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arxiv: 2605.23119 · v1 · pith:D2FTHNV5new · submitted 2026-05-22 · 🪐 quant-ph · cs.IT· math.IT

Construction of EAQECCs with imperfect ebits

Guanmin Guo , Ruihu Li This is my paper

Pith reviewed 2026-05-25 05:01 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords entanglement-assisted quantum error correctionnoisy ebitsq-ary stabilizer codessymplectic geometryadditive codes over finite fieldsqudit systemsfault-tolerant quantum computation
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The pith

q-ary entanglement-assisted codes with noisy ebits can outperform standard stabilizer codes under specific noise conditions.

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

The paper generalizes the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits from binary to qudit systems over any prime-power dimension q. It builds a unified construction method from the generalized Pauli group and symplectic geometry over finite fields, then produces explicit families of such codes. Performance comparisons show these codes can correct errors more effectively than ordinary stabilizer codes that use the same number of resources when the shared entanglement experiences particular forms of noise. The work targets practical fault tolerance in higher-dimensional quantum hardware where maintaining perfect entanglement is difficult.

Core claim

By extending the binary stabilizer formalism to the q-ary case via the generalized Pauli group over F_q and symplectic geometry over F_q^{2n}, the authors obtain equivalent descriptions as additive codes over F_q^{2n}. They construct several families of q-ary EAQECCs-Ne and demonstrate that, under certain noise models on the ebits, these codes achieve better error-correcting performance than optimal stabilizer codes with equivalent capability.

What carries the argument

The unified framework that translates EAQECCs-Ne into additive codes over F_q^{2n} using symplectic geometry, allowing explicit constructions for qudits with imperfect shared entanglement.

If this is right

  • Explicit families of q-ary EAQECCs-Ne exist that meet or exceed the error-correction power of stabilizer codes when ebit noise matches the analyzed models.
  • The symplectic and additive-code formulations give multiple equivalent ways to search for new codes.
  • High-dimensional quantum systems can use noisy entanglement as a resource rather than requiring perfect ebits.
  • The constructions apply directly to fault-tolerant protocols in qudit hardware.

Where Pith is reading between the lines

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

  • Similar constructions might apply to other entanglement-assisted settings such as continuous-variable systems or network coding.
  • The performance edge could reduce the overhead needed to reach a target logical error rate in near-term qudit devices.
  • Classical coding theorists working over finite fields may find new problems by translating these quantum constructions back to additive codes.

Load-bearing premise

The noise models on the imperfect ebits are such that the extra flexibility from the noisy entanglement yields a measurable advantage over stabilizer codes of the same strength.

What would settle it

Numerical simulation or explicit distance calculation for a small q, block length, and one of the paper's noise models showing that the constructed EAQECCs-Ne achieve strictly lower logical error rates than the best stabilizer code with the same number of physical qudits and the same number of logical qudits.

read the original abstract

We generalize the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits (EAQECCs-Ne) from the binary case to the general $q$-ary case, where $q$ is a prime power. By leveraging the structure of the generalized Pauli group over $\mathbb{F}_q$ and symplectic geometry over $\mathbb{F}_q^{2n}$, we establish a unified framework for constructing EAQECCs-Ne for qudit systems. Equivalent formulations in terms of symplectic geometry over $\mathbb{F}_q$ and additive codes over $\mathbb{F}_q^{2n}$ are derived. We further construct several families of $q$-ary EAQECCs with noise ebits and analyze their performance compared to optimal stabilizer codes. Our results demonstrate that under certain noise conditions, the proposed EAQECCs-Ne can outperform standard stabilizer codes with equivalent error-correcting capability, offering a promising approach for fault-tolerant quantum computation in high-dimensional quantum systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript generalizes the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits (EAQECCs-Ne) from the binary case to the q-ary case where q is a prime power. It leverages the generalized Pauli group over F_q and symplectic geometry over F_q^{2n} to establish a unified framework, derives equivalent formulations in terms of symplectic geometry and additive codes over F_q^{2n}, constructs several families of q-ary EAQECCs-Ne, and analyzes their performance, claiming that under certain noise conditions the proposed codes can outperform standard stabilizer codes with equivalent error-correcting capability.

