 ms · 2026-05-10T20:20:44.845342+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

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

  1. Scalable quantum error correction tailored for a heavy-hex qubit array

    quant-ph 2026-04 unverdicted novelty 6.0

    The dynamic compass code achieves competitive thresholds on heavy-hex lattices with an experimental distance-5 implementation showing up to 38.3% logical error rate reduction through noise-informed decoding using ACES...

Reference graph

Works this paper leans on

70 extracted references · 10 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Every logical operatorLof the original stabilizer code can be transformed (by multiplying with an el- ement of⟨X q, Zq⟩and possibly with elements ofA) into an operator ˜Lthat commutes with the whole original stabilizer group and acts as identity on qubitq. Consequently, the restriction of ˜Lto the remaining qubits yields an operator that commutes withS ne...

  2. [2]

    Polynomial-time algorithms for prime fac- torization and discrete logarithms on a quantum com- puter.SIAM review, 41(2):303–332, 1999

    Peter W Shor. Polynomial-time algorithms for prime fac- torization and discrete logarithms on a quantum com- puter.SIAM review, 41(2):303–332, 1999

    1999

  3. [3]

    Quantum simulation.Reviews of Modern Physics, 86(1): 153–185, 2014

    Iulia M Georgescu, Sahel Ashhab, and Franco Nori. Quantum simulation.Reviews of Modern Physics, 86(1): 153–185, 2014

    2014

  4. [4]

    Harnessing quantum power: Revolution- izing materials design through advanced quantum com- putation.Materials Genome Engineering Advances, 2(4): e73, 2024

    Zikang Guo, Rui Li, Xianfeng He, Jiang Guo, and Shenghong Ju. Harnessing quantum power: Revolution- izing materials design through advanced quantum com- putation.Materials Genome Engineering Advances, 2(4): e73, 2024

    2024

  5. [5]

    A fast quantum mechanical algorithm for database search

    Lov K Grover. A fast quantum mechanical algorithm for database search. InProceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219, 1996

    1996

  6. [6]

    A quan- tum engineer’s guide to superconducting qubits.Applied physics reviews, 6(2), 2019

    Philip Krantz, Morten Kjaergaard, Fei Yan, Terry P Or- lando, Simon Gustavsson, and William D Oliver. A quan- tum engineer’s guide to superconducting qubits.Applied physics reviews, 6(2), 2019

    2019

  7. [7]

    Quantum error correction for quan- tum memories.Reviews of Modern Physics, 87(2):307– 346, 2015

    Barbara M Terhal. Quantum error correction for quan- tum memories.Reviews of Modern Physics, 87(2):307– 346, 2015

    2015

  8. [8]

    CRC Press, 2008

    Frank Gaitan.Quantum error correction and fault toler- ant quantum computing. CRC Press, 2008

    2008

  9. [9]

    Roads towards fault-tolerant universal quantum computation.Nature, 549(7671):172–179, 2017

    Earl T Campbell, Barbara M Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation.Nature, 549(7671):172–179, 2017

    2017

  10. [10]

    How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits

    Craig Gidney and Martin Eker˚ a. How to factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits. Quantum, 5:433, 2021

    2048

  11. [11]

    Superconduct- ing qubits: Current state of play.Annual Review of Condensed Matter Physics, 11(1):369–395, 2020

    Morten Kjaergaard, Mollie E Schwartz, Jochen Braum¨ uller, Philip Krantz, Joel I-J Wang, Simon Gustavsson, and William D Oliver. Superconduct- ing qubits: Current state of play.Annual Review of Condensed Matter Physics, 11(1):369–395, 2020

    2020

  12. [12]

    Resolving the positions of defects in superconducting quantum bits.Scientific reports, 10(1): 3090, 2020

    Alexander Bilmes, Anthony Megrant, Paul Klimov, Georg Weiss, John M Martinis, Alexey V Ustinov, and J¨ urgen Lisenfeld. Resolving the positions of defects in superconducting quantum bits.Scientific reports, 10(1): 3090, 2020

    2020

  13. [13]

