Hardware-in-the-Loop Syndrome-to-Decoder Validation for Repetition, Surface, CSS-LDPC, and Digitized-GKP Codes

Chinonso Onah; Dennis Delali Kwesi Wayo; Leonardo Goliatt; Rodrigo Alves Dias; Sven Groppe

REVIEW 3 major objections 4 minor 39 references

Hardware-in-the-Loop Syndrome-to-Decoder Validation for Repetition, Surface, CSS-LDPC, and Digitized-GKP Codes

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Hardware syndrome records can be parsed and audited against the intended check matrix, with repetition and CSS-LDPC runs preserving the expected corrections and the surface run falling back to target-containing localization.

desk verdict A useful, honest syndrome-to-decoder validation pipeline with clean small-code evidence; the surface branch's bit-order mapping is unverified and the GKP parameters are undisclosed, but both are fixable. read the letter →

arxiv 2607.19447 v1 pith:OEMLPFBY submitted 2026-07-21 quant-ph

Dennis Delali Kwesi Wayo , Chinonso Onah , Rodrigo Alves Dias , Leonardo Goliatt , Sven Groppe This is my paper · ORCID

classification quant-ph

keywords syndromeextractionquantumerrorcorrectiondecoderinterfacehardware-in-the-loopsurfacecoderepetitionCSS-LDPCGKPreadout

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that the boundary between quantum hardware readout and a classical decoder—the place where bit-order, check-indexing, and transpilation errors hide—can be made auditable and executable. It builds three hardware syndrome-extraction circuits (a five-qubit repetition code, a 40-data-qubit distance-five surface-code Z-check layer, and the Steane CSS code as a compact CSS-LDPC benchmark) plus an off-hardware digitized Gottesman-Kitaev-Preskill (GKP) proxy, and replays 4,096-shot syndrome streams through three decoder policies. For repetition and CSS-LDPC, the dominant measured syndrome equals the predicted column of the parity-check matrix for every injected error, and hardware localization stays near 0.82–0.87. The routed 56-qubit surface branch produces broad check activation, so exact localization drops to 0.003–0.108 while target-containing localization remains 0.279–0.642. The paper's conclusion is an auditable syndrome-to-decoder interface, not a threshold claim.

What carries the argument

The carrying mechanism is a replay pipeline that expands per-shot hardware counts into syndrome records, constructs a decoder request from each record plus code metadata, and scores corrections with three policies (minimum-weight perfect matching as the baseline, union-find, and hard-decision belief propagation). The load-bearing identity is the linear check relation s = H e (mod 2): each injected target predicts one column of H, and exact localization is counted when the decoded correction equals that target. The pipeline audits bit order, check order, and decoder dispatch simultaneously.

What would settle it

Run one low-weight surface-code injection with the same transpilation and retrieval, and check whether the most frequent measured syndrome matches the predicted column of the 16-by-40 Z-check matrix. A mismatch with the predicted column—beyond the observed noise background—would show the interface mapping is wrong; the paper reports no dominant-syndrome match for the surface branch, so this remains untested.

Watch

Extended reading notes

Core claim

For repetition and CSS-LDPC circuits executed on a live superconducting processor, the most common syndrome after injecting a known single-qubit X error is exactly the corresponding column of the intended parity-check matrix, and the minimum-weight decoder returns the injected site as the dominant correction. For the distance-five surface code, the same parser and decoder replay preserve metadata alignment, but the measured syndromes are so broadly activated that exact single-target localization is only 0.003–0.108, which the paper attributes to hardware-induced syndrome activation rather than to a check-mapping error. The digitized-GKP branch shows that analog quadrature readouts, after bei

Load-bearing premise

The surface-code branch's interface claim depends on the assumption that the 56 measured classical bits from the transpiled circuit map exactly into the parser's assumed order of 40 data qubits and 16 syndrome-measurement ancillas, and a mapping error would look exactly like the broad noise the paper reports.

