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REVIEW 3 major objections 4 minor 67 references

A simple binomial fault model estimates logical error rates across quantum codes and hardware and finds when distributing error correction pays off.

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2026-07-11 21:48 UTC pith:PNTDUUYZ

load-bearing objection A clean, usable binomial-tail screen for modern QEC codes and distributed sweet spots; absolute numbers rest on an unvalidated malignancy fraction, but the qualitative tool is solid and worth engaging. the 3 major comments →

arxiv 2607.04082 v1 pith:PNTDUUYZ submitted 2026-07-05 quant-ph cs.ET

A Cross-Platform Analysis of High-Performance Quantum Error Correction Codes

Bryan Pan , Yufeng Xin This is my paper
classification quant-ph cs.ET
keywords quantum error correctionlogical error ratebinomial fault modeldistributed quantum computingsurface codeqLDPC codesbiased noisecircuit volume
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.

Physical qubits remain too few and too noisy for large-scale quantum computers, so error-correcting codes must be chosen carefully for each hardware platform and for multi-chip systems. This paper claims that the two dominant drivers of logical failure are the code’s fault threshold and the effective number of two-qubit operations per correction cycle, and that a binomial-tail probability on those quantities already gives useful leading-order estimates. The same model is extended to split local versus interconnect operations and to biased noise, allowing rapid comparison of topological, subsystem, concatenated, qLDPC and Floquet codes on trapped-ion, superconducting and neutral-atom machines. Qualitative orderings seen in expensive full-stack simulations reappear analytically, and the framework locates concrete “sweet-spot” regions—numbers of chips and code distances—where distribution improves logical error despite noisier links. Designers therefore gain a fast way to decide which codes are practical and when multi-chip architectures become advantageous without running full simulations for every candidate.

Core claim

Logical error rate is bounded, to leading order, by the probability that the total number of independent physical faults in one correction cycle exceeds the code’s correctable threshold κ = t + 1. Once circuit volume (native two-qubit gates plus routing, measurement and interconnect overhead) and per-location error rates are known, this single binomial (or multi-class Poisson-binomial) tail reproduces the main performance trends of advanced QEC codes across platforms and identifies the QPU-count and distance at which distributed surface-code or bivariate-bicycle constructions become worthwhile.

What carries the argument

The binomial-tail bound p_L ≤ P[X ≥ κ], where X is the sum of independent fault indicators (or separate binomials for intra- and inter-QPU classes) and κ is the minimum number of faults that can produce a logical error. Circuit volume N_loc and physical error rates enter only through the parameters of X; everything else is abstracted away.

Load-bearing premise

Every set of at least κ independent physical faults is treated as a logical failure, so the model ignores decoder success, error degeneracy, ancilla-only faults and correlated noise.

What would settle it

Run full-stack syndrome-extraction simulations (or real-device experiments) for the same code instances and physical error rates used in the paper’s tables; if the measured logical error rates reverse the predicted ordering of codes or show no sweet-spot improvement near the claimed q ≈ 9, d = 29 surface-code point, the leading-order claim fails.

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

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 / 4 minor

Summary. The paper introduces a lightweight binomial-tail framework for estimating logical error rates of advanced QEC codes (topological, subsystem, concatenated, qLDPC, Floquet) across trapped-ion, superconducting and neutral-atom platforms as well as distributed multi-QPU systems. Logical failure is bounded by p_L ≤ P[X ≥ κ] where X is the number of independent physical faults per cycle and κ = t + 1; N_loc is taken primarily from two-qubit gate counts (with extensions for routing, inter-QPU links and biased noise). Code parameters are drawn from the ECCentric study; the model is used to rank codes, reproduce qualitative simulation trends, and locate distributed “sweet spots” (e.g., q ≈ 9, d = 29 for a surface-code-like scaling at interconnect ratio r = 10).

Significance. If the leading-order picture is reliable, the framework supplies a transparent, computationally cheap screening tool that can identify which code–hardware combinations are even plausible before expensive full-stack simulation, and that can map the design region in which distributing a code across multiple QPUs improves rather than degrades logical performance. The explicit separation of intra- versus inter-QPU faults, the inclusion of bias-tailored distances, and the concrete numerical sweet-spot tables are useful contributions for architecture-level exploration. The authors correctly flag the model’s dual optimism/pessimism and do not claim quantitative decoder-level accuracy.

