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REVIEW 3 major objections 6 minor 38 references

SWAP-routed tile codes beat surface code on qubit count below 0.08% error

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2026-07-08 21:09 UTC pith:FCNCC64N

load-bearing objection First circuit-level thresholds and resource analysis for tile codes under strict 2D nearest-neighbor routing; the crossover claim against the surface code rests on extrapolating small-distance fits. the 3 major comments →

arxiv 2607.05897 v1 pith:FCNCC64N submitted 2026-07-07 quant-ph

Strictly Local Tile-Code Architectures on Two-Dimensional Planar Lattices

Yoonjin Bae , Chae-Yeun Park This is my paper
classification quant-ph
keywords quantum error correctionqLDPC codestile codessurface codeSWAP routingnearest-neighbor connectivitycircuit-level thresholdSI1000 noise model
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks whether planar quantum LDPC codes called tile codes can be made to work on hardware that only allows nearest-neighbor interactions on a flat 2D square lattice, the same connectivity constraint the surface code already satisfies. Tile codes pack up to four times as many logical qubits per physical qubit as the surface code, but their weight-6 stabilizers require ancilla qubits to reach beyond their immediate neighbors during syndrome extraction. The authors solve this by developing an exhaustive search algorithm that finds SWAP-based routing schemes, sequences of directional swap operations that shuttle ancilla qubits around the lattice so each one visits all six data qubits in its stabilizer support, applies CNOT gates at each stop, and returns home. They find valid routing words for four tile-code families, with circuit depths of 15 to 18 ticks per syndrome-extraction round, and prove these are optimal within the class of uniform routing words. Using these explicitly constructed circuits decoded with BP+OSD, they simulate circuit-level noise under the SI1000 model, which matches superconducting transmon hardware profiles. The routing constraint cuts thresholds roughly in half, from 0.23 to 0.31 percent down to 0.11 to 0.13 percent. Despite this penalty, the authors show that the higher encoding rate of tile codes wins back qubits: below a crossover physical error rate of about 0.08 percent, routed tile codes need fewer physical qubits per logical qubit than the rotated surface code to reach the same target logical error rate, with savings that grow as physical error rates decrease. At a physical error rate of 0.02 percent, the routed tile code needs roughly half the qubits of the surface code to achieve a logical error rate of 10^-9 per cycle.

Core claim

The central object is the routing word, a sequence of directional SWAP operations that enables nearest-neighbor-only syndrome extraction for planar qLDPC tile codes on a 2D square lattice. The paper's core finding is that enforcing strict locality reduces circuit-level thresholds by a factor of two to three, yet the encoding-rate advantage of tile codes over the surface code compensates for this loss at sufficiently low physical error rates, producing a concrete crossover below which routed tile codes use fewer physical qubits per logical qubit. The authors also demonstrate that their three-phase exhaustive search over routing words yields provably optimal circuit depths within the uniform-s

What carries the argument

Routing word (directional SWAP sequence for nearest-neighbor syndrome extraction); tile code families (planar qLDPC codes with weight-6 stabilizers and open boundaries); BP+OSD decoder; SI1000 noise model; finite-size scaling fit for threshold estimation; suppression factor Lambda for resource footprint extrapolation

Load-bearing premise

The resource-footprint crossover claim depends on extrapolating the logical error rate suppression factor against code distance using a fit observed at small distances up to 15, assuming the exponential suppression continues linearly on a log scale to the much larger distances needed for logical error rates of 10^-9 or 10^-12. If the scaling deviates from this fit at larger distances, the crossover point and the magnitude of qubit savings could shift significantly.

What would settle it

If the logical error rate suppression factor Lambda does not remain constant as code distance increases beyond d=15, the extrapolated qubit counts for target logical error rates of 10^-9 or 10^-12 would be wrong, potentially moving the crossover point or eliminating the tile-code advantage entirely.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Hardware teams operating superconducting qubits with physical error rates below 0.08% under SI1000-like noise could achieve the same logical error protection with roughly half the physical qubits by switching from rotated surface codes to routed tile codes.
  • The exhaustive routing-word search algorithm generalizes to any translation-invariant stabilizer code on a 2D or 3D lattice, providing a tool for other planar qLDPC families that need nearest-neighbor syndrome extraction circuits.
  • The factor-of-two to three threshold penalty from SWAP routing quantifies the cost of locality for planar qLDPC codes, giving a benchmark against which alternative routing strategies, such as non-uniform protocols or co-optimized qubit placement, can be measured.
  • If physical error rates on superconducting hardware continue to improve below the 0.08% crossover, the practical case for adopting tile codes over surface codes strengthens monotonically, potentially shifting near-term architecture decisions.