Editorial extensions

If this is right

Future quantum error-correction experiments can use the same request boundary to compare decoders on identical recorded syndrome streams.
Repeated rounds and calibration-aware weights can be added to the same interface without changing the audit structure.
GKP-style bosonic readouts can enter the same binary decoding pipeline after explicit digitization.
The surface-code branch provides a concrete scale at which routing and measurement noise dominate one-round exact localization.
The methodology shifts reported QEC results from threshold claims toward verifiable interface correctness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The surface-code branch does not by itself confirm the intended check-matrix mapping: with exact localization at 0.003–0.108, a bit-order or check-index error would be indistinguishable from broad hardware noise.
A dedicated mapping test—running a single low-weight surface injection and comparing the dominant measured syndrome to the predicted 16-bit column—would separate mapping failure from noise; the current data cannot.
The recorded surface streams are a ready benchmark for weighted or calibration-aware decoders, which could raise target-containing localization beyond the unweighted minimum-weight baseline.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Hi — quick take on arXiv:2607.19447. Worth your time if you work on the syndrome-to-decoder boundary in QEC experiments. It contributes a reproducible LiDMaS+ pipeline and runs four code families through it: five-qubit repetition, Steane CSS-LDPC Z-checks, distance-five surface Z-checks, and a digitized-GKP proxy. The clean result is that repetition and Steane branches preserve the expected dominant syndrome and correction on ibm_fez hardware, with Wilson intervals. That's credible evidence the parser, bit ordering, and check indexing work on small circuits. The 56-qubit surface branch is honestly reported as noise-dominated, with exact localization 0.003–0.108.

What's new is the combined four-branch study behind a single interface, with code and data shipped. That's a useful infrastructure contribution for auditing syndrome streams before they reach a decoder, and the authors don't oversell it: no threshold claims.

Soft spots: the surface branch's mapping from 56 transpiled classical bits to the 16 Z-check order is load-bearing and never independently verified. With localization that low, a systematic bit-order perm would look identical to the broad activation they attribute to hardware noise. The matched local simulator shares the same parser, so it cannot rule out a shared misordering. The paper calls the run a scaling test, but the scaling test only means something if the mapping is known right. They need a sanity check that's sensitive to ordering — for example, injecting a fault that produces a multi-bit syndrome pattern with a unique signature, or comparing the clean hardware syndrome distribution against a permutation-mismatched reference. Also the local surface baseline itself isn't explained: how does 'local' get 0.72 exact localization? Ideal circuit? Noise model? That should be stated.

Table IV's zero-residual audit is circular: Algorithm 1 closes any residual syndrome before recording, so nonzero residuals are zero by construction. They do disclose the closure step, but the table is easy to over-read as decoder validation rather than interface-closure validation.

The digitized-GKP branch is off-hardware and under-specified. The text gives decision-window and injected shift sizes, but not the noise parameters (shift noise sigma, half-cell jump rate, flip probability). The appendix has make commands but the actual parameter values are in code. As reported, those localization numbers aren't reproducible. Fixable.

Overall: a careful, honest paper that delivers a reproducible toolchain and solid small-code evidence. The surface mapping and GKP parameter disclosure need to be addressed; then I'd be comfortable citing it as an interface-validation reference. It deserves a serious referee — conditional accept, not reject. I'd bring it to the reading group.

Referee Report

3 major / 4 minor

Summary. The paper reports a four-branch syndrome-to-decoder validation study: a five-qubit repetition code, a distance-five rotated-surface-code Z-check layer, the Z-check half of the Steane CSS code as a compact CSS-LDPC benchmark, and a PennyLane-backed digitized-GKP companion study. For each branch, the authors build clean and injected circuits, execute 4096-shot streams on IBM hardware (or sample PennyLane Gaussian-CV records for GKP), parse measured bits into LiDMaS+ requests, replay MWPM, UF, and BP policies, and report correction-localization rates with Wilson confidence intervals. The two smaller hardware families preserve the expected dominant syndromes; the 56-qubit surface run shows broad hardware-induced activation with exact localization 0.003--0.108; the GKP branch demonstrates that analog readouts can be binned into the same binary interface. The paper explicitly positions itself as an auditable interface methodology rather than a threshold demonstration.