    Fluctuations of energy-relaxation times in superconducting qubits.Phys- ical review letters, 121(9):090502, 2018

    Paul V Klimov, Julian Kelly, Zijun Chen, Matthew Nee- ley, Anthony Megrant, Brian Burkett, Rami Barends, Kunal Arya, Ben Chiaro, Yu Chen, et al. Fluctuations of energy-relaxation times in superconducting qubits.Phys- ical review letters, 121(9):090502, 2018

    2018

  14. [14]

    Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator.Nature Communications, 5(1):4119, 2014

    Jonathan Burnett, Lara Faoro, Ilana Wisby, Vladimir L Gurtovoi, Alexey V Chernykh, Gennady M Mikhailov, Vyacheslav A Tulin, Rais Shaikhaidarov, Vladimir Antonov, Phil J Meeson, et al. Evidence for interacting two-level systems from the 1/f noise of a superconducting resonator.Nature Communications, 5(1):4119, 2014

    2014

  15. [15]

    Electric field spectroscopy of material defects in transmon qubits.npj Quantum Information, 5(1):105, 2019

    J¨ urgen Lisenfeld, Alexander Bilmes, Anthony Megrant, Rami Barends, Julian Kelly, Paul Klimov, Georg Weiss, John M Martinis, and Alexey V Ustinov. Electric field spectroscopy of material defects in transmon qubits.npj Quantum Information, 5(1):105, 2019

    2019

  16. [16]

    Scaling superconducting quantum computers with chiplet architectures

    Kaitlin N Smith, Gokul Subramanian Ravi, Jonathan M Baker, and Frederic T Chong. Scaling superconducting quantum computers with chiplet architectures. In2022 55th IEEE/ACM International Symposium on Microar- chitecture (MICRO), pages 1092–1109. IEEE, 2022

    2022

  17. [17]

    Surface codes: Towards practical large-scale quantum computation.Physical Re- view A—Atomic, Molecular, and Optical Physics, 86(3): 032324, 2012

    Austin G Fowler, Matteo Mariantoni, John M Marti- nis, and Andrew N Cleland. Surface codes: Towards practical large-scale quantum computation.Physical Re- view A—Atomic, Molecular, and Optical Physics, 86(3): 032324, 2012

    2012

  18. [18]

    Fault- tolerance thresholds for the surface code with fabrication errors.Physical Review A, 96(4):042316, 2017

    James M Auger, Hussain Anwar, Mercedes Gimeno- Segovia, Thomas M Stace, and Dan E Browne. Fault- tolerance thresholds for the surface code with fabrication errors.Physical Review A, 96(4):042316, 2017

    2017

  19. [19]

    Quantum computing is scalable on a planar ar- ray of qubits with fabrication defects.Physical Review Applied, 19(6):064081, 2023

    Armands Strikis, Simon C Benjamin, and Benjamin J Brown. Quantum computing is scalable on a planar ar- ray of qubits with fabrication defects.Physical Review Applied, 19(6):064081, 2023

    2023

  20. [20]

    Adaptive surface code for quantum error correction in the presence of temporary or permanent de- fects.Quantum, 7:1065, 2023

    Adam Siegel, Armands Strikis, Thomas Flatters, and Si- mon Benjamin. Adaptive surface code for quantum error correction in the presence of temporary or permanent de- fects.Quantum, 7:1065, 2023. 22

    2023

  21. [21]

    Codesign of quantum error-correcting codes and modular chiplets in the pres- ence of defects

    Sophia Fuhui Lin, Joshua Viszlai, Kaitlin N Smith, Gokul Subramanian Ravi, Charles Yuan, Frederic T Chong, and Benjamin J Brown. Codesign of quantum error-correcting codes and modular chiplets in the pres- ence of defects. InProceedings of the 29th ACM Interna- tional Conference on Architectural Support for Program- ming Languages and Operating Systems, V...