major comments (3)
  1. [III.A, Eq. (1)–(2); IV.B, Tables IV–V] Section III.A Eq. (1) and the distributed extension Eq. (2) treat every configuration of ≥κ independent faults as a logical failure. The authors themselves note (after Table III) that many such sets are non-malignant (ancilla-only, correctly decoded, degenerate or non-logical). All numerical results—Table III, Fig. 2, Tables IV–VIII and the reported sweet spots (q = 9, d = 29, p_L ≈ 6.94 × 10^{-10})—are generated from the raw binomial tail with no malignancy fraction or decoder model. Because the absolute p_L values and the precise (q, d) coordinates of the optima rest on the untested assumption that this fraction is roughly constant across N_inter and code family, a sensitivity study (or an explicit statement that only qualitative ordering is claimed) is required for the central distributed-QEC claim to be load-bearing.
  2. [IV.B.1–2, Tables V, VII–VIII] The distributed surface-code and BB scaling studies introduce free parameters α = 4 (boundary overhead), λ = 0.25 (BB embedding factor), and the prefactors N_2Q(d) ≈ 11 d^{2} / 18 d^{2}. Tables V, VII and VIII show that the location of the sweet spot moves dramatically with r and λ. Without a systematic sweep or justification that these particular values are representative rather than illustrative, the concrete “sweet-spot design region” advertised in the abstract and Section V cannot be regarded as robust.
  3. [IV.A, after Table III] The only quantitative comparison with ECCentric is an average p_L at a single physical error rate p = 0.004 (text after Table III). Per-code ordering, distance dependence and the effect of routing overhead are not checked. Because the paper’s claim is that the analytic model “reproduces qualitative trends observed in larger-scale simulations,” a more granular side-by-side comparison (or an explicit list of which trends are reproduced) is needed to substantiate that assertion.
minor comments (4)
  1. [Table II] Table II lists “Gauge Color (2D) ≈ 11”; the approximate distance should be stated precisely or the source of the approximation clarified.
  2. [Fig. 1] Figure 1 caption and axis labels do not indicate whether the curves are the exact binomial tail or a Poisson approximation; a one-line clarification would help reproducibility.
  3. [II.B, III] The notation N_loc is used both for the ideal two-qubit count and for the inflated count that includes SWAPs/measurements; a consistent subscript convention (e.g., N_loc^(eff)) would reduce ambiguity in Sections II–III.
  4. [IV, Table II] Reference [7] (ECCentric) is cited as the source of all circuit volumes; a short appendix table reproducing the exact N_2Q and n_phys numbers extracted from that work would make the present manuscript self-contained.

Circularity Check

0 steps flagged

No circularity: binomial-tail estimates and distributed sweet spots are direct numerical consequences of independently sourced code parameters, hardware rates, and stated scalings, not fits or self-definitional reductions.

full rationale

The central construction (Section III.A Eq. (1), distributed extension Eq. (2), biased-noise Eq. (4)) is the elementary binomial (or Poisson-binomial) tail p_L ≤ P[X ≥ κ] with X the count of independent faults. Code instances (Table II: N_2Q, d, κ) are taken verbatim from the external ECCentric simulation study [7]; physical rates p_2Q are taken from published hardware numbers (Apollo, Willow, Flamingo, Infleqtion). Distributed scalings (n_phys(d)≈2d^{2}, N_2Q(d)≈11d^{2}, N_inter=αd(q-1) with α=4; analogous BB forms with free embedding factor λ) are stated as modeling assumptions, not fitted to any target p_L. The resulting Table III values, Fig. 2/3 curves, and sweet-spot coordinates (e.g., q=9, d=29 at r=10) are therefore ordinary numerical evaluations of those inputs; they do not re-state a fitted quantity as a prediction, nor do they rest on a self-citation uniqueness claim or an ansatz smuggled from the authors’ prior work. Self-citations [1],[2] appear only for background on DQC and are not load-bearing for the error-rate formulas. The acknowledged pessimism of the raw tail (non-malignant faults) is a modeling limitation, not a circular reduction. Hence the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central numerical claims rest on a short list of modeling choices and externally supplied numbers rather than on new physical postulates. The free parameters are the hand-chosen scaling coefficients and noise ratios used in the distributed sweeps; the axioms are the standard independent-fault and threshold-failure assumptions of the binomial model; no new entities are invented.

