Pith. sign in

REVIEW 2 major objections 5 minor 80 references

Achieving computational gains with quantum error-correction primitives: Generation of long-range entanglement enhanced by error detection

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Applying quantum error-correction primitives to unencoded qubits yields net gains on current superconducting processors: over 85% fidelity for CNOT gates across up to 40 sites, and genuine multipartite entanglement in a 75-qubit GHZ state.

desk verdict The long-range CNOT protocol is a real contribution and the comparison is credible; the 75-qubit GME record claim is not supported because MQC fidelity above 0.5 does not certify genuine multipartite entanglement when phase errors are present, as the authors' own 80-qubit control shows. read the letter →

arxiv 2411.14638 v2 pith:GJXFEGNN submitted 2024-11-22 quant-ph

Haoran Liao , Gavin S. Hartnett , Ashish Kakkar , Adrian Tan , Michael Hush , Pranav S. Mundada , Michael J. Biercuk , Yuval Baum This is my paper · ORCID
classification quant-ph
keywords quantumerrorcorrectionprimitivesdetectionwithoutencodingunitaryentangle-disentangleprotocollong-rangeCNOTgateGHZstatepreparationgenuinemultipartiteentanglementmultiple-quantumcoherencefidelitysuperconductingprocessors
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

This paper argues that the ingredients of quantum error correction can deliver net computational gains on today's small superconducting processors when used as error-detection primitives on unencoded qubits, with only a modest overhead. It introduces a unitary entangle-disentangle protocol for long-range CNOT gates: a linear-chain GHZ state is created, reduced to a Bell pair, and used to teleport the gate, while the disentangled intermediate qubits double as error-detection flags. In experiments, this protocol sustains average gate fidelities above 85% across up to 40 lattice sites and consistently outperforms the best measurement-based alternative without extra ancillas. The same error-detection idea, applied as sparse stabilizer parity checks with at most nine flag qubits, produces a 75-qubit GHZ state whose reported fidelity exceeds the 0.5 threshold for genuine multipartite entanglement, which the paper identifies as the largest such state reported. If these claims hold, near-term users can gain from QEC building blocks before full fault-tolerant encoding becomes practical.

What carries the argument

The load-bearing object is the GHZ state as a reusable entanglement resource, together with the unitary entangle-disentangle circuit that prepares and reduces it. Step A grows a linear-chain GHZ state along the shortest path between control and target; step B applies local CNOT gates that rotate the n-qubit GHZ state into a small residual GHZ state times $|0\rangle^{\otimes(n-3)}$, so that the disentangled qubits are guaranteed to read all-0 in the absence of error; step C projectively measures the one remaining 'root' qubit to fix the Bell-pair parity; step D completes the teleported CNOT by local operations and classical communication. For direct GHZ generation, the machinery is the stabilizer structure of the GHZ state: ancilla 'type-0' flag qubits measure selected ZZ parity checks without any SWAP overhead on a heavy-hex topology, and any odd-parity outcome marks the shot as failed. The two uses share one principle: convert the errors a device is most prone to into classical flag outcomes, then discard the flagged shots.

What would settle it

Run the 75-qubit preparation and measure the imaginary part of the n-th Fourier mode of the MQC parity oscillation (the term that is exactly zero for a real, positive GHZ coherence). If that imaginary part is statistically nonzero, the reported fidelity is not the true fidelity and the genuine-multipartite-entanglement claim is unsupported; equivalently, a simulation with a known systematic phase error that pushes F=(C+P)/2 above 0.5 while the true fidelity stays below 0.5 would refute the certification step.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that error detection without logical encoding, implemented inside the circuits themselves, improves performance beyond what the same hardware achieves with the best alternative strategies. The long-range CNOT result rests on a protocol where every disentangled GHZ qubit ideally ends in |0>, so reading those qubits flags any bit-flip or amplitude-damping error that occurred during gate teleportation; with error detection enabled, effective gate infidelity falls by about half, and the protocol beats the measurement-based benchmark at every tested distance up to 40 qubits. The GHZ result rests on sparse parity checks, where a small set of ancilla flags non-destructively measures ZZ stabilizers of the GHZ state and shots with odd parity are discarded; combined with deterministic error suppression, this yields MQC fidelities above 0.78 up to 54 qubits and above 0.55 at 75 qubits, at discard fractions no higher than 78%.