    2024

  22. [22]

    Surf-deformer: Mitigating dynamic defects on surface code via adaptive deformation

    Keyi Yin, Xiang Fang, Travis S Humble, Ang Li, Yunong Shi, and Yufei Ding. Surf-deformer: Mitigating dynamic defects on surface code via adaptive deformation. In2024 57th IEEE/ACM International Symposium on Microar- chitecture (MICRO), pages 750–764. IEEE, 2024

    2024

  23. [23]

    Low-overhead defect- adaptive surface code with bandage-like super-stabilizers

    Zuolin Wei, Tan He, Yangsen Ye, Dachao Wu, Yim- ing Zhang, Youwei Zhao, Weiping Lin, He-Liang Huang, Xiaobo Zhu, and Jian-Wei Pan. Low-overhead defect- adaptive surface code with bandage-like super-stabilizers. npj Quantum Information, 11(1):75, 2025

    2025

  24. [24]

    Snakes and ladders: Adapt- ing the surface code to defects.PRX Quantum, 6(4): 040302, 2025

    Catherine Leroux, Sophia F Lin, Przemyslaw Bienias, Krishanu R Sankar, Asmae Benhemou, Aleksander Ku- bica, and Joseph K Iverson. Snakes and ladders: Adapt- ing the surface code to defects.PRX Quantum, 6(4): 040302, 2025

    2025

  25. [25]

    Luci in the surface code with dropouts.Quantum, 9:1936, 2025

    Dripto M Debroy, Matt McEwen, Craig Gidney, Noah Shutty, and Adam Zalcman. Luci in the surface code with dropouts.Quantum, 9:1936, 2025

    1936

  26. [26]

    Diamond circuits for surface codes

    Dripto M Debroy. Diamond circuits for surface codes. arXiv preprint arXiv:2502.10355, 2025

    work page arXiv 2025

  27. [27]

    Handling fabrication de- fects in hex-grid surface codes

    Oscar Higgott, Benjamin Anker, Matt McEwen, and Dripto M Debroy. Handling fabrication defects in hex- grid surface codes.arXiv preprint arXiv:2508.08116, 2025

    work page arXiv 2025

  28. [28]

    Halma: a routing-based technique for defect mitigation in quantum error correction.arXiv preprint arXiv:2412.21000, 2024

    Runshi Zhou, Fang Zhang, Linghang Kong, and Jianxin Chen. Halma: a routing-based technique for defect mitigation in quantum error correction.arXiv preprint arXiv:2412.21000, 2024

    work page arXiv 2024

  29. [29]

    Two-dimensional color-code quantum computation.Physical Review A—Atomic, Molecular, and Optical Physics, 83(4):042310, 2011

    Austin G Fowler. Two-dimensional color-code quantum computation.Physical Review A—Atomic, Molecular, and Optical Physics, 83(4):042310, 2011

    2011

  30. [30]

    Transversal clifford gates on folded surface codes.Physical Review A, 94(4):042316, 2016

    Jonathan E Moussa. Transversal clifford gates on folded surface codes.Physical Review A, 94(4):042316, 2016

    2016

  31. [31]

    Universal transversal gates with color codes: A simplified approach

    Aleksander Kubica and Michael E Beverland. Universal transversal gates with color codes: A simplified approach. Physical Review A, 91(3):032330, 2015

    2015

  32. [32]

    Low-overhead magic state distillation with color codes.PRX Quantum, 6(3):030317, 2025

    Seok-Hyung Lee, Felix Thomsen, Nicholas Fazio, Ben- jamin J Brown, and Stephen D Bartlett. Low-overhead magic state distillation with color codes.PRX Quantum, 6(3):030317, 2025

    2025

  33. [33]

    Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes.PRX Quantum, 2(2):020341, 2021

    Michael E Beverland, Aleksander Kubica, and Krysta M Svore. Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes.PRX Quantum, 2(2):020341, 2021

    2021

  34. [34]

    Scaling and logic in the colour code on a superconducting quantum processor.Nature, 645(8081):614–619, 2025