Significance. If the central interface claim is sound, the paper offers a useful and reproducible benchmark: the code is released, the pipeline steps are documented with make targets, shot counts are fixed, Wilson intervals are reported, and the surface-code failure mode is described honestly rather than dressed up as a threshold result. The repetition and CSS-LDPC branches are clean positive evidence that the parser/check-matrix mapping can be validated on small circuits. However, the surface branch's claimed semantic alignment rests on an unverified bit-order/check-index mapping, and the 'replay audit' in Table IV is circular because Algorithm 1 closes residual syndromes before recording them. These two issues are load-bearing for the paper's full claim, so the manuscript needs revision before the 56-qubit-scale interface claim can be accepted.

major comments (3)

[Algorithm 1 / Table IV] The pipeline explicitly includes a syndrome-closing step ('close any residual syndrome diagnosed by H_F e_d != s_r') before recording policy diagnostics. Table IV's zero residual count for every MWPM/UF/BP row is therefore guaranteed by construction and cannot be used as evidence that the decoder policies correctly close the measured syndromes. This weakens the 'replay audit' language in Section III.E and Section IV. Please report pre-closure residuals, or state plainly that residual zero is a post-closure invariant rather than an independent audit.
[Section III.B, Fig. 7] For the 56-qubit surface branch, the mapping from the transpiled circuit's 56 measured classical bits to the parser's logical order (40 data + 16 Z-check ancillas) is not independently verified. The local simulator uses the same parser, so a systematic bit-order or check-index permutation would make both the hardware and local heatmaps look column-like, and in the hardware data the broad activation produced by such a permutation would be indistinguishable from the reported 'hardware-induced syndrome activation'. With exact localization only 0.003--0.108, Fig. 7 contains no unambiguous column structure to certify the mapping. The sentence 'consistent with hardware-induced syndrome activation ... rather than a check-indexing error' is an assertion, not a test. Please provide an explicit check of the backend measurement ordering against H_surf,Z, for example per-target per-Z-check expected-
[Eq. (4) / Sections III.B and III.C] The minimum-weight objective in Eq. (4) can have multiple minimizers for the degenerate surface-code syndrome, but the paper does not specify a tie-breaking rule for the MWPM/minimum-weight baseline. Exact localization ('e_hat = {i}') is therefore not uniquely defined without the code's tie-breaking convention; for the surface branch the reported 0.003--0.108 range could depend strongly on that choice. Please state the tie-breaking rule and, ideally, report the rate at which the intended singleton is one of the minimum-weight corrections, not merely whether it is the selected representative.

minor comments (4)

[Section II.C] The GKP digitization uses an ad hoc decision window |y_j| <= 0.25 sqrt(pi), a q-shift of 0.56 sqrt(pi), and a Gaussian proxy rather than finite-energy non-Gaussian grid states. The paper is appropriately candid that this is a proxy, but it should explicitly state these are model choices and report sensitivity to the window half-width and noise parameters.
[Title/Abstract] The title 'Hardware-in-the-Loop ... and Digitized-GKP Codes' may overstate the GKP branch, which is a PennyLane model study rather than a hardware-in-the-loop experiment. Consider adding a qualifier such as 'companion digitized-GKP study' in the title or abstract for accuracy.
[Appendix A] The make-based reproducibility protocol is welcome. For a stable benchmark, include a commit hash or versioned release tag, and record the IBM Runtime/backend snapshot date and calibration information so the hardware results can be interpreted and reproduced by others.
[Section IV / Fig. 12] The study-level comparison across code families in Fig. 12 aggregates cases with very different degeneracy and syndrome structure. The comparison is useful as an interface check, but the caption should caution that aggregate localization is not a decoder-quality comparison across code families.

Circularity Check

1 steps flagged · score 2.0 of 10

One tautological replay-audit metric; central interface claims remain empirically grounded.

self definitional [Algorithm 1 (Sec. II.B) and Table IV / Sec. III.E / Sec. IV]
"compute a correction ê_d and close any residual syndrome diagnosed by H_F ê_d ≠ s_r (mod 2) ... The residual column counts rows whose correction fails to close the measured syndrome after the policy-specific closure step."