free parameters (4)
  • α (inter-QPU boundary overhead) = 4
    Set by hand to α=4 in the surface-code distributed model; controls how fast N_inter grows with q and d and therefore moves the sweet-spot location.
  • λ (BB embedding factor) = 0.25 (default)
    Fraction of two-qubit gates that become inter-QPU under partitioning; swept over {0.05,0.10,0.25,0.50,1.00} and strongly affects whether distribution helps BB codes.
  • r = p_inter / p_intra = 10 (main case)
    Interconnect noise ratio; representative values 1,3,10,30 are chosen by hand to map the design space.
  • N_2Q(d) scaling prefactors = 11 and 18
    Surface N_2Q≈11d² and BB N_2Q≈18d² are normalized to the single benchmark instances of Table II rather than derived from first principles.
axioms (4)
  • domain assumption Faults at distinct circuit locations are independent Bernoulli trials, so the total fault count is binomial (or Poisson-binomial).
    Stated in Section III.A and used for every p_L bound; standard but known to be violated by crosstalk and leakage.
  • ad hoc to paper Logical failure occurs whenever the number of faults reaches or exceeds κ = t+1, independent of decoder, syndrome structure or degeneracy.
    Eq. (1) and the distributed extensions; the paper itself notes this is a pessimistic upper bound.
  • domain assumption Two-qubit gate count (plus simple routing/interconnect additives) is a sufficient leading-order proxy for total fault locations.
    Section III opening paragraphs; measurement, idle and SPAM errors are declared negligible.
  • domain assumption Code parameters (d, N_2Q) taken from the ECCentric simulation study are representative of the code families.
    Table II and surrounding text; the paper does not re-derive or re-compile the circuits.

pith-pipeline@v1.1.0-grok45 · 24170 in / 3028 out tokens · 30704 ms · 2026-07-11T21:48:31.831876+00:00 · methodology

0 comments
read the original abstract

The theory of quantum error correction was established decades ago. Yet the limitation of the quantum computing platforms in terms of noise level and available physical qubit count persists, which greatly hinders the development of scalable quantum computing systems. In this paper, we present analytical estimates of logical error rates of advanced QEC codes across leading hardware platforms and distributed quantum computing systems using a simple but unified framework. The analysis captures two dominant contributors to logical error: code structure and two-qubit gate overhead. The framework provides a fast estimate of logical error rates and identification of dominating factors in different hardware platforms, such as circuit volume, routing overhead, inter-QPU operations, or asymmetric noise protection. We show that several qualitative trends observed in larger-scale simulations can be reproduced and interpreted analytically within this framework. We further demonstrate that the framework can be used to find the sweet spot design region of distributed QEC, which is critical for the design of distributed quantum computing systems.

Figures

Figures reproduced from arXiv: 2607.04082 by Bryan Pan, Yufeng Xin.

Figure 1
Figure 1. Figure 1: Logical error estimates per round vs. two-qubit error rate for different QEC codes optimize code distance under a fixed physical-qubit budget, rather it compares fixed benchmark instances. By contrast, the distributed-QPU sweeps in Section IV-B include a simplified capacity constraint, so any improvement from distribution should be interpreted as capacity-enabled distance growth rather than an intrinsic ad… view at source ↗
Figure 2
Figure 2. Figure 2: Distributed-QPU logical error estimate as a function of QPU count. Each QPU assumed to support 200 physical qubits, nphys ≈ 2d 2 , N2Q ≈ 11d 2 , and Ninter = 4d(q − 1). where Xintra ∼ Binomial(Nintra, pintra) Xinter ∼ Binomial(Ninter, pinter) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the resulting logical error estimates as a function of QPU count for several interconnect noise ratios using pintra = 10−4 and λ = 0.25. When r = 1, inter-QPU operations are as reliable as local gates, so increasing the number of QPUs improves performance by allowing a larger￾distance BB code. When r = 3, there is still a distribution benefit, but the sweet spot occurs at a smaller QPU count. When r … view at source ↗
Figure 4
Figure 4. Figure 4: Sensitivity of the distributed BB analytical model to the nonlocal embedding factor λ for fixed pintra = 10−4 and r = 10. contrast, BB codes require either highly reliable interconnects or a highly modular layout to benefit from distribution. To test the importance of the embedding assumption, Fig￾ure 4 fixes r = 10. and varies λ. Smaller values of λ correspond to better partitioning, fewer cross-QPU check… view at source ↗
Figure 5
Figure 5. Figure 5: Logical error estimates per round pL vs bias η for a fixed two-qubit error rate of ploc = 10−4 . and bias-tailored topological codes (XZZX). This set lets us isolate how noise bias favors codes with enhanced Z distance relative to standard symmetric ones. It is clear that within this model, codes tailored to more Z errors by having a higher Z distance vastly outperform traditional codes at high levels of η… view at source ↗

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Works this paper leans on

67 extracted references · 3 canonical work pages

  1. [1]

    Topological codesencode logical information into a lattice and have both high error thresholds and a natural fit for hardware platforms with local connectivity, such as superconducting circuits. Topological codes typically require a large number of stabilizer measurements per cycle, resulting in high circuit volume but strong error suppression in sufficie...