Significance. If the explicit constructions, derivations, and performance comparisons hold, the work extends binary EAQECC results to qudit systems using standard tools of the field (generalized Pauli operators and symplectic geometry). This could offer a route to improved fault tolerance in high-dimensional quantum systems under specific noise models, with the qualified performance claim aligning with existing literature on entanglement-assisted codes.

major comments (1)
  1. [Abstract] Abstract: the central performance claim that EAQECCs-Ne 'can outperform standard stabilizer codes with equivalent error-correcting capability' under 'certain noise conditions' is stated without any explicit constructions, generator matrices, error bounds, noise models, or comparative data. This is load-bearing for the main result and prevents verification of the outperformance assertion.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for recognizing the generalization of EAQECCs-Ne to the q-ary setting. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central performance claim that EAQECCs-Ne 'can outperform standard stabilizer codes with equivalent error-correcting capability' under 'certain noise conditions' is stated without any explicit constructions, generator matrices, error bounds, noise models, or comparative data. This is load-bearing for the main result and prevents verification of the outperformance assertion.

    Authors: The abstract is a concise summary of the paper's contributions and results. The explicit constructions of multiple families of q-ary EAQECCs-Ne (via symplectic geometry over F_q^{2n} and additive codes), the corresponding generator matrices, the specific noise models (including the imperfect ebit noise parameters), error bounds, and the comparative performance data versus optimal stabilizer codes are all provided in the body of the manuscript, particularly in Sections 4 and 5. These sections include concrete examples, parameter tables, and numerical evaluations showing outperformance under the stated noise conditions. Verification is therefore possible from the full text, consistent with standard practice for abstracts in the field. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract and described framework generalize the binary EAQECC stabilizer formalism to prime-power q using the pre-existing generalized Pauli group over F_q and symplectic geometry over F_q^{2n}. These are standard external mathematical structures, not defined or derived inside the paper. Equivalent additive-code formulations follow directly from these known equivalences. Explicit family constructions and performance comparisons (qualified to 'certain noise conditions' and 'equivalent error-correcting capability') introduce no fitted parameters renamed as predictions, no self-citations as load-bearing premises, and no self-definitional loops. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are specified.

pith-pipeline@v0.9.0 · 5692 in / 1050 out tokens · 46146 ms · 2026-05-25T05:01:45.139669+00:00 · methodology

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Scheme for reducing decoherence in quantum computer memory,

    P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A, vol. 52, no. 4, pp. 2493–2496, Oct. 1995

    work page 1995

  2. [2]

    Error correcting codes in quantum theory,

    A. M. Steane, “Error correcting codes in quantum theory,”Phys. Rev. Lett., vol. 77, no. 5, pp. 793–797, Aug. 1996

    work page 1996

  3. [3]

    Stabilizer codes and quantum error correction,

    D. Gottesman, “Stabilizer codes and quantum error correction,” Ph.D. dissertation, California Inst. Technol., Pasadena, CA, USA, 1997

    work page 1997

  4. [4]

    Quantum error correction via codes over GF(4),

    A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction via codes over GF(4),”IEEE Trans. Inf. Theory, vol. 44, no. 4, pp. 1369–1387, Jul. 1998

    work page 1998

  5. [5]

    Nonbinary stabilizer codes over finite fields,

    A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “Nonbinary stabilizer codes over finite fields,”IEEE Trans. Inf. Theory, vol. 52, no. 11, pp. 4892–4914, Nov. 2006. Construction of EAQECCs with imperfect ebits 11

    work page 2006

  6. [6]

    Entanglement required in achieving entanglement-assisted channel capacities,

    G. Bowen, “Entanglement required in achieving entanglement-assisted channel capacities,”Phys. Rev. A, vol. 66, no. 5, p. 052313, Nov. 2002

    work page 2002

  7. [7]

    Correcting quantum errors with entanglement,

    T. Brun, I. Devetak, and M.-H. Hsieh, “Correcting quantum errors with entanglement,” Science, vol. 314, no. 5798, pp. 436–439, Oct. 2006. doi:10.1126/science.1131563

    work page doi:10.1126/science.1131563 2006

  8. [8]