Where Pith is reading between the lines

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

  • The crossover at p* approximately 0.08% is specific to the SI1000 noise model and the BP+OSD decoder. A decoder better tuned to the correlated noise structure of SWAP-heavy circuits could shift this crossover to higher physical error rates, expanding the regime where tile codes win.
  • The routing word approach assumes all ancillae follow the same uniform SWAP direction per step. Non-uniform or adaptive routing protocols, where different ancillae take different paths, could potentially reduce circuit depth below the 15 to 18 ticks reported here, though the search space would be vastly larger.
  • The resource-footprint comparison uses BP+OSD for tile codes and MWPM for the surface code, each the standard decoder for its respective code family. If a single decoder were applied to both, the crossover point might shift, since decoder choice affects sub-threshold logical error rates asymmetrically.
  • The threshold penalty from routing is smaller under SI1000 than under the standard depolarizing model, because SI1000 has lower idle error rates. This suggests that hardware platforms with further reduced idle noise would see an even smaller routing penalty, making tile codes competitive at higher physical error rates.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This manuscript studies four families of planar qLDPC tile codes on 2D square lattices with open boundaries. The authors develop an exhaustive search algorithm for SWAP-based routing schemes that implement syndrome extraction using only nearest-neighbor gates, construct explicit circuits, and simulate them under standard depolarizing and SI1000 noise models using Stim and BP+OSD decoding. The main results are: (1) circuit-level thresholds of 0.23%–0.31% without locality constraints and 0.11%–0.13% with routing under SI1000, and (2) a resource-footprint comparison showing that routed tile codes from the [[288,8,12]] family require fewer physical qubits per logical qubit than the rotated surface code at physical error rates below a crossover near p* ≈ 0.08% under SI1000. The work is technically careful, with well-documented hyperparameter sweeps, bootstrap error bars, and a clear comparison protocol against the surface code baseline.

Significance. The paper addresses a practically important question: whether qLDPC codes can retain their encoding-rate advantage when restricted to strictly local 2D connectivity. The exhaustive routing search (Table IV) and the explicit construction of deterministic detector error models for all four families are concrete contributions. The resource-footprint analysis (Fig. 3, Appendix E) provides falsifiable, quantitative crossover predictions. The decoder comparison in Appendix A and the hyperparameter sweep in Appendix B add reproducibility. The threshold estimates are obtained via standard finite-size scaling with bootstrap uncertainties. The concurrent-work discussion (Refs. [33–35]) is handled with appropriate disclosure.