Load-bearing premise

The headline 75-qubit entanglement record assumes that the measured multiple-quantum-coherence figure F=(C+P)/2 equals the true fidelity of the prepared state; that equality is guaranteed only when the GHZ coherence is real and positive, and a systematic phase error would make the reported number an upper bound rather than a certification.

Editorial extensions

If this is right

  • Long-range CNOTs with over 85% average fidelity across up to 40 qubits become practical primitives for quantum Fourier transforms, fermionic simulations, and non-local qLDPC codes on current devices.
  • Because the entangle-disentangle construction trades readout measurements for two-qubit gates, it is better matched to present hardware, where readout errors typically exceed two-qubit-gate errors by a factor of three to six.
  • A 75-qubit GHZ state prepared with 9 flag ancillas is reported to exceed the 0.5 fidelity threshold for genuine multipartite entanglement with a 78% worst-case discard fraction, and the paper identifies it as the largest such state reported to date.
  • The same circuit family interpolates between the linear-depth unitary limit and the constant-depth measurement-based limit, so future implementations can be tuned to a device's specific balance of gate and readout errors.

Reading between the lines

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

  • The overhead accounting suggests a design rule the paper only gestures at: on near-term devices, the most cost-effective QEC primitive is the one that turns the device's dominant error channel into a detectable flag, not the one offering the most complete logical protection. A testable corollary is that the same entangle-disentangle construction should improve other controlled-unitary gates, with
  • The 75-qubit record rests on identifying the MQC fidelity estimate with the true state fidelity. If systematic phase errors make the GHZ coherence complex rather than real and positive, the reported F=(C+P)/2 overestimates the true fidelity; a direct measurement of the imaginary part of the n-th Fourier mode of the parity oscillation would settle whether the entanglement claim survives.
  • The measured saturation of fidelity with flag count (for example, the 45-qubit case saturating around three flags) indicates residual error is dominated by Z-dephasing, which ZZ parity checks cannot see. An extension that adds refocusing or dynamical-decoupling layers and checks whether the saturation point moves would separate dephasing from other error sources in this preparation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper reports two experimental demonstrations of low-overhead error detection ('QEC primitives') on IBM superconducting processors. In the first part, a 'unitary entangle-disentangle' protocol implements a long-range CNOT by unitarily preparing a linear-chain GHZ state along the path between control and target, then unitarily disentangling all but two of the intermediate qubits into a Bell pair; the disentangled qubits are measured as error-detection flags at no additional ancilla cost. The authors report average gate fidelities above 85% for teleportation distances up to 40 qubits, consistently outperforming the measurement-based protocol of Bäumer et al. on the same device, in both shot-by-shot and readout-mitigated post-processing modes. In the second part, following Mooney et al., the paper prepares GHZ states with sparse type-0 parity-check flags, combined with a deterministic error-suppression pipeline, reporting MQC fidelities above 0.78 for up to 54 qubits and above 0.55 for 75 qubits, with a maximum discarded fraction of 78% at 75 qubits. The paper claims that the 75-qubit state exhibits genuine multipartite entanglement (GME) and is the largest reported to date, and it argues that the combination of error suppression and physical-level error detection offers a favorable overhead trade-off relative to full logical encoding.

Significance. If upheld, the results are significant: the flag-based long-range CNOT design is elegant, the error-detection capability of the disentangled qubits costs no additional ancillas, and the sparse parity-check layout adds only one two-qubit-gate layer (43 vs. 42) to the 75-qubit GHZ circuit. The experimental methodology is careful: both CNOT protocols were run consecutively on ibm_fez to control drift, error bars are bootstrapped, the error-suppression pipeline is applied symmetrically to both arms, and the paper reports honest negative results (type-1 flags are not beneficial; error detection alone does not suffice). The overhead accounting (Table I; 27-qubit vs. encoded 24-qubit comparisons) is transparent. The two load-bearing weaknesses are the GME certification at 75 qubits, which rests on an MQC fidelity threshold that is not rigorously established as presented and is contradicted in practice by the paper's own 80-qubit control, and the 'largest reported to date' claim, which appears to conflict with a larger certified GHZ state in a cited reference.