    Optimal entanglement formulas for entanglement-assisted quantum coding,

    M. M. Wilde and T. A. Brun, “Optimal entanglement formulas for entanglement-assisted quantum coding,” Phys. Rev. A, vol. 77, no. 6, p. 064302, Jun. 2008

    work page 2008

  9. [9]

    Entanglement increases the error-correcting ability of quantum error- correcting codes,

    C.-Y . Lai and T. A. Brun, “Entanglement increases the error-correcting ability of quantum error- correcting codes,” Phys. Rev. A, vol. 88, no. 1, p. 012320, Jul. 2013

    work page 2013

  10. [10]

    Linear Plotkin bound for entanglement-assisted quantum codes,

    L. Guo and R. Li, “Linear Plotkin bound for entanglement-assisted quantum codes,”Phys. Rev. A, vol. 87, no. 3, p. 032309, Mar. 2013

    work page 2013

  11. [11]

    Entanglement-assisted quantum zero-error capacity,

    B. Shaw, M. M. Wilde, O. Oreshkov, I. Kremsky, and D. A. Lidar, “Entanglement-assisted quantum zero-error capacity,”Phys. Rev. A, vol. 78, no. 1, p. 012337, Jul. 2008

    work page 2008

  12. [12]

    Entanglement boosts quantum turbo codes,

    M. M. Wilde and M.-H. Hsieh, “Entanglement boosts quantum turbo codes,” in 2011 IEEE Int. Symp. Inf. Theory (ISIT) , St. Petersburg, Russia, Jul. 31–Aug. 5, 2011, pp. 445–449. doi:10.1109/ISIT.2011.6034165

    work page doi:10.1109/isit.2011.6034165 2011

  13. [13]

    Entanglement-assisted quantum error-correcting codes with imperfect ebits,

    C. Y . Lai and T. A. Brun, “Entanglement-assisted quantum error-correcting codes with imperfect ebits,” Phys. Rev. A, vol. 86, no. 3, p. 032319, Sep. 2012

    work page 2012

  14. [14]

    Entanglementassisted concatenated quantum codes,

    J. Fan, J. Li, Y . Zhou, M.-H. Hsieh, and H. V . Poor, “Entanglementassisted concatenated quantum codes,” Proc. Natl. Acad. Sci. USA , vol. 119, no. 24, p. e2202235119, Jun. 2022. doi:10.1073/pnas.2202235119

    work page doi:10.1073/pnas.2202235119 2022

  15. [15]

    Wan, Geometry of Classical Groups over Finite Fields

    Z. Wan, Geometry of Classical Groups over Finite Fields . Lund, Sweden: Studentlitteratur, 1993

    work page 1993

  16. [16]

    Chen, Quantum Error-Correcting Codes: Theory and Applications

    F. Chen, Quantum Error-Correcting Codes: Theory and Applications . Singapore: World Scientific, 2015

    work page 2015

  17. [17]

    Entanglement-assisted quantum quasi-cyclic low-density parity-check codes,

    M.-H. Hsieh, T. A. Brun, and I. Devetak, “Entanglement-assisted quantum quasi-cyclic low-density parity-check codes,” Phys. Rev. A 79, 032340 (2009)

    work page 2009

  18. [18]

    Symplectic geometry-based construction of quantum error- correcting codes,

    R. Li, Y . Ren, C. Guan and Y . Liu, “Symplectic geometry-based construction of quantum error- correcting codes, ” arXiv:2501.15465v2 [quant-ph], Aug. 2025

    work page arXiv 2025

  19. [19]

    All the quantum codes of distance 3,

    J. Bierbrauer, “All the quantum codes of distance 3,” IEEE Trans. Inf. Theory , vol. 54, no. 11, pp. 5235–5245, Nov. 2008

    work page 2008

  20. [20]

    Entanglement-assisted quantum error-correcting codes achieve the quan- tum Hamming bound,

    R. Li, L. Guo, and Z. Xu, “Entanglement-assisted quantum error-correcting codes achieve the quan- tum Hamming bound,” Phys. Rev. A, vol. 110, no. 2, p. 022415, Aug. 2024

    work page 2024