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REVIEW 3 major objections 4 minor 82 references

Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that a single algebraic condition, Theorem 1's Eq. (2), governs error correction for every entanglement-assisted quantum code, unifying EAQEC, EAOQEC, and EACQ.

desk verdict The unification theorem is wrong for subsystem codes: Theorem 1 misclassifies gauge-type errors, contradicting the paper's own Theorem 2 for EAOQEC. read the letter →

arxiv 2411.14389 v1 pith:FHHP2LK3 submitted 2024-11-21 quant-ph

Priya J. Nadkarni , Serge Adonsou , Guillaume Dauphinais , David W. Kribs , Michael Vasmer This is my paper · ORCID
classification quant-ph MSC 81P7094B60 PACS 03.67.Pp
keywords entanglement-assistedquantumerrorcorrectionoperatoralgebrastabilizerformalismsubsystemcodeshybridclassical-quantumcodedistanceEAOAQEC
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 introduces a single framework for entanglement-assisted quantum error correction, called EAOAQEC, that contains the previously separate EAQEC, EAOQEC, and EACQ code families. The central claim is that every such code can be viewed as an operator-algebra quantum error correcting code built from an extended stabilizer, and that one algebraic condition — membership of $E_a^\dagger E_b$ in the set of Eq. (2) — decides exactly which error sets are correctable. The generality matters because it converts code design, distance computation, and error correction questions for all entanglement-assisted codes into one uniform formalism. The paper then uses that formalism to define a code distance, to characterize which hybrid classical-quantum codes arise as EACQ codes, and to construct new subsystem and hybrid codes that fall outside the older frameworks.

What carries the argument

The device that carries the argument is the EAOAQEC code $C(H,S,G_0,L_0,T_0)$: start with an $n$-qubit Pauli subgroup $H$, add one ebit for each anticommuting pair so the extended operators generate an Abelian stabilizer $S$ on $n+e$ qubits, then add gauge operators $G_0$, logical operators $L_0$, and a subset $T_0$ of normalizer coset representatives to encode subsystem and classical-hybrid structure. The workhorse is the normalizer-coset structure: because each transversal operator can be written $T=T^{(n)}\otimes I$, Bob's ebits stay noiseless, and the correction condition of Eq. (1) collapses to the set membership of Eq. (2), with the code distance of Eq. (3) read off as the minimum weight of operators in the complementary set.

What would settle it

Find or compute a valid EAOAQEC code whose normalizer cosets cannot all be represented by operators of the form $T^{(n)}\otimes I$; for such a code Eq. (2) and the distance formula Eq. (3) would not follow from Eq. (1), and the unified criterion would fail.

Watch

Extended reading notes

Core claim

The paper's central discovery is Theorem 1: for an EAOAQEC code $C(H,S,G_0,L_0,T_0)$, a set of errors $\{E_a\otimes I\}$ with $E_a\in P_n$ is correctable if and only if for all $a,b$, $E_a^\dagger E_b$ belongs to the set $\big(\langle H_I,G_0^{(n)},iI\rangle \cup (P_n\setminus Z(H))\big)\cap\big(P_n\setminus \bigcup_{i\neq j} T_i^{(n)}(T_j^{(n)})^{-1}Z(H)\big)$, where $Z(H)$ is the centralizer of the original Pauli subgroup $H$. This condition is a direct application of the OAQEC stabilizer-formalism correctability test (Eq. (1)) to errors that act trivially on Bob's ebits. The authors state that this single theorem subsumes the error correction theorems of EAQEC, EAOQEC, EACQ, and ordinary OAQEC, and it yields a natural minimum-weight distance for the whole family.

Load-bearing premise

The argument assumes every EAOAQEC code has a coset transversal whose representatives act as the identity on Bob's ebits, which the paper sketches but does not fully prove.