major comments (2)
  1. [Sec. III.A, Eq. (4); App. C, Figs. 13-14] The GME claim for the 75-qubit state rests on the statement in Sec. III.A that 'it is sufficient to show MQC fidelity F > 0.5 [64]'. The threshold of Ref. [64] applies to the true state fidelity, F_true = (P + 2 Re ρ_01)/2 with ρ_01 the GHZ off-diagonal element, whereas the measured quantity is F_mqc = (P + C)/2 with C = 2√I_n (Eq. (4)). F_mqc equals F_true only when the coherence is real and positive; a systematic phase error in ρ_01 makes F_mqc an upper bound on the true fidelity, so F_mqc > 0.5 does not by itself certify GME. This is not academic: App. C (Fig. 14) reports an 80-qubit state with F_mqc = 0.511(2) (C = 0.064(3), P = 0.958(2)), statistically well above 0.5, and explicitly declines to claim GME 'despite its MQC fidelity at F = 0.511(2)'. The 75-qubit state has similar qualitative features — F ≈ 0.55 with strongly suppressed coherence (Fig. 4(e)) and heavily damped parity oscillations (Fig. 13) — and its margin over threshold is only about 0.05. The authors should report the fitted phase of the parity oscillation and give a rigorous lower bound on Re ρ_01 satisfying Re ρ_01 > (1 − P)/2, or certify GME by an independent witness, or downgrade the 75-qubit claim.
  2. [Abstract; Sec. III.B] The abstract and Sec. III.B describe the 75-qubit GHZ state as 'the largest reported to date' and 'a record in the published literature'. This appears to conflict with Ref. [58] (Moses et al., Phys. Rev. X 13 (2023)), which reports the preparation of a 92-qubit GHZ state with a quoted fidelity above the 0.5 GME threshold (approximately 0.75). If that fidelity estimate is accepted, the record claim is incorrect; if the authors believe the certification in Ref. [58] does not meet the standard applied here, they should say so explicitly and justify the difference. Because the record claim is a headline result, a reconciliation with all larger certified GHZ states in the cited literature is required.
minor comments (5)
  1. [Abstract; Sec. I] The abstract claims advantages 'relative to any alternative error-reduction strategy', but the experiments benchmark against one specific alternative (the measurement-based protocol of Ref. [36]) and the authors' own no-error-detection and no-error-suppression baselines; the wording should be tempered to reflect the strategies actually tested.
  2. [Fig. 3 caption; Sec. II.B] The reported gate fidelities are conditional on discarding shots requiring feedforward, a substantial fraction of shots, in addition to the error-detection discards shown in Fig. 10; reporting the total acceptance probability and the fidelity at each post-selection stage would make the 'computational gain' claim measurable.
  3. [Sec. III.B] For each flag count l, the presented MQC fidelity is the maximum over the C(f_max, l) subsets of flags; this makes the plotted curves an upper envelope, and the optimism of this selection should be acknowledged or quantified (for example, by reporting the mean or median over subsets).
  4. [App. C, Fig. 14] The reason for not claiming GME for the 80-qubit state ('the interference pattern is nearly entirely damped') is qualitative; given that F_mqc = 0.511(2) is statistically above 0.5, the authors should state the quantitative criterion that separates the 75-qubit GME claim from the 80-qubit non-claim.
  5. [Abstract; Sec. III.B] The statement that the discarded fraction 'grows no higher than 78%' should be scoped to the sizes for which GME is claimed, since the 80-qubit control in App. C has a discard fraction of about 98%.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claims are externally benchmarked experimental measurements, and the MQC fidelity caveat is a validity concern rather than a construction-level circularity.