Table IV is presented as a 'decoder replay audit' showing MWPM, UF, and BP close every measured syndrome, but the residual is defined after a mandatory closure step. Algorithm 1 explicitly instructs each policy to 'close any residual syndrome diagnosed by H_F ê_d ≠ s_r', so a zero residual count is guaranteed by construction. Therefore Table IV's zero-residual row does not independently validate decoder correctness or interface semantics. This is a minor tautology, not a load-bearing part of the central localization/activation results, which are computed before closure and are grounded in hardware and sampled syndrome records.

full rationale

The paper's central contribution is a reproducible syndrome-to-decoder interface methodology, not a threshold or code-performance claim. The repetition and CSS-LDPC branches report dominant measured syndromes matching the intended check-matrix columns on live hardware; this is an empirical check that could have failed under a parser or bit-order error, and it does not reduce to the paper's definitions. The surface branch honestly reports that exact localization is very low (0.003–0.108) and frames the result as a scaling test rather than a validation of precise correction; the concern that the 56-qubit bit-ordering is unverified is a correctness/risk issue, not a circularity, since the paper does not derive the mapping from the conclusion. The digitized-GKP branch is explicitly scoped as an off-hardware model study and does not claim to predict hardware GKP performance. No load-bearing self-citation or imported uniqueness theorem was found; the reference list consists of external QEC literature, and the LiDMaS+ tool is disclosed with code rather than cited as an authority. The only concrete circular element is the replay-audit residual metric, which is tautological because Algorithm 1 closes all residual syndromes before counting them. That metric is auxiliary and does not affect localization rates or the qualitative interface claim. A score of 2 reflects this single minor definitional circularity while recognizing that the main evidence is independently grounded in measured syndrome statistics.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its quantitative GKP results depend on several hand-set constants (digitization window, injected shift magnitude, unreported noise parameters), and the surface branch's reported ranges depend on a selected target subset. The main background assumptions are standard stabilizer linear algebra and trust in the IBM backend; the GKP binning threshold is ad hoc to this paper.

free parameters (5)

GKP decision window half-width = 0.25√π ≈ 0.443
Used to bin wrapped quadrature coordinate y_j into a binary Z-check syndrome (Eq. 2 and Sec. II.C). No derivation or reference justifies the value; changing it directly changes all GKP localization rates.
Injected GKP q-shift magnitude = 0.56√π
Injected on each GKP target before sampling. The chosen magnitude affects how strongly the target syndrome appears relative to noise; no justification is given.
GKP noise rounds and noise parameters = 3 rounds; widths unspecified
Three rounds of Gaussian shift noise, finite-squeezed q-readout noise, and measurement flips are sampled, but the distribution widths and flip probabilities are not reported, making the exact GKP streams unreproducible.
Surface representative target set Tsurf = {1,5,10,14,17,22,32,37}
Eight of forty possible single-X injections are selected. Localization ranges over this subset may not represent the full qubit population; the choice is stated but its effect on reported ranges is not analyzed.
Local simulator surface localization mechanism = exact localization ≈ 0.72
The local reference for the surface code reports exact localization near 0.72 without a stated noise model. Since a noiseless reference should give 1.0 for unique syndrome columns, this implies an unstated error model, tie-breaking rule, or syndrome degeneracy that readers cannot audit.

assumptions (5)

standard math Syndrome is computed as s = H e mod 2 for the stated parity-check matrices (Eqs. 1 and 3).
Standard linear-algebra model of stabilizer syndromes, used throughout the decoder replay.
domain assumption IBM Runtime Sampler counts faithfully reflect the executed transpiled circuit on ibm_fez.
All hardware analysis trusts the backend retrieval; no independent calibration is provided to verify that counts correspond to the intended logical circuit.
standard math The Steane H_Z matrix of Eq. (3) is the correct Z-check matrix of the Steane code.
Textbook CSS code; used as the CSS-LDPC parity-check matrix for the injected-X targets.
ad hoc to paper Digitizing wrapped quadrature y_j with window |y_j| ≤ 0.25√π is a valid proxy for finite-squeezed GKP readout.
The specific threshold is a free choice with no derivation or reference; it defines the binary syndrome in the GKP branch.
domain assumption GKP modes are interpreted through the outer surface-code Z-check incidence matrix H_surf,Z for 40 modes.
The analog check coordinate is normalized by sqrt(|S_j|) and then binned; this mapping from oscillator modes to outer stabilizer checks is a design choice of the companion study.