Editorial extensions

If this is right

  • All entanglement-assisted error correction theorems (EAQEC, EAOQEC, EACQ) become special cases of Theorem 1, so existing code results can be re-derived and compared through one criterion.
  • The distance defined by Eq. (3) gives a uniform notion of minimum distance for every EAOAQEC code, with dressed and bare variants, and a separate distance when Bob's ebits are noisy.
  • EACQ codes are a proper subclass of EA hybrid subspace codes, characterized by the condition that the represented cosets form a subgroup and by the algebraic containment $H\subseteq Z(Z(Z(T_0^{(n)})\cap H))$.
  • The gauge-fixing, clean-qubits, entanglement-assisted gauge-fixing, and general gauge-fixing constructions produce new EAOAQEC subsystem and hybrid codes, including examples with parameters not captured by the previous frameworks.

Reading between the lines

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

  • Extension: the Eq. (2) criterion could be used as a search oracle, enumerating Pauli subgroups $H$ and transversal sets $T_0$ to classify all EAOAQEC codes by distance directly without first building the code through an older framework.
  • Extension: the clean-qubits construction suggests a two-way trade-off: choosing $e$ linearly independent columns of the check matrix to become ebits always yields an EAOAQEC code with distance at least the original, so the construction may double as a distance-amplification tool.
  • Extension: because EACQ representability is checked algorithmically by the paper's Algorithm 2, one can mechanically test whether a given EA hybrid subspace code admits a classical-quantum stabilizer splitting, which may expose further code families outside EACQ.
  • Extension: the framework's algebraic form strongly suggests the same Eq. (2) structure will carry over to qudits by replacing $P_n$ with the qudit Pauli group, even though the paper only proves the qubit case.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces a framework called EAOAQEC, intended to unify entanglement-assisted quantum error-correcting codes (EAQEC), entanglement-assisted operator quantum error-correcting codes (EAOQEC), and entanglement-assisted hybrid classical-quantum codes (EACQ) by viewing them through operator algebra quantum error correction (OAQEC). The main technical object is a code C(H,S,G0,L0,T0), where H is an n-qubit Pauli subgroup, S is an Abelian extension of part of H on n+e qubits, G0 and L0 are gauge and logical operators, and T0 is a subset of coset representatives. Theorem 1 gives a correctability condition in terms of the centralizer Z(H), Definition 2 defines a code distance from the same set, and subsequent sections specialize this to EAQEC, EAOQEC, EACQ, and provide subsystem-code constructions and characterizations. The paper is built on the stabilizer formalism for OAQEC introduced in [24].

Significance. If the central theorem were correct, the unification would be a useful conceptual and technical contribution: it would reduce correctability questions for all previously known entanglement-assisted code families to one algebraic condition, yield a unified distance notion, and provide new constructions of hybrid and subsystem codes. The paper also contains several worked examples and explicit algorithms, which are valuable for potential applications. However, the central theorem as stated is inconsistent with the paper's own EAOQEC specialization, because the input group H is allowed to contain gauge generators that are not extended into S. This makes the claimed unification currently unsupported. The issue appears fixable within the manuscript's scope by giving H a consistent definition and re-deriving the affected statements, but the correction is substantial.