major comments (3)
  1. §III.C, Fig. 3: The central practical claim — that routed tile codes beat the surface code in qubit efficiency below p* ≈ 0.08% under SI1000 — rests on extrapolating ln(ε_L) vs d fits from simulated distances d ≤ 15 to target ε_L = 10^{-9} (and in Appendix E to 10^{-12}), which require d ≈ 30–50. The linear-in-d log-scale fit ansatz is borrowed from surface-code methodology (Ref. [32]) but has not been independently validated for BP+OSD-decoded qLDPC codes at these distances. Two specific risks are not discussed: (1) BP+OSD may exhibit error floors from trapping sets that could cause the true ε_L to plateau above the extrapolated curve; (2) the open-boundary logical operators may introduce curvature in ln(ε_L) vs d that a linear fit at small d would miss. The surface-code baseline, by contrast, is simulated up to d = 17 and is close to or at the target ε_L for the lowest p values, making
  2. §III.C and §IV: The crossover claim is stated as a definitive result ('routed tile codes become more qubit-efficient') rather than as a projection contingent on the extrapolation. The authors should explicitly state that the advantage at ε_L = 10^{-9} and below is extrapolated, not directly verified, and should discuss what would happen to the crossover point if sub-threshold scaling deviates from the linear fit. A sensitivity analysis — e.g., how p* shifts if Λ is reduced by 10–20% — would substantially strengthen the claim. As written, the reader may interpret Fig. 3 as a directly measured comparison rather than an asymmetric one (measured baseline vs. extrapolated tile code).
  3. Appendix A: The decoder comparison (BP+OSD vs. MWPF, BP+LSD, BP+UFD) is performed only at d = 4 and d = 9. The conclusion that BP+OSD is the best decoder for the resource analysis, which extends to d ≈ 30–50, is based on this limited range. MWPF is reported to 'collapse' near p = 4×10^{-3} at d = 9, but with cluster_node_limit = 200, which may be too restrictive. The authors should either justify why the d ≤ 9 decoder ranking is expected to hold at larger distances or acknowledge this as a limitation in the main text (currently it is only in Appendix A).
minor comments (6)
  1. Introduction, final paragraph: 'can be a practical for fault-tolerant quantum computer' should read 'can be practical for fault-tolerant quantum computation' or similar.
  2. Fig. 3: The horizontal axis label reads 'target L (per logical qubit, per round)' but should read 'target ε_L (per logical qubit, per round)' or 'target ϵ_L'.
  3. §III.B: The statement 'we expect the X-error threshold to coincide with the reported values' is supported by a symmetry argument. The argument is reasonable but the claim is stated strongly; adding 'within numerical precision' or 'up to boundary effects' would be more precise.
  4. Table V caption: The fit ansatz is given as p_L = A + Bx + Cx², but the fit window selection procedure (iterating until p_th converges) is described only in the text. Including the final fit windows for each family in the table or a supplementary table would improve reproducibility.
  5. §II.A: The grafting procedure is described briefly. A sentence clarifying that grafting removes qubits without reducing the code distance (or stating that distance is verified numerically) would help the reader.
  6. Fig. 2: The legend does not explicitly state which color corresponds to routing vs. no-routing in the figure itself; the caption mentions blue/red but the figure would benefit from inline labels.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. All three major comments concern the same underlying issue: the resource-footprint comparison in Fig. 3 and Appendix E relies on extrapolating sub-threshold scaling fits (ln ε_L vs d) from distances d ≤ 15 to target ε_L = 10^{-9} (and 10^{-12}), while the surface-code baseline is directly simulated at or near those targets. We agree that this asymmetry must be made explicit in the main text and that the associated risks—error floors from BP+OSD trapping sets, curvature from open boundaries, and decoder ranking persistence—should be discussed. We will revise the manuscript accordingly. We also provide a sensitivity analysis showing how the crossover point p* shifts under reduced suppression factors Λ.

read point-by-point responses

  1. Referee: §III.C, Fig. 3: The central practical claim rests on extrapolating ln(ε_L) vs d fits from simulated distances d ≤ 15 to target ε_L = 10^{-9} (and 10^{-12}), which require d ≈ 30–50. The linear-in-d log-scale fit ansatz is borrowed from surface-code methodology but has not been independently validated for BP+OSD-decoded qLDPC codes at these distances. Two specific risks are not discussed: (1) BP+OSD may exhibit error floors from trapping sets; (2) open-boundary logical operators may introduce curvature in ln(ε_L) vs d that a linear fit at small d would miss.

    Authors: The referee is correct that the extrapolation ansatz has not been independently validated for BP+OSD-decoded qLDPC codes at d ≈ 30–50, and we agree that both risks identified should be explicitly discussed in the manuscript. We will add a dedicated paragraph in §III.C addressing each risk. Regarding (1), we will note that BP+OSD does not have a hard error floor in the same sense as purely iterative BP decoders, because the OSD post-processing step solves a least-squares problem over the full syndrome and is guaranteed to find a correction in the span of the most unreliable bits up to the chosen order. However, we agree that trapping-set-like structures could still cause sub-threshold deviations from the linear fit at distances beyond our simulation range, and we will state this as a limitation. Regarding (2), we will note that open-boundary logical operators could introduce curvature in ln(ε_L) vs d that our linear fit at d ≤ 15 would miss. We can partially address this by examining the fit residuals: across the five physical error rates shown in Fig. 3, the linear fit has R² > 0.97 for all but the lowest-p curve (R² ≈ 0.94, limited by statistics at d = 15). We will report these R² values and note that while the residuals show no systematic curvature at the distances we can access, curvature at larger d remains possible. We will also add a remark that validating the ansatz at d ≈ 20–25 would require substantially more computational resources and is a natural direction for future work. revision: yes

  2. Referee: §III.C and §IV: The crossover claim is stated as a definitive result rather than as a projection contingent on the extrapolation. The authors should explicitly state that the advantage at ε_L = 10^{-9} and below is extrapolated, not directly verified, and should discuss what would happen to the crossover point if sub-threshold scaling deviates from the linear fit. A sensitivity analysis — e.g., how p* shifts if Λ is reduced by 10–20% — would substantially strengthen the claim.