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Cite this review

Pith. "Pith review of Hardware-in-the-Loop Syndrome-to-Decoder Validation for Repetition, Surface, CSS-LDPC, and Digitized-GKP Codes." pith.science (2026). https://pith.science/paper/OEMLPFBY

@misc{pith2026260719447,
  author       = {Pith},
  title        = {Pith review of: Hardware-in-the-Loop Syndrome-to-Decoder Validation for Repetition, Surface, CSS-LDPC, and Digitized-GKP Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OEMLPFBY}},
  note         = {Machine review of arXiv:2607.19447}
}

abstract

Quantum error-correction experiments increasingly require a verified interface between measured syndrome bits and decoder-native correction requests. We report a four-branch syndrome-to-decoder study spanning three IBM gate-model hardware circuits and one PennyLane-backed digitized-GKP model. The hardware branches implement a five-data-qubit repetition code, a distance-five rotated-surface-code Z-check extraction layer, and the Z-check half of the Steane CSS code as a compact CSS-LDPC benchmark. The GKP branch samples finite-squeezed Gaussian-CV q-readout and injected q-shifts, then bins wrapped quadrature coordinates into the same outer surface-code Z-check interface. All cases use 4096 shots per stream, clean and injected streams, LiDMaS+ request construction, and MWPM/minimum-weight correction as the plotted baseline, with union-find and hard-decision belief-propagation/min-sum policies replayed for interface validation. The correction-volume panels additionally report mean minimum-weight correction weight for each decoded stream. Repetition and CSS-LDPC hardware preserve the dominant expected syndrome and correction for every injected target. The routed 56-qubit surface circuit exhibits broad hardware-induced syndrome activation: exact localization drops to $0.003$--$0.108$, but target-containing localization remains $0.279$--$0.642$. The digitized-GKP study gives exact q-shift localization of $0.350$--$0.495$ and target-containing localization of $0.417$--$0.608$. The results support an auditable syndrome-to-decoder interface rather than a threshold claim.

Figures

Figures reproduced from arXiv: 2607.19447 by the authors.

**Figure 3.** IBM platform rendering of a transpiled repetition [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** IBM platform renderings of representative transpiled distance-five surface-code Z-check circuits. Each circuit uses 40 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (8 more)

**Figure 5.** Figure 5: IBM platform rendering of a transpiled CSS [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: Repetition-code hardware-in-the-loop results on [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: Distance-five surface-code Z-check hardware-in-the [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

**Figure 9.** Figure 9: CSS-LDPC/Steane Z-check hardware-in-the-loop [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]

**Figure 10.** Figure 10: Correction-inclusion confusion maps for the two [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]

**Figure 12.** Figure 12: Aggregate decoder-policy replay comparison across injected streams. MWPM, UF, and BP close every measured [PITH_FULL_IMAGE:figures/full_fig_p008_12.png]

**Figure 13.** Figure 13: Digitized-GKP wrapped-check-coordinate distributions before binary thresholding. Dashed vertical lines mark [PITH_FULL_IMAGE:figures/full_fig_p008_13.png]

**Figure 14.** Figure 14: Empirical surface-code hardware-noise diagnostic from the observed syndrome records. The top panel compares [PITH_FULL_IMAGE:figures/full_fig_p009_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 1 linked inside Pith

[1]