major comments (3)
  1. [Section 4, paragraph after Eq. (1)] The paper defines EAOQEC codes by taking H = <H_I, H_E, H_G>, then extending only H_I and H_E to form S, so H includes the gauge generators H_G. Theorem 1, however, uses Z(H) in its correctability condition, and its proof asserts that E⊗I belongs to N(S) if and only if E belongs to Z(H). For such codes, E⊗I commutes with S if and only if E commutes with <H_I, H_E>; it need not commute with H_G. Hence Z(S) ∩ (P_n ⊗ I) is Z(<H_I,H_E>)⊗I, not Z(H)⊗I, and Theorem 1's condition is wrong for the EAOQEC specialization. Concretely, for an [[3,1;1,1]] code with H_E = <Z1,X1>, H_G = <Z2,X2>, S = <Z1Z4,X1X4>, G0 = <Z2,X2>, and T0 = {I}, the operator E = Z2X3 lies outside Z(H) and is therefore classified as correctable by Theorem 1. But E⊗I commutes with S and is not in G = <S,G0,iI>, so E⊗I is in N(S) \ G and is uncorrectable by Eq. (1) and by Theorem 2. The claimed unification therefore does not hold as stated; the framework needs an unambiguous definition of H (for example, H = <H_I,H_E> for EAOQEC codes) and a re-derivation of Theorem 1 and Definition 2 under that convention.
  2. [Section 4, paragraph after Eq. (1)] The paper asserts that a coset transversal T for an EAOAQEC code can always be chosen so that every T ∈ T has the form T = T^(n) ⊗ I, i.e., acts as the identity on Bob's ebits. This property is load-bearing: Theorem 1 and Definition 2 both rely on it. The support given is only a sketch, based on a set of destabilizer-type operators. Please provide a complete proof or a precise sufficient condition for the existence of such a transversal. If there are valid EA code constructions for which every transversal representative must act nontrivially on the ebits, then Eq. (2) and Eq. (3) would not follow for those codes.
  3. [Section 6.1, Theorem 5 and Algorithm 1] The proof of the backward direction of Theorem 5 relies on Algorithm 1, but the algorithm's update rule 'Tp → Tp T1_{TpHi=-HiTp}j' uses an undefined indicator-style notation, and the proof that the final sets SQgen and SCgen satisfy the claimed generation, coset, and commutation properties is sketched rather than fully demonstrated. Since Theorem 5 is stated as a complete characterization of EACQ representability, this needs a rigorous proof, especially because the argument invokes Corollary 2 and Lemma 3, whose proofs are also terse.
minor comments (4)
  1. [Lemma 3 proof] The phrase 'we perform a symplectic-isotropic decomposition on A and ensure we get the isotropic subgroup to be Z(A)' is confusing: the isotropic subgroup is a subgroup of A, whereas Z(A) generally contains elements outside A. Please clarify the intended decomposition and justify this step.
  2. [Section 7.1, gauge fixing construction] The definition 'H′ = H ∪ {G_Z1, ..., G_Zy}' should read 'H′ = <H, G_Z1, ..., G_Zy>' because H ∪ {generators} is not generally a group.
  3. [Equation (19) and surrounding text] The displayed equation labeled (19) has a stray closing parenthesis after the display, and the phrase 'Equation (19),' at the beginning of the next paragraph should simply be 'Equation (19)'.
  4. [Theorem 4 proof] The notation S_Q^(I) is used in Eqs. (8) through (11) without being defined; it presumably denotes the isotropic subgroup of S_Q, but this should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is a direct specialization of the independently published OAQEC condition (Eq. 1), and the distance, EACQ, and construction results are derived, not assumed.

full rationale

The derivation chain is self-contained given the cited OAQEC stabilizer-formalism theorem [24]. Theorem 1's proof explicitly says "This result comes as a direct application of the conditions given in Eq. (1)," and Eq. (1) is quoted from [24]. Although three authors of the present paper are also authors of [24], that citation is not a circular reduction: [24] is a separately published, parameter-free theorem whose assumptions do not include the EAOAQEC target, and the present paper supplies the specialization to E⊗I errors and the noiseless-on-Bob coset transversal. The distance definition (Eq. 3) is a complement of the correctable set, not a fitted quantity; the distance bounds and EACQ representability criteria are proved from definitions. The main caveats are non-circular: the existence of the noiseless transversal is sketched rather than fully proved, and the §5.1 notation leaves ambiguous whether H in Theorem 1 includes H_G (Z(H) vs Z(⟨H_I,H_E⟩)); if H includes gauge generators, Theorem 1's proof would need revision. These are correctness/rigor concerns, not cases where a conclusion is equivalent to its input by construction. Therefore no circular step is identified.

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

No free parameters are fitted to data; the paper is a pure algebraic framework. No new physical entities, particles, or forces are introduced. The main external axiom is Eq. (1) from the authors' prior OAQEC stabilizer formalism paper [24].

assumptions (5)
  • domain assumption The OAQEC stabilizer error correction condition of Eq. (1) from [24] is assumed as a theorem.
    Theorem 1 is proven by applying Eq. (1) directly; the paper does not derive Eq. (1) itself.
  • standard math The isotropic-symplectic decomposition of Pauli subgroups and the fact that any isomorphism between Pauli subgroups is unitarily implemented up to phases.
    Used in Section 3 to build the Abelian extension S from the non-Abelian H. This is standard in the Pauli group literature.
  • domain assumption A coset transversal for N(S) on the extended space can be chosen with representatives acting as identity on the ebits.
    Stated in Section 4 after Eq. (1) with a sketch; used to simplify Eq. (2) and the distance definition.
  • domain assumption The key physical assumption that Bob's half of the ebits is error-free, so errors are of the form E ⊗ I.
    This is the standard EA channel model stated in Section 4.
  • domain assumption The code-sector construction from [24], including the claim that different normalizer cosets give orthogonal Hilbert-space sectors.
    Used to define hybrid classical-quantum structure via the transversal T0; taken from [24].