    Nathan Lacroix, Alexandre Bourassa, Francisco JH Heras, Lei M Zhang, Johannes Bausch, Andrew W Se- nior, Thomas Edlich, Noah Shutty, Volodymyr Sivak, Andreas Bengtsson, et al. Scaling and logic in the colour code on a superconducting quantum processor.Nature, 645(8081):614–619, 2025

    2025

  35. [35]

    Experimental demonstration of logical magic state distillation.Nature, 645(8081):620–625, 2025

    Pedro Sales Rodriguez, John M Robinson, Paul Niklas Jepsen, Zhiyang He, Casey Duckering, Chen Zhao, Kai- Hsin Wu, Joseph Campo, Kevin Bagnall, Minho Kwon, et al. Experimental demonstration of logical magic state distillation.Nature, 645(8081):620–625, 2025

    2025

  36. [36]

    Establishing a new benchmark in quantum computational advantage with 105-qubit zu- chongzhi 3.0 processor.Physical Review Letters, 134(9): 090601, 2025

    Dongxin Gao, Daojin Fan, Chen Zha, Jiahao Bei, Guo- qing Cai, Jianbin Cai, Sirui Cao, Fusheng Chen, Jiang Chen, Kefu Chen, et al. Establishing a new benchmark in quantum computational advantage with 105-qubit zu- chongzhi 3.0 processor.Physical Review Letters, 134(9): 090601, 2025

    2025

  37. [37]

    Quantum error correction below the surface code thresh- old.Nature, 638(8052):920–926, 2025

    2025

  38. [38]

    Topo- logical quantum distillation.Physical review letters, 97 (18):180501, 2006

    Hector Bombin and Miguel Angel Martin-Delgado. Topo- logical quantum distillation.Physical review letters, 97 (18):180501, 2006

    2006

  39. [39]

    Anyon condensation and the color code

    Markus S Kesselring, Julio C Magdalena de la Fuente, Fe- lix Thomsen, Jens Eisert, Stephen D Bartlett, and Ben- jamin J Brown. Anyon condensation and the color code. PRX Quantum, 5(1):010342, 2024

    2024

  40. [40]

    Facili- tating practical fault-tolerant quantum computing based on color codes.Physical Review Research, 6(3):033086, 2024

    Jiaxuan Zhang, Yu-Chun Wu, and Guo-Ping Guo. Facili- tating practical fault-tolerant quantum computing based on color codes.Physical Review Research, 6(3):033086, 2024

    2024

  41. [41]

    New circuits and an open source decoder for the color code , publisher =

    Craig Gidney and Cody Jones. New circuits and an open source decoder for the color code.arXiv preprint arXiv:2312.08813, 2023

    work page arXiv 2023

  42. [42]

    Realization of an error- correcting surface code with superconducting qubits

    Youwei Zhao, Yangsen Ye, He-Liang Huang, Yiming Zhang, Dachao Wu, Huijie Guan, Qingling Zhu, Zuolin Wei, Tan He, Sirui Cao, et al. Realization of an error- correcting surface code with superconducting qubits. Physical Review Letters, 129(3):030501, 2022

    2022

  43. [43]

    Strong quantum computa- tional advantage using a superconducting quantum pro- cessor.Physical review letters, 127(18):180501, 2021

    Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming- Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, et al. Strong quantum computa- tional advantage using a superconducting quantum pro- cessor.Physical review letters, 127(18):180501, 2021

    2021

  44. [44]

    Suppressing quantum errors by scaling a surface code log- ical qubit.Nature, 614(7949):676–681, 2023

    2023

  45. [45]

    Quantum supremacy using a programmable supercon- ducting processor.nature, 574(7779):505–510, 2019

    Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando GSL Brandao, David A Buell, et al. Quantum supremacy using a programmable supercon- ducting processor.nature, 574(7779):505–510, 2019

    2019

  46. [46]

    Neural network de- coder for topological color codes with circuit level noise

    Paul Baireuther, Marcello D Caio, Ben Criger, Carlo WJ Beenakker, and Thomas E O’Brien. Neural network de- coder for topological color codes with circuit level noise. New Journal of Physics, 21(1):013003, 2019

    2019

  47. [47]

