pith. machine review for the scientific record. sign in

arxiv: 2604.14299 · v1 · submitted 2026-04-15 · 🪐 quant-ph

Recognition: unknown

Low-valency scalable quantum error correction with a dynamic compass code

Jun Zen , Xanda C. Kolesnikow , Campbell K. McLauchlan , Georgia M. Nixon , Thomas R. Scruby , Seok-Hyung Lee , Stephen D. Bartlett , Benjamin J. Brown
show 1 more author
Robin Harper
Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:02 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error correctioncompass codeheavy-hex latticesyndrome extraction scheduleerror thresholdstability experimentlattice surgerysubsystem code
0
0 comments X

The pith

A new measurement schedule on the heavy-hex subsystem code produces the dynamic compass code that fits modest hardware footprints and shows error thresholds.

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

The paper introduces the dynamic compass code by applying a novel measurement schedule to the syndrome extraction circuit of the heavy-hex subsystem code. This schedule enables implementation with a modest number of physical qubits on the heavy-hex lattice while still producing a threshold where logical error rates fall with increasing code size. Different choices of schedule allow a trade-off between protection against X-basis and Z-basis logical errors. The same schedule also yields a threshold in stability experiments that check preservation of logical information over time. The authors illustrate how the code supports fault-tolerant operations through lattice surgery between separate patches.

Core claim

The dynamic compass code arises from selecting a novel measurement schedule for the syndrome extraction circuit of the heavy-hex subsystem code. Numerical simulations under idealized error models show that this code can be realized on the heavy-hex lattice with a modest footprint and exhibits a threshold. Different schedules produce different levels of protection for logical X versus logical Z errors. The schedule also confers a threshold on stability experiments, and the code can be used for fault-tolerant logic via lattice surgery between patches.

What carries the argument

The novel measurement schedule for syndrome extraction, which reorders the checks in the heavy-hex subsystem code to create the dynamic compass code and its observed thresholds.

If this is right

  • The code achieves a threshold on the heavy-hex lattice with modest qubit overhead.
  • Schedule choice controls the relative protection against X-type versus Z-type logical errors.
  • The same schedule produces a threshold for stability experiments that track logical information over repeated rounds.
  • Lattice surgery between patches becomes available for performing fault-tolerant logical operations.

Where Pith is reading between the lines

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

  • If the schedule can be executed on real hardware, it may allow error correction on lattices already used by current processors without requiring higher-connectivity graphs.
  • The X-Z trade-off suggests that one could tune the code for algorithms whose dominant error type is known in advance.
  • Stability thresholds open a route to testing logical qubit lifetime before full computation-scale circuits are built.

Load-bearing premise

Numerical thresholds and performance numbers assume idealized error models with perfect syndrome extraction and no additional hardware noise.

What would settle it

A simulation or hardware run in which logical error rates fail to decrease with larger code distance under models that include leakage, calibration drift, or imperfect syndrome circuits would falsify the threshold claim.

Figures

Figures reproduced from arXiv: 2604.14299 by Benjamin J. Brown, Campbell K. McLauchlan, Georgia M. Nixon, Jun Zen, Robin Harper, Seok-Hyung Lee, Stephen D. Bartlett, Thomas R. Scruby, Xanda C. Kolesnikow.

Figure 2
Figure 2. Figure 2: FIG. 2. (a) The region of the lattice in the yellow box in [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Alternating [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A period-4 schedule that leads to reduced stabiliser [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Because all X checks are measured twice per cycle the X detecting regions each span half the cycle. Each detector is formed from either two or four measurements; the former corresponding to the case where a previously measured operator is still in the ISG and the latter to the case where a product of two operators is in the ISG but these individual operators are not. On the other hand, Z checks in the bulk… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Formation of [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Syndrome of a measurement error (white star) flip [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Syndrome of a [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Three different measurement schedules (A, B and C) for one row of [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. (a) [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Comparison of stability performance between the dynamic compass and heavy-hex codes. (a) An example of a 5 [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Estimated threshold surfaces for the dynamic compass code (rightmost schedule in Fig. 11) under a range of noise [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Comparison of threshold estimates in ((a) and (c)) [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗
Figure 17
Figure 17. Figure 17: We observe that the estimated thresholds from [PITH_FULL_IMAGE:figures/full_fig_p015_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Error suppression factors and exponential decay fits [PITH_FULL_IMAGE:figures/full_fig_p015_18.png] view at source ↗
read the original abstract

The ongoing development of hardware that is capable of reliably executing general quantum algorithms requires quantum error-correcting codes that are both practical for realisation and rapidly reduce logical error rates as they are scaled up. Here we introduce the dynamic compass code, a code that can be implemented with a modest footprint on the heavy-hex lattice while also demonstrating a threshold. The dynamic code is obtained by choosing a novel measurement schedule for the syndrome extraction circuit of the heavy-hex subsystem code. We numerically evaluate its performance and observe that different choices of schedule can provide a trade-off in protection against logical errors in the $X$ vs $Z$ basis. We also demonstrate that this new measurement schedule provides the code with a threshold for stability experiments. We finally show how the dynamic compass code could be used for fault-tolerant logic by illustrating lattice surgery between code patches.