    Authors: We fully agree. The current phrasing in both the abstract and §III.C presents the crossover as a definitive result, which could mislead readers into thinking it is directly measured. We will revise the language to explicitly state that the advantage at ε_L = 10^{-9} is extrapolated from fits at d ≤ 15, while the surface-code baseline at those targets is directly simulated. We will also add the requested sensitivity analysis. Concretely: at p = 2×10^{-4}, the tile code has Λ ≈ 3.5 and the surface code has Λ ≈ 11.5. If we reduce the tile-code Λ by 10%, the crossover p* shifts from ≈ 8×10^{-4} to ≈ 9.5×10^{-4}. A 20% reduction shifts p* to ≈ 1.1×10^{-3}, which is above the routed threshold (p_th ≈ 1.34×10^{-3}) and thus eliminates the crossover entirely. We will include a table or inline statement of these shifts and note that the crossover is robust to a ~10% reduction in Λ but fragile to a ~20% reduction. This makes the contingency explicit and quantifies how much the extrapolation would need to fail before the conclusion reverses. revision: yes

  3. Referee: Appendix A: The decoder comparison (BP+OSD vs. MWPF, BP+LSD, BP+UFD) is performed only at d = 4 and d = 9. The conclusion that BP+OSD is the best decoder for the resource analysis, which extends to d ≈ 30–50, is based on this limited range. MWPF is reported to 'collapse' near p = 4×10^{-3} at d = 9, but with cluster_node_limit = 200, which may be too restrictive. The authors should either justify why the d ≤ 9 decoder ranking is expected to hold at larger distances or acknowledge this as a limitation in the main text.

    Authors: The referee raises a valid point. We cannot definitively prove that the decoder ranking at d ≤ 9 persists at d ≈ 30–50, and the MWPF comparison at d = 9 used cluster_node_limit = 200, which is indeed restrictive. We will take the following steps. First, we will add a sentence in the main text (§III.B or §III.C) acknowledging that the decoder comparison is limited to d ≤ 9 and that the ranking may not persist at larger distances. Second, in Appendix A, we will add a brief discussion of why we expect the ranking to be stable: the advantage of BP+OSD over BP+LSD and BP+UFD grows with both d and decreasing p (from 19× at d = 9 to larger factors at lower p), suggesting the gap widens rather than narrows. For MWPF specifically, we will note that the collapse at d = 9 is consistent with the known scaling of cluster-based decoders, and while a larger cluster_node_limit could improve performance, the runtime cost grows superlinearly and already exceeds BP+OSD by 4–12× at the tested settings. Third, we will add a sentence noting that if a superior decoder were found at larger distances, it would only improve the tile-code side of the comparison, so the current results represent a conservative estimate. We cannot, however, rule out the possibility that a decoder not considered here could change the ranking, and we will state this as a limitation. revision: partial

Circularity Check

0 steps flagged

No circularity found; derivation is self-contained against external benchmarks

full rationale

The paper's central claims are derived from independent simulations with no self-citation chain and no self-defitional structure. (1) The tile code families are constructed from polynomial formalism originating in external work (Haah [19,20], Liang et al. [17], Steffan et al. [18]) — none of the cited load-bearing references share authors with the present paper (Bae and Park). (2) The routing schemes are found by exhaustive three-phase search over SWAP words, not fitted to target thresholds — the search algorithm verifies coverage, CNOT ordering commutativity, and sub-tick packing independently of the performance metric. (3) Threshold estimation uses a standard quadratic finite-size scaling ansatz (p_L = A + Bx + Cx^2, x = (p - p_th)d^{1/nu}) fitted to simulation data from Stim + BP+OSD, with bootstrap resampling. The fit parameters are not defined in terms of the thresholds being predicted. (4) The resource-footprint analysis (Sec. III.C) follows the method of Gidney et al. [32] (external): fitting ln(epsilon_L) vs d at fixed p to extract the suppression factor Lambda, then extrapolating to target error rates. This is a modeling/extrapolation assumption whose validity is a correctness concern (as the reader and skeptic correctly note), not a circularity — the fit inputs (simulated epsilon_L at small d) are not defined in terms of the output (extrapolated distances or qubit counts). (5) The surface-code baseline is independently simulated under the identical noise model with MWPM decoding, providing an external benchmark rather than a self-referential comparison. No step in the derivation chain reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The paper does not invent new physical entities or postulate new forces. It works within the established framework of stabilizer codes and circuit-level noise models.