  2. [2]

    This reduces measurement complexity and can significantly decrease circuit depth

    Subsystem Stabilizer Codesextend stabilizer codes by introducing additional degrees of freedom (gauge qubits), allowing stabilizer measurements to be decomposed into lower weight operators. This reduces measurement complexity and can significantly decrease circuit depth. This construction works well on architectures with flexible connectivity, such as tra...

  3. [3]

    The concatenated Steane code is a canonical example, constructed by recursively encod- ing the[[7,1,3]]Steane code

    Concatenated Codesencode logical qubits recursively by nesting one code within another. The concatenated Steane code is a canonical example, constructed by recursively encod- ing the[[7,1,3]]Steane code. While concatenation obviously improves error suppression, it leads to exponential growth in qubit count and reduced coding rate. Concatenated codes trade...

  4. [4]

    Analogous to classical LDPC codes, it enables efficient de- coding algorithms

    qLDPC Codes(Quantum low-density parity-check codes) are a class of stabilizer codes in which each stabi- lizer generator acts on only a small number of qubits, and each qubit participates in only a small number of stabilizers. Analogous to classical LDPC codes, it enables efficient de- coding algorithms. Unlike topological codes, qLDPC codes are not const...

  5. [5]

    Floquet Codesdo not have fixed stabilizer measure- ments. Instead, they vary periodically in time, measuring sequences of low-weight Pauli operators according to a pre- defined schedule, from which the effective stabilizer struc- 2 Concatenated qLDPC Topological Subsystem Stabilizer Floquet Concatenated Steane Bivariate Bicycle Surface Color XZZXSurface B...

  6. [6]

    As a result, multi-qubit circuits can typically be executed without the need for SW AP- based routing

    Trapped-Ionsystems are characterized by near all-to-all connectivity within a single QPU [24]. As a result, multi-qubit circuits can typically be executed without the need for SW AP- based routing. This high connectivity allows logical circuits to be implemented with minimal overhead, so the physical two-qubit gate count closely matches the logical requir...

  7. [7]

    To model this, we can distinguish between the logical two-qubit gate count required by a QEC code and the physical two-qubit gate count after routing

    SuperconductingQPUs face the connectivity constraint that two-qubit gates can only be performed between adja- cent qubits, and non-local interactions must be implemented through sequences of SW AP operations, so superconducting qubits incur more connectivity overhead [25]. To model this, we can distinguish between the logical two-qubit gate count required...

  8. [8]

    Neutral-AtomQPUs can support flexible connectiv- ity through qubit rearrangement and long-range Rydberg- mediated interactions [26], [27]. However, this flexibility often requires additional atom movement, scheduling constraints, or extra entangling operations, all of which can increase the total number of physical fault locations relative to the logical ...

  9. [9]

    It assumes that each fault location fails independently with probabilityp loc.When a fault occurs, it results in a Pauli error drawn uniformly from{X, Y, Z}

    Symmetric Depolarizing Noiseis a widely used model due to its simplicity. It assumes that each fault location fails independently with probabilityp loc.When a fault occurs, it results in a Pauli error drawn uniformly from{X, Y, Z}. Thus, bit flip and phase flip components occur symmetrically, although they are not strictly independent because aYerror cont...

  10. [10]

    For simplicity, we neglect explicitY- type error correlations

    Biased Noise.In particular, many qubit platforms typ- ically exhibit strongly biased noise where phase errors (Z errors) occur far more frequently than bit-flip errors (X errors) [30], [15]. For simplicity, we neglect explicitY- type error correlations. This approximation is sufficient for comparing how different effectiveXandZdistances respond to increas...

  11. [11]

    Real hardware, however, typically has non- uniform error rates across qubits, couplers, and measurement channels

    Heterogeneous Physical Error Rates.The previous models assume that every fault location has the same er- ror probability. Real hardware, however, typically has non- uniform error rates across qubits, couplers, and measurement channels. To model this, we let each physical locationifail independently with probabilityp i.The total number of faults is then a ...

    1920

  12. [12]

    Distributed Surface-Code Evaluation:We next use the analytical model for an illustrative scaling study to see whether distributed QEC can improve logical error rates when each individual QPU has a limited qubit capacity. We assume each QPU can host approximately200physical qubits and use a surface-code-like scaling modeln phys(d)≈2d 2,[8], [39] so that th...

  13. [13]

    The purpose of this model is to compare a geometrically local topological code with a nonlocal qLDPC code under the same distributed binomial framework

    Distributed Bivariate-Bicycle Code Evaluation:We now construct an analogous distributed-QPU analytical model for the Bivariate Bicycle code. The purpose of this model is to compare a geometrically local topological code with a nonlocal qLDPC code under the same distributed binomial framework. In the surface-code analytical model, the pri- mary source of i...