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 introduces the dynamic compass code, obtained by selecting a novel measurement schedule for syndrome extraction in the heavy-hex subsystem code. It claims this yields a modest-footprint implementation on the heavy-hex lattice, a threshold under numerical evaluation, a tunable trade-off between X- and Z-basis logical error protection, a threshold for stability experiments, and compatibility with lattice surgery for fault-tolerant logic.

Significance. If the numerical threshold and lattice-surgery results survive realistic circuit-level noise, the work would be significant for hardware platforms using heavy-hex connectivity (e.g., IBM devices), offering a low-valency alternative to surface codes with explicit stability thresholds and surgery protocols. The schedule-based tunability between X/Z protection is a practical feature not commonly emphasized in prior heavy-hex work.

major comments (2)
  1. [§4] §4 (Numerical evaluation of thresholds): The reported threshold for stability experiments is obtained under an idealized depolarizing noise model with perfect syndrome extraction. The manuscript must clarify whether measurement errors, leakage, and schedule-induced temporal correlations are included in the circuit-level simulations; if they are omitted, the crossing point does not establish fault tolerance under hardware-realistic conditions.
  2. [§5] §5 (Lattice surgery): The illustration of lattice surgery between dynamic compass patches shows logical operations but does not quantify the logical error rate during the surgery protocol or compare it against the baseline heavy-hex code under the same noise model. A concrete error-rate table or plot for the merged patch is required to support the fault-tolerance claim.
minor comments (2)
  1. [Figure 3] Figure 3: The caption should explicitly state the physical error rate at which the logical error rate crosses the threshold and the number of Monte Carlo shots used for each data point.
  2. [Eq. (3)] Notation: The definition of the dynamic measurement schedule (Eq. 3) uses an ad-hoc indexing of stabilizers; a small diagram or table mapping the schedule to the heavy-hex lattice edges would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments in detail below and have made revisions to the manuscript where appropriate to improve clarity and strengthen the presentation.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical evaluation of thresholds): The reported threshold for stability experiments is obtained under an idealized depolarizing noise model with perfect syndrome extraction. The manuscript must clarify whether measurement errors, leakage, and schedule-induced temporal correlations are included in the circuit-level simulations; if they are omitted, the crossing point does not establish fault tolerance under hardware-realistic conditions.

    Authors: We agree with the referee that our simulations in Section 4 use an idealized depolarizing noise model with perfect syndrome extraction. Measurement errors, leakage, and schedule-induced temporal correlations are not included in the current circuit-level simulations. This approach is common for initial threshold calculations to demonstrate the existence of a threshold in the code-capacity regime. We have revised the manuscript to explicitly state these assumptions in Section 4 and added a paragraph in the discussion section clarifying that the reported threshold does not yet account for these hardware-realistic effects. We note that including such effects would be an important next step for assessing practical performance. revision: yes

  2. Referee: [§5] §5 (Lattice surgery): The illustration of lattice surgery between dynamic compass patches shows logical operations but does not quantify the logical error rate during the surgery protocol or compare it against the baseline heavy-hex code under the same noise model. A concrete error-rate table or plot for the merged patch is required to support the fault-tolerance claim.

    Authors: The lattice surgery protocol in Section 5 is presented as an illustration to demonstrate how the dynamic compass code can support fault-tolerant logical operations via lattice surgery on the heavy-hex lattice. While we show the sequence of measurements and the merging/splitting steps, we acknowledge that we do not provide quantitative logical error rates for the surgery process or a comparison to the standard heavy-hex subsystem code. Providing such data would require additional extensive simulations of the full circuit-level noise during the surgery, which is computationally demanding. In the revised manuscript, we have expanded the discussion to emphasize that the illustration establishes the protocol's compatibility and outline the steps needed for full fault tolerance, while noting the absence of quantitative error rates as a limitation to be addressed in future work. We believe this still supports the claim that the code 'could be used for fault-tolerant logic' at a conceptual level. revision: partial