The local reference and hardware run both use 4096 shots per stream

Repetition Code The repetition-code reproduction uses one clean circuit and all five single- X injected circuits. The local reference and hardware run both use 4096 shots per stream. Af- ter the hardware job completes, the returned counts are expanded into syndrome requests and decoded with the three policies listed in Table II. To reproduce this case, ru...
[2]

The local and hardware runs use the same circuit la- bels and shot count

CSS-LDPC/Steane Code The CSS-LDPC reproduction uses the Steane HZ ma- trix, one clean circuit, and all seven single- X injected cir- cuits. The local and hardware runs use the same circuit la- bels and shot count. The post-processing path is the same as for the repetition code: count expansion, request con- struction, MWPM/UF/BP replay, and summary-statis...
[3]

The local reference is generated for the same clean and injected streams as the hardware run

Surface Code The surface-code reproduction uses distance five and the representative target set Tsurf defined in the Methods section. The local reference is generated for the same clean and injected streams as the hardware run. The hardware counts are then converted to the 16-bit Z-check order before decoder replay and localization analysis. To reproduce ...
[4]

Digitized-GKP Study The digitized-GKP study has no IBM Runtime submis- sion step. It uses the same distance-five outer Z-check support graph as the surface-code experiment, but obtains syndrome bits by digitizing PennyLane Gaussian-CV q- readout samples and injected GKP q-shift samples. The resulting requests are decoded and analyzed with the same MWPM/UF...
[5]

A. Y. Kitaev, Russian Mathematical Surveys52, 1191 (1997)

1997
[6]

A. G. Fowler, A. M. Stephens, and P. Groszkowski, Phys- ical Review A80, 052312 (2009)

2009
[7]

A. G. Fowler, Physical Review Letters109, 180502 (2012)

2012
[8]

D. S. Wang, A. G. Fowler, and L. C. L. Hollenberg, Phys- ical Review A83, 020302 (2011)

2011

Show all 39 references

[9]

D. K. Tuckett, S. D. Bartlett, and S. T. Flammia, Physical Review Letters120, 050505 (2018)

2018
[10]

D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, Physical Review Letters124, 130501 (2020)

2020
[11]

J. P. Bonilla Ataides, D. K. Tuckett, S. D. Bartlett, S. T. Flammia, and B. J. Brown, Nature Communications12, 2172 (2021)

2021
[12]

J. Lee, J. Park, and J. Heo, Quantum Information Pro- cessing20, 231 (2021)

2021
[13]

A. Dua, A. Kubica, L. Jiang, S. T. Flammia, and M. J. Gullans, PRX Quantum5, 010347 (2024)

2024
[14]

A. M. Steane, Physical Review Letters77, 793 (1996)

1996
[15]

A. R. Calderbank and P. W. Shor, Physical Review A54, 1098 (1996)

1996
[16]

Tillich and G

J.-P. Tillich and G. Z´ emor, IEEE Transactions on Infor- mation Theory60, 1193 (2014)

2014
[17]

Bravyi, A

S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, Nature627, 778 (2024)

2024
[18]

Gottesman, A

D. Gottesman, A. Kitaev, and J. Preskill, Physical Review A64, 012310 (2001)

2001
[19]

Fukui, A

K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, Physical Review X8, 021054 (2018)

2018
[20]

Tzitrin, T

I. Tzitrin, T. Matsuura, R. N. Alexander, G. Dauphinais, J. E. Bourassa, K. K. Sabapathy, N. C. Menicucci, and I. Dhand, PRX Quantum2, 040353 (2021)

2021
[21]

L. S. Madsen, F. Laudenbach, M. F. Askarani,et al., Nature606, 75 (2022)

2022
[22]

M. V. Larsen, J. E. Bourassa, S. Kocsis,et al., Nature 642, 587 (2025)

2025
[23]

Krinner, N

S. Krinner, N. Lacroix, A. Remm, and et al., Nature605, 669 (2022)

2022
[24]

Google Quantum AI, Nature614, 676 (2023)

2023
[25]

Google Quantum AI and Collaborators, Nature638, 920 (2025)

2025
[26]

Y. Zhao, Y. Liu, C. Zhang, and et al., Physical Review 12 Letters129, 030501 (2022)

2022
[27]