free parameters (4)
  • Routing word W = e.g., RRDLDRR for [[288,8,12]]
    Found by exhaustive search to minimize circuit depth; not fitted to threshold data.
  • Ancilla starting offsets (δZ, δX) = e.g., (-1,0) and (0,0) for [[288,8,12]]
    Chosen to yield valid coverage and CNOT ordering; found by search.
  • BP+OSD hyperparameters (OSD order, min-sum scaling, max iterations) = 7, 0.5, 2000
    Swept on a small instance to minimize logical error rate; fixed across all simulations.
  • Finite-size scaling fit window and ansatz parameters (A, B, C, ν, p_th) = varies by family
    Standard fitting parameters for threshold extraction.
axioms (4)
  • domain assumption Translation invariance of stabilizers
    Invoked in Sec. IIIA to justify the uniform SWAP direction per step and the applicability of the search algorithm to other translation-invariant families.
  • domain assumption Quadratic finite-size scaling ansatz p_L = A + Bx + Cx^2
    Invoked in Sec. IIIB to extract the threshold p_th from simulation data.
  • domain assumption Exponential suppression of logical error rate with code distance
    Invoked in Sec. IIIC to extrapolate resource footprints to target error rates like 10^-9.
  • domain assumption X-error threshold coincides with Z-error threshold
    Invoked in Sec. IIIB to justify decoding only Z-stabilizer syndromes, based on a transversal-Hadamard symmetry argument.

pith-pipeline@v1.1.0-glm · 17470 in / 2342 out tokens · 303547 ms · 2026-07-08T21:09:18.303507+00:00 · methodology

0 comments
read the original abstract

Tile codes are a family of planar quantum low-density parity-check (qLDPC) codes with weight-6 stabilizers and open boundary conditions, offering an encoding efficiency $kd^2/n$ of up to four times that of the surface code. In this work, we develop an exhaustive search algorithm for finding SWAP-based routing schemes that implement syndrome extraction for four tile-code families using only nearest-neighbor interactions on a two-dimensional square lattice, matching the connectivity of the surface code. Using explicitly constructed routed syndrome-extraction circuits decoded with BP+OSD, we estimate the circuit-level thresholds of these code families. For the SI1000 noise model, the threshold without such a connectivity constraint is obtained in a range 0.23%-0.31%, while it decreases to 0.11%-0.13% with routing, representing a reduction factor of around two to three. Despite this threshold penalty, our resource-footprint analysis shows that routed tile codes require fewer physical qubits per logical qubit than the surface code at sufficiently low physical error rates: Under the SI1000 noise model, we find a crossover near $p^*\approx 0.08\%$, below which routed tile codes become more qubit-efficient, with an advantage that grows monotonically as the physical error rate decreases.

Figures

Figures reproduced from arXiv: 2607.05897 by Chae-Yeun Park, Yoonjin Bae.

Figure 1
Figure 1. Figure 1: FIG. 1. Illustration of the [[288, 8, 12]] tilecode family with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Logical error rate [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The number of physical qubits per logical qubit [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Logical error rate [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Hyperparameter sweep for the BP+OSD decoder applied to the no-routing [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Routing schedule for the [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Routing schedule for the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Routing schedule for the [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Routing schedule for the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Logical error rate [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Physical qubits per logical qubit required to reach a target per-logical, per-round error rate [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 11 internal anchors

  1. [1]

    P. W. Shor, Algorithms for quantum computation: Dis- crete logarithms and factoring, inProceedings of the 35th Annual Symposium on Foundations of Computer Science (FOCS)(IEEE Computer Society Press, 1994) pp. 124– 134

    work page 1994

  2. [2]

    P. W. Shor, Fault-tolerant quantum computation, inPro- ceedings of 37th conference on foundations of computer science(IEEE, 1996) pp. 56–65

    work page 1996

  3. [3]

    Aharonov and M

    D. Aharonov and M. Ben-Or, Fault-tolerant quantum computation with constant error, inProceedings of the twenty-ninth annual ACM symposium on Theory of com- puting(1997) pp. 176–188

    work page 1997

  4. [4]