    Stim: a fast stabilizer circuit simulator

    Craig Gidney. Stim: a fast stabilizer circuit simulator. Quantum, 5:497, 2021

    2021

  48. [48]

    One gate scheme to rule them all: Introducing a complex yet reduced instruction set for quantum computing

    Jianxin Chen, Dawei Ding, Weiyuan Gong, Cupjin Huang, and Qi Ye. One gate scheme to rule them all: Introducing a complex yet reduced instruction set for quantum computing. InProceedings of the 29th ACM In- ternational Conference on Architectural Support for Pro- gramming Languages and Operating Systems, Volume 2, pages 779–796, 2024

    2024

  49. [49]

    Efficient implementation of ar- bitrary two-qubit gates using unified control.Nature Physics, 21(9):1489–1496, 2025

    Zhen Chen, Weiyang Liu, Yanjun Ma, Weijie Sun, Ruixia Wang, He Wang, Huikai Xu, Guangming Xue, Haisheng Yan, Zhen Yang, et al. Efficient implementation of ar- bitrary two-qubit gates using unified control.Nature Physics, 21(9):1489–1496, 2025

    2025

  50. [50]

    Quan- tum computing by color-code lattice surgery.arXiv preprint arXiv:1407.5103, 2014

    Andrew J Landahl and Ciaran Ryan-Anderson. Quan- tum computing by color-code lattice surgery.arXiv preprint arXiv:1407.5103, 2014

    work page arXiv 2014

  51. [51]

    Magic state cultivation: growing T states as cheap as CNOT gates

    Craig Gidney, Noah Shutty, and Cody Jones. Magic state cultivation: growing t states as cheap as cnot gates.arXiv preprint arXiv:2409.17595, 2024

    work page internal anchor Pith review arXiv 2024

  52. [52]

    Landahl and Jonas T

    Andrew J Landahl, Jonas T Anderson, and Patrick R Rice. Fault-tolerant quantum computing with color 23 codes.arXiv preprint arXiv:1108.5738, 2011

    work page arXiv 2011

  53. [53]

    Low-overhead quantum comput- ing with the color code.Physical Review Research, 6(4): 043125, 2024

    Felix Thomsen, Markus S Kesselring, Stephen D Bartlett, and Benjamin J Brown. Low-overhead quantum comput- ing with the color code.Physical Review Research, 6(4): 043125, 2024

    2024

  54. [54]

    Lattice surgery for dummies.Sensors, 25(6):1854, 2025

    Avimita Chatterjee, Subrata Das, and Swaroop Ghosh. Lattice surgery for dummies.Sensors, 25(6):1854, 2025

    2025

  55. [55]

    A sat scalpel for lattice surgery: representa- tion and synthesis of subroutines for surface-code fault- tolerant quantum computing

    Daniel Bochen Tan, Murphy Yuezhen Niu, and Craig Gidney. A sat scalpel for lattice surgery: representa- tion and synthesis of subroutines for surface-code fault- tolerant quantum computing. In2024 ACM/IEEE 51st Annual International Symposium on Computer Architec- ture (ISCA), pages 325–339. IEEE, 2024

    2024

  56. [56]

    Lattice surgery with a twist: Simplifying clifford gates of surface codes

    Daniel Litinski and Felix von Oppen. Lattice surgery with a twist: Simplifying clifford gates of surface codes. Quantum, 2:62, 2018

    2018

  57. [57]

    A game of surface codes: Large-scale quantum computing with lattice surgery.Quantum, 3: 128, 2019

    Daniel Litinski. A game of surface codes: Large-scale quantum computing with lattice surgery.Quantum, 3: 128, 2019

    2019

  58. [58]

    Magic teleporta- tion with generalized lattice surgery.arXiv preprint arXiv:2503.19758, 2025

    Yifei Wang and Yingfei Gu. Magic teleporta- tion with generalized lattice surgery.arXiv preprint arXiv:2503.19758, 2025

    work page arXiv 2025

  59. [59]

    Topolog- ical and subsystem codes on low-degree graphs with flag qubits.Physical Review X, 10(1):011022, 2020