Takeda, A

K. Takeda, A. Noiri, T. Nakajima, T. Kobayashi, and S. Tarucha, Nature608, 682 (2022)

2022
[28]

Z. Ni, S. Li, X. Deng, and et al., Nature616, 56 (2023)

2023
[29]

Aghaee Rad, T

H. Aghaee Rad, T. Ainsworth, R. N. Alexander,et al., Nature638, 912 (2025)

2025
[30]

Higgott and C

O. Higgott and C. Gidney, Quantum5, 551 (2021)

2021
[31]

Delfosse and N

N. Delfosse and N. H. Nickerson, Quantum5, 595 (2021), originally circulated as arXiv:1709.06218 (2017)

2021 arXiv
[32]

Duclos-Cianci and D

G. Duclos-Cianci and D. Poulin, Physical Review Letters 104, 050504 (2010)

2010
[33]

Higgott, T

O. Higgott, T. C. Bohdanowicz, A. Kubica, S. T. Flammia, and E. T. Campbell, Physical Review X13, 031007 (2023)

2023
[34]

Skoric, D

L. Skoric, D. E. Browne, K. M. Barnes, N. I. Gillespie, and E. T. Campbell, Nature Communications14, 7040 (2023)

2023
[35]

X. Tan, F. Zhang, R. Chao, Y. Shi, and J. Chen, PRX Quantum4, 040344 (2023)

2023
[36]

Q. Xu, N. Mannucci, A. Seif, A. Kubica, S. T. Flammia, and L. Jiang, Physical Review Research5, 013035 (2023)

2023
[37]

A. S. Darmawan, B. J. Brown, A. L. Grimsmo, D. K. Tuckett, and S. Puri, PRX Quantum2, 030345 (2021)

2021
[38]

R´ eglade, A

U. R´ eglade, A. Bocquet, R. Gautier, and et al., Nature 629, 778 (2024)

2024
[39]

A. Z. Ding, B. L. Brock, A. Eickbusch, and et al., Nature Communications16, 5279 (2025)

2025

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

The local reference and hardware run both use 4096 shots per stream

Repetition Code The repetition-code reproduction uses one clean circuit and all five single- X injected circuits. The local reference and hardware run both use 4096 shots per stream. Af- ter the hardware job completes, the returned counts are expanded into syndrome requests and decoded with the three policies listed in Table II. To reproduce this case, ru...

[2] [2]

The local and hardware runs use the same circuit la- bels and shot count

CSS-LDPC/Steane Code The CSS-LDPC reproduction uses the Steane HZ ma- trix, one clean circuit, and all seven single- X injected cir- cuits. The local and hardware runs use the same circuit la- bels and shot count. The post-processing path is the same as for the repetition code: count expansion, request con- struction, MWPM/UF/BP replay, and summary-statis...

[3] [3]

The local reference is generated for the same clean and injected streams as the hardware run

Surface Code The surface-code reproduction uses distance five and the representative target set Tsurf defined in the Methods section. The local reference is generated for the same clean and injected streams as the hardware run. The hardware counts are then converted to the 16-bit Z-check order before decoder replay and localization analysis. To reproduce ...

[4] [4]

Digitized-GKP Study The digitized-GKP study has no IBM Runtime submis- sion step. It uses the same distance-five outer Z-check support graph as the surface-code experiment, but obtains syndrome bits by digitizing PennyLane Gaussian-CV q- readout samples and injected GKP q-shift samples. The resulting requests are decoded and analyzed with the same MWPM/UF...

[5] [5]

A. Y. Kitaev, Russian Mathematical Surveys52, 1191 (1997)

1997

[6] [6]

A. G. Fowler, A. M. Stephens, and P. Groszkowski, Phys- ical Review A80, 052312 (2009)

2009

[7] [7]

A. G. Fowler, Physical Review Letters109, 180502 (2012)

2012

[8] [8]

D. S. Wang, A. G. Fowler, and L. C. L. Hollenberg, Phys- ical Review A83, 020302 (2011)

2011