  14. [14]

    Qtest: A simulation testbed for distributed quantum computing and networking with deep reinforcement learning,

    H. Chen, B. Du, and R. Li, “Qtest: A simulation testbed for distributed quantum computing and networking with deep reinforcement learning,” IEEE Internet Computing, vol. 30, no. 1, pp. 26–37, 2026

    2026

  15. [15]

    Towards a high-performance quantum data center network architecture,

    Y . Xin and L. Zhang, “Towards a high-performance quantum data center network architecture,” inICC 2025 - IEEE International Conference on Communications, 2025, pp. 2400–2405

    2025

  16. [16]

    A single quantum cannot be cloned,

    W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature, vol. 299, no. 5886, pp. 802–803, 1982. [Online]. Available: https://doi.org/10.1038/299802a0

    doi:10.1038/299802a0 1982

  17. [17]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010

    2010

  18. [18]

    Stabilizer codes and quantum error correction,

    D. Gottesman, “Stabilizer codes and quantum error correction,” 1997. [Online]. Available: https://arxiv.org/abs/quant-ph/9705052

    Pith/arXiv arXiv 1997

  19. [19]

    Quantum error correction: an introductory guide,

    J. Roffe, “Quantum error correction: an introductory guide,” Contemporary Physics, vol. 60, no. 3, p. 226–245, Jul. 2019. [Online]. Available: http://dx.doi.org/10.1080/00107514.2019.1667078

    doi:10.1080/00107514.2019.1667078 2019

  20. [20]

    Eccentric: An empirical analysis of quantum error correction codes,

    A. ´Swierkowska, J. Pflieger, E. Giortamis, and P. Bhatotia, “Eccentric: An empirical analysis of quantum error correction codes,” 2025. [Online]. Available: https://arxiv.org/abs/2511.01062

    arXiv 2025

  21. [21]

    Surface codes: Towards practical large-scale quantum computation,

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: Towards practical large-scale quantum computation,” Physical Review A, vol. 86, no. 3, Sep. 2012. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.86.032324

    doi:10.1103/physreva.86.032324 2012

  22. [22]

    Topological and subsystem codes on low-degree graphs with flag qubits,

    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. X, vol. 10, p. 011022, Jan 2020. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevX.10.011022

    doi:10.1103/physrevx.10.011022 2020

  23. [23]

    High-threshold and low-overhead fault-tolerant quantum memory,

    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,”Nature, vol. 627, no. 8005, p. 778–782, Mar. 2024. [Online]. Available: http://dx.doi.org/10.1038/s41586-024-07107-7

    doi:10.1038/s41586-024-07107-7 2024

  24. [24]

    Constructions and performance of hyperbolic and semi-hyperbolic floquet codes,

    O. Higgott and N. P. Breuckmann, “Constructions and performance of hyperbolic and semi-hyperbolic floquet codes,” PRX Quantum, vol. 5, no. 4, Nov. 2024. [Online]. Available: http://dx.doi.org/10.1103/PRXQuantum.5.040327

    doi:10.1103/prxquantum.5.040327 2024

  25. [25]

    Fault-tolerant quantum computation by anyons,

    A. Kitaev, “Fault-tolerant quantum computation by anyons,”Annals of Physics, vol. 303, no. 1, p. 2–30, Jan. 2003. [Online]. Available: http://dx.doi.org/10.1016/S0003-4916(02)00018-0

    doi:10.1016/s0003-4916(02)00018-0 2003

  26. [26]

    Topological quantum memory,

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory,”Journal of Mathematical Physics, vol. 43, no. 9, p. 4452–4505, Sept. 2002. [Online]. Available: http://dx.doi.org/10.1063/1.1499754

    doi:10.1063/1.1499754 2002

  27. [27]

    Topological quantum distillation,

    H. Bombin and M. A. Martin-Delgado, “Topological quantum distillation,”Physical Review Letters, vol. 97, no. 18, Oct. 2006. [Online]. Available: http://dx.doi.org/10.1103/PhysRevLett.97.180501

    doi:10.1103/physrevlett.97.180501 2006

  28. [28]

    The xzzx surface code,

    J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, “The xzzx surface code,”Nature Communications, vol. 12, no. 1, Apr. 2021. [Online]. Available: http://dx.doi.org/10.1038/s41467- 021-22274-1

    doi:10.1038/s41467- 2021

  29. [29]

    Operator quantum error-correcting subsystems for self-correcting quantum memories,

    D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories,”Physical Review A, vol. 73, no. 1, Jan. 2006. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.73.012340

    doi:10.1103/physreva.73.012340 2006

  30. [30]