    Christopher Chamberland, Guanyu Zhu, Theodore J Yo- der, Jared B Hertzberg, and Andrew W Cross. Topolog- ical and subsystem codes on low-degree graphs with flag qubits.Physical Review X, 10(1):011022, 2020

    2020

  60. [60]

    Magic state injection on ibm quantum processors above the distillation threshold.Scientific Reports, 2026

    Younghun Kim, Martin Sevior, and Muhammad Usman. Magic state injection on ibm quantum processors above the distillation threshold.Scientific Reports, 2026

    2026

  61. [61]

    Demonstration of low-overhead quantum error correction codes.Nature Physics, pages 1–7, 2026

    Ke Wang, Zhide Lu, Chuanyu Zhang, Gongyu Liu, Ji- achen Chen, Yanzhe Wang, Yaozu Wu, Shibo Xu, Xuhao Zhu, Feitong Jin, et al. Demonstration of low-overhead quantum error correction codes.Nature Physics, pages 1–7, 2026

    2026

  62. [62]

    Decoding across the quantum low-density parity-check code landscape.Physical Review Research, 2(4):043423, 2020

    Joschka Roffe, David R White, Simon Burton, and Earl Campbell. Decoding across the quantum low-density parity-check code landscape.Physical Review Research, 2(4):043423, 2020

    2020

  63. [63]

    Sparse blossom: cor- recting a million errors per core second with minimum- weight matching.Quantum, 9:1600, 2025

    Oscar Higgott and Craig Gidney. Sparse blossom: cor- recting a million errors per core second with minimum- weight matching.Quantum, 9:1600, 2025

    2025

  64. [64]

    Learning high-accuracy error decod- ing for quantum processors.Nature, 635(8040):834–840, 2024

    Johannes Bausch, Andrew W Senior, Francisco JH Heras, Thomas Edlich, Alex Davies, Michael Newman, Cody Jones, Kevin Satzinger, Murphy Yuezhen Niu, Sam Blackwell, et al. Learning high-accuracy error decod- ing for quantum processors.Nature, 635(8040):834–840, 2024

    2024

  65. [65]

    Stability experiments: The overlooked dual of memory experiments.Quantum, 6:786, 2022

    Craig Gidney. Stability experiments: The overlooked dual of memory experiments.Quantum, 6:786, 2022

    2022

  66. [66]

    Awesome quantum comput- ing experiments: Benchmarking experimental progress towards fault-tolerant quantum computation.arXiv preprint arXiv:2507.03678, 2025

    Fran¸ cois-Marie Le R´ egent. Awesome quantum comput- ing experiments: Benchmarking experimental progress towards fault-tolerant quantum computation.arXiv preprint arXiv:2507.03678, 2025

    work page arXiv 2025

  67. [67]

    Stability of noisy quantum computing devices,

    Samudra Dasgupta and Travis S Humble. Stability of noisy quantum computing devices.arXiv preprint arXiv:2105.09472, 2021

    work page arXiv 2021

  68. [68]

    Characteriz- ing the stability of nisq devices

    Samudra Dasgupta and Travis S Humble. Characteriz- ing the stability of nisq devices. In2020 IEEE Inter- national Conference on Quantum Computing and Engi- neering (QCE), pages 419–429. IEEE, 2020

    2020

  69. [69]

    Optimal number of stabilizer measurement rounds in an idling surface code patch.Quantum, 9:1767, 2025

    ´Aron M´ arton and J´ anos K Asb´ oth. Optimal number of stabilizer measurement rounds in an idling surface code patch.Quantum, 9:1767, 2025

    2025

  70. [70]

    Empirical overhead of the adapted surface code on defective qubit arrays

    Sophia Lin, Joshua Viszlai, Kaitlin Smith, Gokul Subra- manian Ravi, Charles Yuan, Frederic Chong, and Ben- jamin Brown. Empirical overhead of the adapted surface code on defective qubit arrays. InAPS March Meeting Abstracts, volume 2024, pages N51–004, 2024

    2024