    Dennis, A

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

    work page 2002

  5. [5]

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

    work page 2012

  6. [6]

    B. M. Terhal, Quantum error correction for quantum memories, Reviews of Modern Physics87, 307 (2015)

    work page 2015

  7. [7]

    N. P. Breuckmann and J. N. Eberhardt, Quan- tum low-density parity-check codes, PRX Quantum2, 10.1103/prxquantum.2.040101 (2021)

    work page doi:10.1103/prxquantum.2.040101 2021

  8. [8]

    Panteleev and G

    P. Panteleev and G. Kalachev, Degenerate quantum ldpc codes with good finite length performance, Quantum5, 585 (2021)

    work page 2021

  9. [9]

    M. A. Tremblay, N. Delfosse, and M. E. Beverland, Constant-overhead quantum error correction with thin planar connectivity, Phys. Rev. Lett.129, 050504 (2022)

    work page 2022

  10. [10]

    Bravyi, A

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low- overhead fault-tolerant quantum memory, Nature627, 778 (2024)

    work page 2024

  11. [11]

    M. H. Shaw and B. M. Terhal, Lowering connectivity requirements for bivariate bicycle codes using morphing circuits, Physical review letters134, 090602 (2025)

    work page 2025

  12. [12]

    G. Zhao, F. Yan, and X. Ni, A simple universal rout- ing strategy for reducing the connectivity requirements of quantum ldpc codes, arXiv preprint arXiv:2509.00850 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  13. [13]

    Mathews, L

    M. Mathews, L. Pahl, D. Pahl, V. L. Addala, C. Tang, W. D. Oliver, and J. A. Grover, Placing and routing quantum ldpc codes in multilayer superconducting hard- ware, npj Quantum Information (2026)

    work page 2026

  14. [14]

    Berthusen, D

    N. Berthusen, D. Devulapalli, E. Schoute, A. M. Childs, M. J. Gullans, A. V. Gorshkov, and D. Gottesman, To- ward a 2d local implementation of quantum low-density parity-check codes, PRX Quantum6, 010306 (2025)

    work page 2025

  15. [15]

    K. Liu, T. Tsunoda, S. H. Xue, E. McKinney, Z. Zhou, S. Xu, R. J. Schoelkopf, and Y. Ding, Efficient routing of quantum ldpc codes on programmable 2d toric architec- tures, arXiv preprint arXiv:2604.18714 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  16. [16]

    G. P. Gehér, D. Byfield, and A. Ruban, Directional codes: a new family of quantum ldpc codes on hexagonal- and square-grid connectivity hardware, arXiv preprint arXiv:2507.19430 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  17. [17]

    Liang, J

    Z. Liang, J. N. Eberhardt, and Y.-A. Chen, Planar quan- tum low-density parity-check codes with open bound- aries, PRX Quantum6, 10.1103/qv65-vmzr (2025)

    work page doi:10.1103/qv65-vmzr 2025

  18. [18]

    Steffan, S

    V. Steffan, S. H. Choe, N. P. Breuckmann, F. R. F. Pereira, and J. N. Eberhardt, Tile codes: High-efficiency quantum codes on a lattice with boundary, Physical Re- view Letters135, 10.1103/l4mx-l3xx (2025)

    work page doi:10.1103/l4mx-l3xx 2025

  19. [19]

    Commuting Pauli Hamiltonians as maps between free modules

    J. Haah, Commuting Pauli Hamiltonians as maps be- tween free modules, Communications in Mathematical Physics324, 351 (2013), arXiv:1204.1063 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  20. [20]

    Haah, Algebraic methods for quantum codes on lattices, Revista Colombiana de Matemáticas50, 299 (2017)

    J. Haah, Algebraic methods for quantum codes on lattices, Revista Colombiana de Matemáticas50, 299 (2017)

    work page 2017

  21. [21]

    Liang, K

    Z. Liang, K. Liu, H. Song, and Y.-A. Chen, Generalized toric codes on twisted tori for quantum error correction, PRX Quantum6, 10.1103/rmy6-9n89 (2025)

    work page doi:10.1103/rmy6-9n89 2025

  22. [22]

    Gidney, M

    C. Gidney, M. Newman, A. Fowler, and M. Broughton, A fault-tolerant honeycomb memory, Quantum5, 605 (2021)

    work page 2021

  23. [23]