full rationale

The paper's central derivations are experimental benchmarks, not fitted predictions. Long-range CNOT fidelities are measured via Monte Carlo process certification against the ideal CNOT using an external procedure from Refs. [36,51], with no parameter fitted to the compared protocol; the unitary entangle-disentangle protocol is a new circuit whose performance is compared directly with the independent measurement-based protocol of Bäumer et al. The GHZ fidelity is measured from raw parity-oscillation and population data, with error-detection post-selection, so F=(C+P)/2 is a characterization quantity rather than a quantity defined to equal the claim. The error-suppression pipeline is imported from the authors' prior independent publications [19,25] and is applied symmetrically to both arms of the comparison, so it does not force the outcome. The MQC fidelity threshold F>0.5 for genuine multipartite entanglement is cited from external Ref. [64], not derived in this paper. The paper's own Appendix C statement about the 80Q state (F=0.511(2) but no GME claim) is a legitimate caveat about the sufficiency of that threshold under strong dephasing, but it is a correctness/validity concern about the estimator, not circularity: the MQC fidelity is not defined in terms of the GME claim, and no equation in the paper reduces to its own input. Accordingly, the derivation chain is self-contained and no circular step is identifiable.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new free parameters, entities, or forces. The main axiological burden is the MQC fidelity-to-GME inference, which is a domain assumption shared with prior literature but not rigorously tied to the estimator used. The error suppression pipeline is a prior, independently published tool.

assumptions (3)
  • domain assumption The MQC parity oscillation is band-limited to order n, so the n-th Fourier mode In isolates the GHZ coherence.
    Used in App. C, Eq. (C4), to extract coherence C = 2 sqrt(In). For strongly dephased states this band-limited assumption may break down, biasing the fidelity estimate.
  • domain assumption MQC fidelity F > 0.5 implies genuine multipartite entanglement.
    Invoked in Sec. III.A citing Ref. [64]. The theorem applies to the true fidelity, not necessarily the MQC estimate; the authors' 80Q result suggests they do not fully trust the threshold at scale.
  • domain assumption The error suppression pipeline (Refs. [19,25]) reduces coherent and readout errors without biasing the comparisons.
    Both competing protocols are run with this pipeline; the pipeline's details come from the authors' own prior publications, and the comparison assumes it does not interact differently with the two protocols.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Achieving computational gains with quantum error-correction primitives: Generation of long-range entanglement enhanced by error detection." pith.science (2026). https://pith.science/paper/GJXFEGNN

@misc{pith2026241114638,
  author       = {Pith},
  title        = {Pith review of: Achieving computational gains with quantum error-correction primitives: Generation of long-range entanglement enhanced by error detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GJXFEGNN}},
  note         = {Machine review of arXiv:2411.14638}
}
read the original abstract

The resource overhead required to achieve net computational benefits from quantum error correction (QEC) limits its utility while current systems remain constrained in size, despite exceptional progress in experimental demonstrations. In this paper, we demonstrate that the strategic application of QEC primitives without logical encoding can yield significant advantages on superconducting processors--relative to any alternative error-reduction strategy--while only requiring a modest overhead. We first present a novel protocol for implementing long-range CNOT gates that relies on a unitarily prepared Greenberger-Horne-Zeilinger (GHZ) state as well as a unitary disentangling step; the protocol natively introduces an error-detection process using the disentangled qubits as flags. We demonstrate that it achieves state-of-the-art gate fidelities of over 85% across up to 40 lattice sites, significantly and consistently outperforming the best alternative measurement-based protocol without introducing any additional ancilla qubits. We then apply sparse stabilizer measurements to generate large GHZ states by detecting bit-flip and amplitude-damping errors. Employing this technique in combination with deterministic error suppression, we generate a 75-qubit GHZ state exhibiting genuine multipartite entanglement, the largest reported to date. The generation requires no more than 9 ancilla qubits and the fraction of samples discarded due to errors grows no higher than 78%, far lower than previous discard fractions required for tests using comparable numbers of fully encoded qubits. This work in total represents compelling evidence that adopting QEC primitives on current-generation devices can deliver substantial net benefits.

Figures

Figures reproduced from arXiv: 2411.14638 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 6
Figure 6. Figure 6: Ia in Ref. [36]). 3. Long-range quantum fan-out gates and Toffoli gates [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

80 extracted references · 51 canonical work pages

  1. [64]

    K. X. Wei, I. Lauer, S. Srinivasan, N. Sundaresan, D. T. McClure, D. Toyli, D. C. McKay, J. M. Gambetta, and S. Sheldon, Physical Review A101, 032343 (2020)

    work page 2020

  2. [58]

    Allcock, J

    J. Allcock, J. Bao, J. F. Doriguello, A. Luongo, and M. Santha, Quantum8, 1530 (2024)

    work page 2024

  3. [1]