    Topological subsystem codes,

    H. Bombin, “Topological subsystem codes,”Physical Review A, vol. 81, no. 3, Mar. 2010. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.81.032301

    doi:10.1103/physreva.81.032301 2010

  31. [31]

    Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes,

    H. Bomb ´ın, “Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes,”New Journal of Physics, vol. 17, no. 8, p. 083002, Aug. 2015

    2015

  32. [32]

    Multiple-particle interference and quantum error correction,

    A. M. Steane, “Multiple-particle interference and quantum error correction,”Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 452, pp. 2551 – 2577, 1996. [Online]. Available: https://api.semanticscholar.org/CorpusID:8246615

    1996

  33. [33]

    Quantum ldpc codes with positive rate and minimum distance proportional to the square root of the blocklength,

    J.-P. Tillich and G. Zemor, “Quantum ldpc codes with positive rate and minimum distance proportional to the square root of the blocklength,” IEEE Trans. Inf. Theor ., vol. 60, no. 2, p. 1193–1202, Feb. 2014. [Online]. Available: https://doi.org/10.1109/TIT.2013.2292061

    doi:10.1109/tit.2013.2292061 2014

  34. [34]

    Tour de gross: A modular quantum computer based on bivariate bicycle codes,

    T. J. Yoder, E. Schoute, P. Rall, E. Pritchett, J. M. Gambetta, A. W. Cross, M. Carroll, and M. E. Beverland, “Tour de gross: A modular quantum computer based on bivariate bicycle codes,” 2025. [Online]. Available: https://arxiv.org/abs/2506.03094

    Pith/arXiv arXiv 2025

  35. [35]

    Distributed quantum error correction based on hyperbolic floquet codes,

    E. Sutcliffe, B. Jonnadula, C. L. Gall, A. E. Moylett, and C. M. Westoby, “Distributed quantum error correction based on hyperbolic floquet codes,” 2025. [Online]. Available: https://arxiv.org/abs/2501.14029

    arXiv 2025

  36. [36]

    Fault-tolerant hyperbolic floquet quantum error correcting codes,

    A. Fahimniya, H. Dehghani, K. Bharti, S. Mathew, A. J. Koll ´ar, A. V . Gorshkov, and M. J. Gullans, “Fault-tolerant hyperbolic floquet quantum error correcting codes,” 2024. [Online]. Available: https://arxiv.org/abs/2309.10033

    Pith/arXiv arXiv 2024

  37. [37]

    Trapped-ion quantum computing: Progress and challenges,

    C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, “Trapped-ion quantum computing: Progress and challenges,”Applied Physics Reviews, vol. 6, no. 2, May 2019. [Online]. Available: http://dx.doi.org/10.1063/1.5088164

    doi:10.1063/1.5088164 2019

  38. [38]

    Superconducting qubits: Current state of play,

    M. Kjaergaard, M. E. Schwartz, J. Braum ¨uller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, “Superconducting qubits: Current state of play,”Annual Review of Condensed Matter Physics, vol. 11, no. 1, p. 369–395, Mar. 2020. [Online]. Available: http://dx.doi.org/10.1146/annurev-conmatphys-031119-050605

    doi:10.1146/annurev-conmatphys-031119-050605 2020

  39. [39]

    Quantum computing with neutral atoms,

    L. Henriet, L. Beguin, A. Signoles, T. Lahaye, A. Browaeys, G.-O. Reymond, and C. Jurczak, “Quantum computing with neutral atoms,”Quantum, vol. 4, p. 327, Sept. 2020. [Online]. Available: http://dx.doi.org/10.22331/q-2020-09-21-327

    doi:10.22331/q-2020-09-21-327 2020

  40. [40]

    Logical quantum processor based on reconfigurable atom arrays,

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V . Vuleti ´c, and M. D. Lukin, “Logical quantum processor based on reconfigurable atom arrays,” Nature, vol...

    doi:10.1038/s41586-023-06927-3 2023

  41. [41]

    High-fidelity universal gate set for 9Be+ ion qubits,

    J. P. Gaebler, T. R. Tan, Y . Lin, Y . Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett., vol. 117, p. 060505, Aug. 2016. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevLett.117.060505

    work page doi:10.1103/physrevlett.117.060505 2016

  42. [42]

    Atomique: A quantum compiler for reconfigurable neutral atom arrays,

    H. Wang, P. Liu, D. B. Tan, Y . Liu, J. Gu, D. Z. Pan, J. Cong, U. A. Acar, and S. Han, “Atomique: A quantum compiler for reconfigurable neutral atom arrays,” inProceedings of the 51st Annual International Symposium on Computer Architecture, ser. ISCA ’24. IEEE Press, 2025, p. 293–309. [Online]. Available: https://doi.org/10.1109/ISCA59077.2024.00030

    doi:10.1109/isca59077.2024.00030 2025

  43. [43]