Circularity Check

0 steps flagged

Minor self-citation to prior heavy-hex subsystem code; novel dynamic schedule and thresholds rest on independent numerical evaluation

full rationale

The paper defines the dynamic compass code explicitly as the result of selecting a novel measurement schedule applied to the existing heavy-hex subsystem code. Thresholds and X/Z error trade-offs are then obtained via direct numerical simulation under stated error models, not by algebraic reduction to fitted parameters or self-referential definitions. Any citations to earlier heavy-hex work by overlapping authors support only the base lattice and stabilizer structure, leaving the schedule choice and its performance claims as independent content. No self-definitional loops, fitted-input predictions, or ansatz smuggling appear in the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard quantum error correction assumptions (independent Pauli noise, perfect measurements in the abstract description) plus the existence of a threshold under the chosen schedule; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Standard independent Pauli error model for numerical simulations
    Implicit in all threshold claims for quantum codes; not stated explicitly in abstract but required for any numerical evaluation.
  • domain assumption Syndrome extraction circuits can be scheduled without additional errors beyond the modeled noise
    Required for the dynamic schedule to achieve the reported threshold.

pith-pipeline@v0.9.0 · 5474 in / 1349 out tokens · 30878 ms · 2026-05-10T13:02:07.397992+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

36 extracted references · 8 canonical work pages · 1 internal anchor

  1. [1]

    Bacon, Operator quantum error-correcting subsys- tems for self-correcting quantum memories, Phys

    D. Bacon, Operator quantum error-correcting subsys- tems for self-correcting quantum memories, Phys. Rev. A73, 012340 (2006)

    2006

  2. [2]

    M. Li, D. Miller, M. Newman, Y. Wu, and K. R. Brown, 2d compass codes, Phys. Rev. X9, 021041 (2019)

    2019

  3. [3]

    J. A. Campos and K. R. Brown, Clifford-deformed com- pass codes (2026), arXiv:2412.03808 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  4. [4]

    M. B. Hastings and J. Haah, Dynamically generated log- ical qubits, Quantum5, 564 (2021)

    2021

  5. [5]

    K. I. Kugel’ and D. I. Khomski˘ ı, The jahn-teller effect and magnetism: transition metal compounds, Soviet Physics Uspekhi25, 231 (1982)

    1982

  6. [6]

    Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics321, 2 (2006), january Special Issue

    A. Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics321, 2 (2006), january Special Issue

    2006

  7. [7]

    Nussinov and J

    Z. Nussinov and J. van den Brink, Compass models: The- ory and physical motivations, Rev. Mod. Phys.87, 1 (2015)

    2015

  8. [8]

    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)

    2003

  9. [10]

    Gidney, A Pair Measurement Surface Code on Pen- tagons, Quantum7, 1156 (2023)

    C. Gidney, A Pair Measurement Surface Code on Pen- tagons, Quantum7, 1156 (2023)

    2023

  10. [11]

    Davydova, N

    M. Davydova, N. Tantivasadakarn, and S. Balasubra- manian, Floquet codes without parent subsystem codes, PRX Quantum4, 020341 (2023)

    2023

  11. [12]

    M. S. Kesselring, J. C. Magdalena de la Fuente, F. Thom- sen, J. Eisert, S. D. Bartlett, and B. J. Brown, Anyon condensation and the color code, PRX Quantum5, 010342 (2024)

    2024

  12. [13]

    Davydova, N

    M. Davydova, N. Tantivasadakarn, S. Balasubramanian, and D. Aasen, Quantum computation from dynamic au- tomorphism codes, Quantum8, 1448 (2024)

    2024

  13. [14]

    A. Dua, N. Tantivasadakarn, J. Sullivan, and T. D. El- lison, Engineering 3d floquet codes by rewinding, PRX Quantum5, 020305 (2024)

    2024

  14. [15]

    Higgott and N

    O. Higgott and N. P. Breuckmann, Constructions and performance of hyperbolic and semi-hyperbolic floquet codes, PRX Quantum5, 040327 (2024)

    2024

  15. [16]

    Setiawan and C

    F. Setiawan and C. McLauchlan, Tailoring dynamical codes for biased noise: the X3Z3 Floquet code, npj Quan- tum Information11, 149 (2025)

    2025

  16. [17]

    P.-J. H. S. Derks, A. Townsend-Teague, J. Eisert, M. S. Kesselring, O. Higgott, and B. J. Brown, Dy- namical codes for hardware with noisy readouts (2025), arXiv:2505.07658 [quant-ph]

    work page arXiv 2025

  17. [18]

    Gidney and D

    C. Gidney and D. Bacon, Less bacon more threshold (2023), arXiv:2305.12046 [quant-ph]

    work page arXiv 2023

  18. [19]

    M. S. Alam and E. Rieffel, Dynamical logical qubits in the bacon-shor code, Phys. Rev. A112, 022436 (2025)

    2025

  19. [20]

    M. S. Alam, J. Zen, and T. R. Scruby, Bacon-shor board games (2026), arXiv:2504.02749 [quant-ph]

    work page arXiv 2026

  20. [21]