    II and depicted in Fig

    Fully unitary disentangling step for long-range CNOT We recall from the unitary entangle-disentangle proto- col described in Sec. II and depicted in Fig. 2 that the intermediate qubits in the GHZ state are disentangled unitarily following a circuit that mirrors the entangling one—the direction of the local CNOT gates are all to- wards the ends of the gate...

    work page

  4. [2]

    This quantity should ideally exhibit a sinusoidal behavior as a function ofϕ (see Fig

    Experimentally one mea- sures the interference pattern between the prepared non- phase-rotated and phase-rotated states,ρ and ρϕ, respec- tively, defined asSϕ := Tr(ρ ρϕ). This quantity should ideally exhibit a sinusoidal behavior as a function ofϕ (see Fig. 4(b) for the corresponding measurement circuit). Specifically, the coherenceC is related to then-t...

    work page

  5. [3]

    Designing circuits for general error budgets and scaling up We continue our focus on the long-range CNOT gate and illustrate a general strategy to more flexibly bal- ance measurement errors and gate errors, resulting in a family of circuits interpolating between the linear-depth, unitary entangle-disentangle protocol and the constant- depth, measurement-b...

    work page

  6. [4]

    Long-range quantum fan-out gates and Toffoli gates /gid00009 /gid00027/gid00018/gid01087 /gid00018/gid01088 /gid00018/gid01089 /gid00018/gid01090 /gid00018/gid01091 /gid00018/gid01092 /gid00018/gid01093 /gid00018/gid01094 /gid00025 /gid00025 /gid00009 /gid00025 /gid00009 /gid00018/gid01095 /gid00018/gid01096 /gid00002 /gid00003/gid00004 /gid00005 /gid0001...

    work page

  7. [5]

    II and depicted in Fig

    Alternative derivation of the measurement-based protocol for long-range CNOT We observe that the measurement-based protocol for the long-range CNOT [36] also follows the three-step general framework of teleporting CNOT gates described in Sec. II and depicted in Fig. 2(a). The protocol can be understood as fusing a linear chain of local Bell pairs (and an ...

    work page

  8. [6]

    (B1) As such, we estimate the process fidelity using Monte Carlo process certification [36, 51]

    Average gate fidelity The average output fidelity, defined as the fidelity be- tween the output states produced by two quantum chan- nels Λ and ˜Λ on a d-dimensional space averaged Haar- uniformly over all pure input states, can be shown to relate to the process fidelity as [67] Fgate(Λ, ˜Λ) = d Fprocess(Λ, ˜Λ) + 1 d + 1 . (B1) As such, we estimate the pr...

    work page

Show all 80 references
  1. [7]

    Additional experimental results In evaluating the quality of the long-range CNOT, it is also essential to examine the fraction of discarded shots. In Fig. 10, we present the discarded fraction of shots as a function of the number of qubits (or path length for the long-range CN...

  2. [8]

    The measurement-based preparation of n-qubit GHZ states [36, 56, 58–60]. The superposition state before simultaneous measurements ofO(n) qubits is uniform, since each measurement on an ancilla has equal probabilities to project the corresponding pair of data qubits into the±1-...

  3. [9]

    IIIA, especially with the full-weight, non-Z- type stabilizers which constitute half of all the2n stabilizers

    Fidelity estimation of an n-qubit GHZ state by sampling its stabilizers [36, 50, 59, 70] discussed in Sec. IIIA, especially with the full-weight, non-Z- type stabilizers which constitute half of all the2n stabilizers. We remark that anyX-basis or Y-basis measurement on a GHZ d...

  4. [10]

    The measurement-based protocol of the long-range 17 Data qubit T ype-0 T ype-1 flag (a) (c) (b) /gid00008/gid00009/gid00027 /gid00021/gid00052/gid00043/gid00032/gid01162/gid01088/gid00001/gid00033/gid00039/gid00028/gid00034 /gid00008/gid00009/gid00027 /gid00021/gid00052/gid000...