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Cite this review

Pith. "Pith review of Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction." pith.science (2026). https://pith.science/paper/FHHP2LK3

@misc{pith2026241114389,
  author       = {Pith},
  title        = {Pith review of: Unified and Generalized Approach to Entanglement-Assisted Quantum Error Correction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FHHP2LK3}},
  note         = {Machine review of arXiv:2411.14389}
}
read the original abstract

We introduce a framework for entanglement-assisted quantum error correcting codes that unifies the three original frameworks for such codes called EAQEC, EAOQEC, and EACQ under a single umbrella. The unification is arrived at by viewing entanglement-assisted codes from the operator algebra quantum error correction perspective, and it is built upon a recently established extension of the stabilizer formalism to that setting. We denote the framework by EAOAQEC, and we prove a general error correction theorem for such codes, derived from the algebraic perspective, that generalizes each of the earlier results. This leads us to a natural notion of distance for such codes, and we derive a number of distance results for subclasses of the codes. We show how EACQ codes form a proper subclass of the entanglement-assisted subspace codes defined by EAOAQEC. We identify and construct new classes of entanglement-assisted subsystem codes and entanglement-assisted hybrid classical-quantum codes that are found outside of the earlier approaches.

Figures

Figures reproduced from arXiv: 2411.14389 by the authors.

Figure 1
Figure 1. Hierarchy of entanglement-assisted error correction frameworks. Arrows indicate proper inclusions of code types. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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Works this paper leans on

82 extracted references · 32 canonical work pages

  1. [24]

    Kribs, and Michael Vasmer

    Guillaume Dauphinais, David W. Kribs, and Michael Vasmer. Stabilizer formalism for operator algebra quantum error correction.Quantum, 8:1261, 2024. doi:10.22331/q-2024-02-21-1261

    doi:10.22331/q-2024-02-21-1261 2024

  2. [1]

    Improved simulation of stabilizer circuits

    Scott Aaronson and Daniel Gottesman. Improved simulation of stabilizer circuits. Phys. Rev. A, 70:052328, 2004

    2004

  3. [2]

    The black hole interior from non-isometric codes and complexity.arXiv preprint arXiv:2207.06536, 2022

    Chris Akers, Netta Engelhardt, Daniel Harlow, Geoff Penington, and Shreya Vardhan. The black hole interior from non-isometric codes and complexity.arXiv preprint arXiv:2207.06536, 2022. doi: 10.48550/arXiv.2207.06536

    arXiv doi:10.48550/arxiv.2207.06536 2022

  4. [3]

    Quantum minimal surfaces from quantum error correction.SciPost Phys., 12:157, 2022

    Chris Akers and Geoff Penington. Quantum minimal surfaces from quantum error correction.SciPost Phys., 12:157, 2022. doi:10.21468/SciPostPhys.12.5.157

    doi:10.21468/scipostphys.12.5.157 2022

  5. [4]

    Holographic renyi entropy from quantum error correction

    Chris Akers and Pratik Rath. Holographic renyi entropy from quantum error correction. JHEP, 2019(5):52, 2019. doi:10.1007/JHEP05(2019)052. 30

    work page doi:10.1007/jhep05(2019)052 2019

  6. [5]

    Holographic quantum error correction and the projected black hole interior.arXiv preprint arXiv:1810.02055, 2018

    Ahmed Almheiri. Holographic quantum error correction and the projected black hole interior.arXiv preprint arXiv:1810.02055, 2018. doi:10.48550/arXiv.1810.02055

    arXiv doi:10.48550/arxiv.1810.02055 2018

  7. [6]