    Tailoring surface codes for highly biased noise,

    D. K. Tuckett, A. S. Darmawan, C. T. Chubb, S. Bravyi, S. D. Bartlett, and S. T. Flammia, “Tailoring surface codes for highly biased noise,” Physical Review X, vol. 9, no. 4, Nov. 2019. [Online]. Available: http://dx.doi.org/10.1103/PhysRevX.9.041031

    doi:10.1103/physrevx.9.041031 2019

  44. [44]

    Ultrahigh error threshold for surface codes with biased noise,

    D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, “Ultrahigh error threshold for surface codes with biased noise,”Physical Review Letters, vol. 120, no. 5, Jan. 2018. [Online]. Available: http://dx.doi.org/10.1103/PhysRevLett.120.050505

    doi:10.1103/physrevlett.120.050505 2018

  45. [45]

    Fault-tolerant computing with biased-noise superconducting qubits: a case study,

    P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,”New Journal of Physics, vol. 11, no. 1, p. 013061, Jan. 2009. [Online]. Available: http://dx.doi.org/10.1088/1367- 2630/11/1/013061

    doi:10.1088/1367- 2009

  46. [46]

    Improving trapped-ion-qubit memories via code-mediated error-channel balancing,

    Y . Seis, B. J. Brown, A. S. Sørensen, and J. F. Goodwin, “Improving trapped-ion-qubit memories via code-mediated error-channel balancing,” Physical Review A, vol. 107, no. 5, May. 2023. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.107.052417

    work page doi:10.1103/physreva.107.052417 2023

  47. [47]

    Leveraging biased noise for more efficient quantum error correction at the circuit-level with two-level qubits,

    J. E. Martinez, P. Schnabl, J. O. del Moral, R. Dastbasteh, P. M. Crespo, and R. M. Otxoa, “Leveraging biased noise for more efficient quantum error correction at the circuit-level with two-level qubits,”

  48. [48]

    Available: https://arxiv.org/abs/2505.17718

    [Online]. Available: https://arxiv.org/abs/2505.17718

    Pith/arXiv arXiv

  49. [49]

    Fault-tolerant quantum computation with constant error rate,

    D. Aharonov and M. Ben-Or, “Fault-tolerant quantum computation with constant error rate,”SIAM Journal on Computing, vol. 38, no. 4, pp. 1207–1282, 2008. [Online]. Available: https://doi.org/10.1137/S0097539799359385

    doi:10.1137/s0097539799359385 2008

  50. [50]

    Quantum accuracy threshold for concatenated distance-3 codes,

    P. Aliferis, D. Gottesman, and J. Preskill, “Quantum accuracy threshold for concatenated distance-3 codes,” 2005. [Online]. Available: https://arxiv.org/abs/quant-ph/0504218

    Pith/arXiv arXiv 2005

  51. [51]

    Noise tailoring for scalable quantum computation via randomized compiling,

    J. J. Wallman and J. Emerson, “Noise tailoring for scalable quantum computation via randomized compiling,”Physical Review A, vol. 94, no. 5, Nov. 2016. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.94.052325

    doi:10.1103/physreva.94.052325 2016

  52. [52]

    Freely scalable quantum technologies using cells of 5-to- 50 qubits with very lossy and noisy photonic links,

    N. H. Nickerson, J. F. Fitzsimons, and S. C. Benjamin, “Freely scalable quantum technologies using cells of 5-to- 50 qubits with very lossy and noisy photonic links,”Phys. 12 Rev. X, vol. 4, p. 041041, Dec 2014. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevX.4.041041

    doi:10.1103/physrevx.4.041041 2014

  53. [53]

    Fault- tolerant connection of error-corrected qubits with noisy links,

    J. Ramette, J. Sinclair, N. P. Breuckmann, and V . Vuleti ´c, “Fault- tolerant connection of error-corrected qubits with noisy links,”npj Quantum Information, vol. 10, p. 58, 2024. [Online]. Available: https://www.nature.com/articles/s41534-024-00855-4

    2024

  54. [54]

    On computing the distribution function for the poisson binomial distribution,

    Y . Hong, “On computing the distribution function for the poisson binomial distribution,”Computational Statistics and Data Analysis, vol. 59, pp. 41–51, 2013. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0167947312003568

    2013

  55. [55]