  5. [11]

    Krinner, N

    S. Krinner, N. Lacroix, A. Remm, A. Di Paolo, E. Genois, C. Leroux, C. Hellings, S. Lazar, F. Swiadek, J. Herrmann, et al., Nature605, 669 (2022)

    2022

  6. [12]

    Y. Zhao, Y. Ye, H.-L. Huang, Y. Zhang, D. Wu, H. Guan, Q. Zhu, Z. Wei, T. He, S. Cao,et al., Physical Review Letters 129, 030501 (2022)

    2022

  7. [13]

    Acharya, I

    R. Acharya, I. Aleiner, R. Allen, T. I. Andersen, M. Ans- mann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, R. Bab- bush, et al., Nature614, 676 (2023)

    2023

  8. [14]

    R. S. Gupta, N. Sundaresan, T. Alexander, C. J. Wood, S. T. Merkel, M. B. Healy, M. Hillenbrand, T. Jochym- O’Connor, J. R. Wootton, T. J. Yoder,et al., Nature625, 259 (2024)

    2024

  9. [15]

    Bluvstein, S

    D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, et al., Nature626, 58 (2024)

    2024

  10. [16]

    Acharya, L

    R. Acharya, L. Aghababaie-Beni, I. Aleiner, T. I. Ander- sen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, N. As- trakhantsev, J. Atalaya,et al., Nature638, 920 (2025)

    2025

  11. [17]

    B. W. Reichardt, D. Aasen, R. Chao, A. Chernoguzov, W. van Dam, J. P. Gaebler, D. Gresh, D. Lucchetti, M. Mills, S. A. Moses,et al., arXiv:2409.04628 (2024)

    2024 arXiv

  12. [18]

    V. G. Matsos, C. H. Valahu, M. J. Millican, T. Nav- ickas, X. C. Kolesnikow, M. J. Biercuk, and T. R. Tan, arXiv:2409.05455 (2024)

    2024 arXiv

  13. [19]

    Lachance-Quirion, M.-A

    D. Lachance-Quirion, M.-A. Lemonde, J. O. Simoneau, L. St-Jean, P. Lemieux, S. Turcotte, W. Wright, A. Lacroix, J. Fréchette-Viens, R. Shillito, et al. , arXiv:2310.11400 (2023)

    2023 arXiv

  14. [20]

    B. W. Reichardt, A. Paetznick, D. Aasen, I. Basov, J. M. Bello-Rivas, P. Bonderson, R. Chao, W. van Dam, M. B. Hastings, A. Paz,et al., arXiv:2411.11822 (2024)

    2024 arXiv

  15. [21]

    Babbush, J

    R. Babbush, J. R. McClean, M. Newman, C. Gidney, S. Boixo, and H. Neven, PRX Quantum2, 010103 (2021)

    2021

  16. [22]

    Litinski, Quantum3, 128 (2019)

    D. Litinski, Quantum3, 128 (2019)

    2019

  17. [23]

    Dutkiewicz, S

    A. Dutkiewicz, S. Polla, M. Scheurer, C. Gogolin, W. J. Huggins, and T. E. O’Brien, arXiv:2410.05369 (2024)

    2024

  18. [24]

    Sundaresan, I

    N. Sundaresan, I. Lauer, E. Pritchett, E. Magesan, P. Ju- rcevic, and J. M. Gambetta, PRX Quantum1, 020318 (2020)

    2020

  19. [25]

    Carroll, S

    M. Carroll, S. Rosenblatt, P. Jurcevic, I. Lauer, and A. Kandala, npj Quantum Information8, 132 (2022)

    2022

  20. [26]

    M. Bal, A. A. Murthy, S. Zhu, F. Crisa, X. You, Z. Huang, T. Roy, J. Lee, D. v. Zanten, R. Pilipenko,et al., npj Quantum Information10, 43 (2024)

    2024

  21. [27]

    E. A. Sete, V. Tripathi, J. A. Valery, D. Lidar, and J. Y. Mutus, Physical Review Applied22, 014059 (2024)

    2024

  22. [28]

    P. D. Nation, H. Kang, N. Sundaresan, and J. M. Gam- betta, PRX Quantum2, 040326 (2021)

    2021

  23. [29]

    P. S. Mundada, A. Barbosa, S. Maity, Y. Wang, T. Merkh, T. Stace, F. Nielson, A. R. Carvalho, M. Hush, M. J. Bier- cuk, et al., Physical Review Applied20, 024034 (2023)

    2023

  24. [30]

    Kandala, K

    A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Nature567, 491–495 18 (2019)