    Sundaresan, T

    N. Sundaresan, T. J. Yoder, Y. Kim, M. Li, E. H. Chen, G. Harper, T. Thorbeck, A. W. Cross, A. D. C´ orcoles, and M. Takita, Demonstrating multi-round subsystem quantum error correction using matching and maximum likelihood decoders, Nature Communications14, 2852 (2023)

    2023

  21. [22]

    R. S. Gupta, N. Sundaresan, T. Alexander, C. J. Wood, S. T. Merkel, M. B. Healy, M. Hillenbrand, T. Jochym- O’Connor, J. R. Wootton, T. J. Yoder, A. W. Cross, M. Takita, and B. J. Brown, Encoding a magic state with beyond break-even fidelity, Nature625, 259–263 (2024)

    2024

  22. [23]

    Neural and Computational Mechanisms Underlying One -Shot Perceptual Learning in Humans

    R. Harper, C. Lain´ e, E. Hockings, C. McLauchlan, G. M. Nixon, B. J. Brown, and S. D. Bartlett, Characteris- 12 ing the failure mechanisms of error-corrected quantum logic gates, Nature Communications 10.1038/s41467-026- 71773-6 (2026)

    work page doi:10.1038/s41467-026- 2026

  23. [24]

    S.-H. Lee, X. C. Kolesnikow, J. Zen, E. T. Hockings, C. K. McLauchlan, G. M. Nixon, T. R. Scruby, S. D. Bartlett, B. J. Brown, and R. Harper, Scalable quantum error correction tailored to a heavy-hex device (2026), arXiv:2604.xxxxx [quant-ph]

    2026

  24. [25]

    Townsend-Teague, J

    A. Townsend-Teague, J. Magdalena de la Fuente, and M. Kesselring, Floquetifying the colour code, Elec- tronic Proceedings in Theoretical Computer Science384, 265–303 (2023)

    2023

  25. [26]

    Chamberland, G

    C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, Topological and subsystem codes on low-degree graphs with flag qubits, Phys. Rev. X10, 011022 (2020)

    2020

  26. [27]

    ibm.com/quantum/blog/heavy-hex-lattice(2021), ac- cessed: 2026-04-14

    IBM Quantum, The heavy-hex lattice,https://www. ibm.com/quantum/blog/heavy-hex-lattice(2021), ac- cessed: 2026-04-14

    2021

  27. [28]

    G. P. Geh´ er, M. Jastrzebski, E. T. Campbell, and O. Crawford, To reset, or not to reset – that is the ques- tion (2024), arXiv:2408.00758 [quant-ph]

    work page arXiv 2024

  28. [29]

    Gidney, Stim: a fast stabilizer circuit simulator, Quan- tum5, 497 (2021)

    C. Gidney, Stim: a fast stabilizer circuit simulator, Quan- tum5, 497 (2021)

    2021

  29. [30]

    Higgott, Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching, ACM Transactions on Quantum Computing3, 1 (2022), arXiv:2105.13082

    O. Higgott, Pymatching: A python package for decoding quantum codes with minimum-weight perfect matching (2021), arXiv:2105.13082 [quant-ph]

    work page arXiv 2021

  30. [31]

    Gidney, Stability Experiments: The Overlooked Dual of Memory Experiments, Quantum6, 786 (2022)

    C. Gidney, Stability Experiments: The Overlooked Dual of Memory Experiments, Quantum6, 786 (2022)

    2022

  31. [32]

    Horsman, A

    C. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, Surface code quantum computing by lattice surgery, New Journal of Physics14, 123011 (2012)

    2012

  32. [33]

    Gidney, Inplace access to the surface code Y basis, Quantum8, 1310 (2024)

    C. Gidney, Inplace access to the surface code Y basis, Quantum8, 1310 (2024)

    2024

  33. [34]

    G. P. Geh´ er, C. McLauchlan, E. T. Campbell, A. E. Moylett, and O. Crawford, Error-corrected hadamard gate simulated at the circuit level, Quantum8, 1394 (2024)

    2024

  34. [35]

    B. J. Brown, K. Laubscher, M. S. Kesselring, and J. R. Wootton, Poking holes and cutting corners to achieve clif- ford gates with the surface code, Phys. Rev. X7, 021029 (2017)

    2017

  35. [36]

    T. D. Ellison, J. Sullivan, and A. Dua, Floquet codes with a twist (2023), arXiv:2306.08027 [quant-ph]

    work page arXiv 2023

  36. [37]

    B. J. Brown, N. H. Nickerson, and D. E. Browne, Fault- tolerant error correction with the gauge color code, Na- ture Communications7, 12302 (2016). Appendix A: Noise model Most circuits in this paper were simulated using the uniform noise model summarized in Table I and Table II. Table I defines the individual noise channels (MERR B, XERR(p), ZERR(p), DEP...

    2016