    Toward a 2d local implementation of quantum low-density parity-check codes,

    N. Berthusen, D. Devulapalli, E. Schoute, A. M. Childs, M. J. Gullans, A. V . Gorshkov, and D. Gottesman, “Toward a 2d local implementation of quantum low-density parity-check codes,” PRX Quantum, vol. 6, no. 1, Jan. 2025. [Online]. Available: http://dx.doi.org/10.1103/PRXQuantum.6.010306

    doi:10.1103/prxquantum.6.010306 2025

  56. [56]

    Distributed quantum error correction with bivariate bicycle codes in a modular architecture,

    N. K. Chandra, E. Kaur, R. Nejabati, and K. P. Seshadreesan, “Distributed quantum error correction with bivariate bicycle codes in a modular architecture,” 2026. [Online]. Available: https://arxiv.org/abs/2605.04663

    Pith/arXiv arXiv 2026

  57. [57]

    Distributed fault-tolerant quantum memories over a 2xl array of qubit modules,

    E. Tham, M. Ye, I. Khait, J. Gamble, and N. Delfosse, “Distributed fault-tolerant quantum memories over a 2xl array of qubit modules,”

  58. [58]

    Available: https://arxiv.org/abs/2508.01879

    [Online]. Available: https://arxiv.org/abs/2508.01879

    Pith/arXiv arXiv

  59. [59]

    Lowering connectivity requirements for bivariate bicycle codes using morphing circuits,

    M. H. Shaw and B. M. Terhal, “Lowering connectivity requirements for bivariate bicycle codes using morphing circuits,”Physical Review Letters, vol. 134, no. 9, Mar. 2025. [Online]. Available: http://dx.doi.org/10.1103/PhysRevLett.134.090602

    doi:10.1103/physrevlett.134.090602 2025

  60. [60]

    Fault-tolerant quantum computation against biased noise,

    P. Aliferis and J. Preskill, “Fault-tolerant quantum computation against biased noise,”Physical Review A, vol. 78, no. 5, Nov. 2008. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.78.052331

    doi:10.1103/physreva.78.052331 2008

  61. [61]

    Removing leakage-induced correlated errors in superconducting quantum error correction,

    M. McEwen, D. Kafri, Z. Chen, J. Atalaya, K. J. Satzinger, C. Quintana, P. V . Klimov, D. Sank, C. Gidney, A. G. Fowler, F. Arute, K. Arya, B. Buckley, B. Burkett, N. Bushnell, B. Chiaro, R. Collins, S. Demura, A. Dunsworth, C. Erickson, B. Foxen, M. Giustina, T. Huang, S. Hong, E. Jeffrey, S. Kim, K. Kechedzhi, F. Kostritsa, P. Laptev, A. Megrant, X. Mi,...

    doi:10.1038/s41467-021-21982-y 2021

  62. [62]

    Leakage mitigation for quantum error correction using a mixed qubit scheme,

    N. C. Brown and K. R. Brown, “Leakage mitigation for quantum error correction using a mixed qubit scheme,”Physical Review A, vol. 100, no. 3, Sept. 2019. [Online]. Available: http://dx.doi.org/10.1103/PhysRevA.100.032325

    work page doi:10.1103/physreva.100.032325 2019

  63. [63]

    Tensor networks and quantum error correction,

    A. J. Ferris and D. Poulin, “Tensor networks and quantum error correction,”Physical Review Letters, vol. 113, no. 3, July 2014. [Online]. Available: http://dx.doi.org/10.1103/PhysRevLett.113.030501

    doi:10.1103/physrevlett.113.030501 2014

  64. [64]

    General tensor network decoding of 2d pauli codes,

    C. T. Chubb, “General tensor network decoding of 2d pauli codes,”

  65. [65]

    Available: https://arxiv.org/abs/2101.04125

    [Online]. Available: https://arxiv.org/abs/2101.04125

    Pith/arXiv arXiv

  66. [66]

    Quantum error correction below the surface code threshold,

    Google, “Quantum error correction below the surface code threshold,” Nature, vol. 638, no. 8052, pp. 920–926, Feb. 2025. [Online]. Available: https://doi.org/10.1038/s41586-024-08449-y

    doi:10.1038/s41586-024-08449-y 2025

  67. [67]

    Learning high-accuracy error decoding for quantum processors,

    J. Bausch, A. W. Senior, F. J. H. Heras, T. Edlich, A. Davies, M. Newman, C. Jones, K. Satzinger, M. Y . Niu, S. Blackwell, G. Holland, D. Kafri, J. Atalaya, C. Gidney, D. Hassabis, S. Boixo, H. Neven, and P. Kohli, “Learning high-accuracy error decoding for quantum processors,”Nature, vol. 635, no. 8040, p. 834–840, Nov. 2024. [Online]. Available: http:/...

    doi:10.1038/s41586-024-08148-8 2024