    2019

  25. [31]

    W. J. Huggins, S. McArdle, T. E. O’Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Physical Review X11 (2021)

    2021

  26. [32]

    E.vandenBerg, Z.K.Minev, A.Kandala, andK.Temme, Nature Physics19, 1116–1121 (2023)

    2023

  27. [33]

    Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Nature618, 500 (2023)

    2023

  28. [34]

    H. Liao, D. S. Wang, I. Sitdikov, C. Salcedo, A. Seif, and Z. K. Minev, Nature Machine Intelligence6, 1478 (2024)

    2024

  29. [35]

    Coote, R

    P. Coote, R. Dimov, S. Maity, G. S. Hartnett, M. J. Biercuk, and Y. Baum, PRX Quantum6, 010332 (2025)

    2025

  30. [36]

    A. Seif, H. Liao, V. Tripathi, K. Krsulich, M. Malekakhlagh, M. Amico, P. Jurcevic, and A. Javadi-Abhari, inACM/IEEE 51st Annual Interna- tional Symposium on Computer Architecture (ISCA) (2024)

    2024

  31. [37]

    Tripathi, H

    V. Tripathi, H. Chen, M. Khezri, K.-W. Yip, E. Levenson- Falk, and D. A. Lidar, Physical Review Applied 18, 024068 (2022)

    2022

  32. [38]

    Pokharel, N

    B. Pokharel, N. Anand, B. Fortman, and D. A. Lidar, Physical Review Letters121, 220502 (2018)

    2018

  33. [39]

    A. R. R. Carvalho, H. Ball, M. J. Biercuk, M. R. Hush, and F. Thomsen, Physical Review Applied15, 064054 (2021)

    2021

  34. [40]

    P. D. Nation and M. Treinish, PRX Quantum4, 010327 (2023)

    2023

  35. [41]

    G. S. Hartnett, A. Barbosa, P. S. Mundada, M. Hush, M. J. Biercuk, and Y. Baum, Quantum8, 1542 (2024)

    2024

  36. [42]

    Sachdeva, G

    N. Sachdeva, G. S. Hartnett, S. Maity, S. Marsh, Y. Wang, A. Winick, R. Dougherty, D. Canuto, M. J. Biercuk, Y. Baum, et al., arXiv:2406.01743 (2024)

    2024 arXiv

  37. [43]

    By QEC primitives, we refer to core components or sub- routines employed in QEC, such as encoding, parity checks, non-destructive measurements via ancillas, and decoding

  38. [44]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information (Cambridge University Press, 2011)

    2011

  39. [45]

    G. J. Mooney, G. A. White, C. D. Hill, and L. C. Hol- lenberg, Journal of Physics Communications5, 095004 (2021)

    2021

  40. [46]

    Bäumer, V

    E. Bäumer, V. Tripathi, D. S. Wang, P. Rall, E. H. Chen, S. Majumder, A. Seif, and Z. K. Minev, PRX Quantum 5, 030339 (2024)

    2024

  41. [47]

    Holmes, S

    A. Holmes, S. Johri, G. G. Guerreschi, J. S. Clarke, and A. Y. Matsuura, Quantum Science and Technology5, 025009 (2020)

    2020

  42. [48]

    Bravyi, A

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

    2024

  43. [49]

    Berthusen, D

    N. Berthusen, D. Devulapalli, E. Schoute, A. M. Childs, M. J. Gullans, A. V. Gorshkov, and D. Gottesman, PRX Quantum 6, 010306 (2025)

    2025

  44. [50]

    Gottesman and I

    D. Gottesman and I. L. Chuang, Nature402, 390–393 (1999)

    1999

  45. [51]

    Eisert, K

    J. Eisert, K. Jacobs, P. Papadopoulos, and M. B. Plenio, Physical Review A62 (2000)

    2000

  46. [52]

    Piveteau and D

    C. Piveteau and D. Sutter, IEEE Transactions on Infor- mation Theory70, 2734–2745 (2024)

    2024

  47. [53]

    A. C. Vazquez, C. Tornow, D. Ristè, S. Woerner, M. Takita, and D. J. Egger, Nature (2024)

    2024

  48. [54]

    McEwen, D

    M. McEwen, D. Bacon, and C. Gidney, Quantum7, 1172 (2023)

    2023

  49. [55]