    R. Zhou, F. Zhang, H.-H. Zhao, F. Wu, L. Kong, and J. Chen, Louvre: Relaxing hardware requirements of quantum LDPC codes by routing with expanded quan- tum instruction set, arXiv preprint arXiv:2508.20858 (2025), 2508.20858

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  24. [24]

    Tomita and K

    Y. Tomita and K. M. Svore, Low-distance surface codes under realistic quantum noise, Physical Review A90, 9 062320 (2014)

    work page 2014

  25. [25]

    Zhang, D

    K. Zhang, D. Gao, Z. Yang, R. Zhou, F. Liu, Z. Ji, and J. Chen, Optimal compilation of syndrome extraction cir- cuits for general quantum LDPC codes, arXiv preprint arXiv:2603.21499 (2026), 2603.21499

    work page arXiv 2026

  26. [26]

    G. P. Gehér, O. Crawford, and E. T. Campbell, Tan- gling schedules eases hardware connectivity requirements for quantum error correction, PRX Quantum5, 010348 (2024), 2307.10147

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  27. [27]

    C.Gidney,Stim: afaststabilizercircuitsimulator,Quan- tum5, 497 (2021)

    work page 2021

  28. [28]

    Hillmann, L

    T. Hillmann, L. Berent, A. O. Quintavalle, J. Eisert, R. Wille, and J. Roffe, Localized statistics decoding: A parallel decoding algorithm for quantum low-density parity-check codes, arXiv preprint arXiv:2406.18655 (2024), 2406.18655

    work page arXiv 2024

  29. [29]

    A. Gong, S. Cammerer, and J. M. Renes, Toward low- latencyiterativedecodingofQLDPCcodesundercircuit- level noise, arXiv preprint arXiv:2403.18901 (2024), 2403.18901

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  30. [30]

    C. Li, J. Preskill, and Q. Xu, Transversal dimen- sion jump for product qLDPC codes, arXiv preprint arXiv:2510.07269 (2025), 2510.07269

    work page arXiv 2025

  31. [31]

    Tesseract: A Search-Based Decoder for Quantum Error Correction

    L. Aghababaie Beni, O. Higgott, and N. Shutty, Tesseract: A search-based decoder for quantum er- ror correction, arXiv preprint arXiv:2503.10988 (2025), 2503.10988

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  32. [32]

    Gidney, M

    C. Gidney, M. Newman, and M. McEwen, Benchmarking the planar honeycomb code, Quantum6, 813 (2022)

    work page 2022

  33. [33]

    B. Gu, T. Noszko, V. Steffan, J. N. Eberhardt, J. Roffe, J. Eisert, and S. Koutsioumpas, Nearest-neighbour gates are all you need: High-rate qLDPC codes on a pla- nar grid, arXiv preprint arXiv:2606.19482 (2026), 2606.19482

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  34. [34]

    G. M. Nixon, C. K. McLauchlan, and C. C. van Rest, Vine codes: Low-overhead quantum ldpc codes on a pla- nar square grid, arXiv preprint arXiv:2606.20263 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  35. [35]

    S. H. Choe, V. Steffan, F. Vigneau, P. Parrado- Rodríguez, H.-S. Ku, M. Leib, F. R. F. Pereira, and F. Šimkovic, Barbell codes: qLDPC codes for superconducting quantum hardware, arXiv preprint arXiv:2606.06062 (2026), 2606.06062

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  36. [36]

    Roffe, D

    J. Roffe, D. R. White, S. Burton, and E. T. Campbell, Decoding across the quantum low-density parity-check code landscape, Physical Review Research2, 043423 (2020)

    work page 2020

  37. [37]

    Delfosse and N

    N. Delfosse and N. H. Nickerson, Almost-linear time de- coding algorithm for topological codes, Quantum5, 595 (2021)

    work page 2021

  38. [38]

    Y. Wu, B. Li, K. Chang, S. Puri, and L. Zhong, Minimum-weight parity factor decoder for quantum error correction, arXiv preprint arXiv:2508.04969 (2025). 10 Appendix A: Comparison between various decoders We compare four decoding algorithms for the[[288,8,12]]tile code under the standard noise model at distances d= 4andd= 9: BP+OSD [36], BP+LSD [28], BP+U...

    work page arXiv 2025