    Bäumer and S

    E. Bäumer and S. Woerner, arXiv:2408.03064 (2024)

    2024 arXiv

  50. [56]

    Pham and K

    P. Pham and K. M. Svore, Quantum Information & Com- putation 13, 937–962 (2013)

    2013

  51. [57]

    Høyer and R

    P. Høyer and R. Špalek, Theory of Computing1, 81 (2005)

    2005

  52. [59]

    S. Xu, C. T. Hann, B. Foxman, S. M. Girvin, and Y. Ding, in 56th Annual IEEE/ACM International Symposium on Microarchitecture(2023) p. 526–538

    2023

  53. [60]

    S. T. Flammia and Y.-K. Liu, Physical Review Letters 106, 230501 (2011)

    2011

  54. [61]

    M. P. da Silva, O. Landon-Cardinal, and D. Poulin, Physical Review Letters107 (2011)

    2011

  55. [62]

    C. L. Degen, F. Reinhard, and P. Cappellaro, Reviews of Modern Physics89, 035002 (2017)

    2017

  56. [63]

    Hillery, V

    M. Hillery, V. Bužek, and A. Berthiaume, Physical Re- view A59, 1829 (1999)

    1999

  57. [65]

    J. F. Kam, H. Kang, C. D. Hill, G. J. Mooney, and L. C. L. Hollenberg, Physical Review Research6, 033155 (2024)

    2024

  58. [66]

    C. K. Andersen, A. Remm, S. Lazar, S. Krinner, N. Lacroix, G. J. Norris, M. Gabureac, C. Eichler, and A. Wallraff, Nature Physics16, 875–880 (2020)

    2020

  59. [67]

    B. Yang, R. Raymond, and S. Uno, Physical Review A 106, 012423 (2022)

    2022

  60. [68]

    S. A. Moses, C. H. Baldwin, M. S. Allman, R. Ancona, L. Ascarrunz, C. Barnes, J. Bartolotta, B. Bjork, P. Blan- chard, M. Bohn,et al., Physical Review X13 (2023)

    2023

  61. [69]

    E. H. Chen, G.-Y. Zhu, R. Verresen, A. Seif, E. Bäumer, D. Layden, N. Tantivasadakarn, G. Zhu, S. Sheldon, A. Vishwanath, et al., Nature Physics21, 161 (2025)

    2025

  62. [70]

    Hashim, M

    A. Hashim, M. Yuan, P. Gokhale, L. Chen, C. Juenger, N. Fruitwala, Y. Xu, G. Huang, L. Jiang, and I. Siddiqi, arXiv:2403.18768 (2024)

    2024

  63. [71]

    T.Monz, P.Schindler, J.T.Barreiro, M.Chwalla, D.Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, Physical Review Letters106, 130506 (2011)

    2011

  64. [72]

    Chamberland, G

    C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, Physical Review X10, 011022 (2020)

    2020

  65. [73]

    Z. Bao, S. Xu, Z. Song, K. Wang, L. Xiang, Z. Zhu, J. Chen, F. Jin, X. Zhu, Y. Gao,et al., Nature Commu- nication 15 (2024)

    2024

  66. [74]

    Leibfried, E

    D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost, C. Langer,et al., Nature438, 639 (2005)

    2005

  67. [75]

    de Bone, R

    S. de Bone, R. Ouyang, K. Goodenough, and D. Elk- ouss, IEEE Transactions on Quantum Engineering1, 1–10 (2020)

    2020

  68. [76]

    Delfosse, M

    N. Delfosse, M. E. Beverland, and M. A. Tremblay, arXiv:2109.14599 (2021)

    2021 arXiv

  69. [77]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Physical Review A60, 1888 (1999)

    1999

  70. [78]

    M. A. Nielsen, Physics Letters A303, 249–252 (2002)

    2002

  71. [79]

    J. A. Smolin, J. M. Gambetta, and G. Smith, Physical Review Letters108, 070502 (2012)

    2012

  72. [80]

    S. Cao, B. Wu, F. Chen, M. Gong, Y. Wu, Y. Ye, C. Zha, H. Qian, C. Ying, S. Guo, Q. Zhu,et al., Nature619, 1 (2023)

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.