    Bulk locality and quantum error correction in AdS/CFT

    Ahmed Almheiri, Xi Dong, and Daniel Harlow. Bulk locality and quantum error correction in AdS/CFT. JHEP, 2015(4):163, 2015. doi:10.1007/JHEP04(2015)163

    doi:10.1007/jhep04(2015)163 2015

  8. [7]

    Aly and Andreas Klappenecker

    Salah A. Aly and Andreas Klappenecker. Subsystem code constructions. In2008 IEEE International Symposium on Information Theory (ISIT), pages 369–373, 2008.doi:10.1109/ISIT.2008.4595010

    work page arXiv 2008

Show all 82 references
  1. [8]

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

    Dave Bacon. Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A, 73:012340, 2006. doi:10.1103/PhysRevA.73.012340

    2006 doi

  2. [9]

    Cédric Bény, Achim Kempf, and David W. Kribs. Generalization of quantum error correction via the Heisenberg picture. Phys. Rev. Lett., 98:100502, 2007. doi:10.1103/PhysRevLett.98.100502

    2007 doi

  3. [10]

    Cédric Bény, Achim Kempf, and David W. Kribs. Quantum error correction of observables.Phys. Rev. A, 76:042303, 2007. doi:10.1103/PhysRevA.76.042303

    2007 doi

  4. [11]

    Quantum error correction on infinite-dimensional Hilbert spaces

    Cédric Bény, Achim Kempf, and David W Kribs. Quantum error correction on infinite-dimensional Hilbert spaces. J. Math. Phys., 50(6):062108, 2009. doi:10.1063/1.3155783

    2009 doi

  5. [12]

    Single-shot fault-tolerant quantum error correction

    Héctor Bombín. Single-shot fault-tolerant quantum error correction. Phys. Rev. X, 5:031043, 2015. doi:10.1103/PhysRevX.5.031043

    2015 doi

  6. [13]

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

    Héctor Bombín. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes. New J. Phys., 17(8):083002, 2015. doi:10.1088/1367-2630/17/8/083002

    2015 doi

  7. [14]

    Eli Bourassa, Rafael N

    J. Eli Bourassa, Rafael N. Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q. Baragiola, Saikat Guha, Guillaume Dauphinais, Krishna K. Sabapathy, Nicolas C. Menicucci, and Ish Dhand. Blueprint for a scalable photonic fault-tolerant quan...

    2021 doi

  8. [15]

    GHZ extraction yield for multipartite stabilizer states

    Sergey Bravyi, David Fattal, and Daniel Gottesman. GHZ extraction yield for multipartite stabilizer states. J. Math. Phys., 47(6):062106, 2006. doi:10.1063/1.2203431

    2006 doi

  9. [16]

    Correcting quantum errors with entanglement.Science, 314(5798):436–439, 2006

    Todd Brun, Igor Devetak, and Min-Hsiu Hsieh. Correcting quantum errors with entanglement.Science, 314(5798):436–439, 2006. doi:10.1126/science.1131563

    2006 doi

  10. [17]

    Catalytic quantum error correction.IEEE Trans

    Todd A Brun, Igor Devetak, and Min-Hsiu Hsieh. Catalytic quantum error correction.IEEE Trans. Inf. Theory, 60(6):3073–3089, 2014. doi:10.1109/TIT.2014.2313559

    2014

  11. [18]

    A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane. Quantum error correction and orthogonal geometry.Phys. Rev. Lett., 78:405–408, 1997. doi:10.1103/PhysRevLett.78.405

    1997 doi

  12. [19]

    A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. Phys. Rev. A, 54:1098–1105, 1996. doi:10.1103/PhysRevA.54.1098

    1996 doi

  13. [20]

    Higher rank matricial ranges and hybrid quantum error correction.Linear Multilinear Alg., 69(5):827–839,

    Ningping Cao, David W Kribs, Chi-Kwong Li, Mike I Nelson, Yiu-Tung Poon, and Bei Zeng. Higher rank matricial ranges and hybrid quantum error correction.Linear Multilinear Alg., 69(5):827–839,

  14. [21]

    Quantum teleportation in the commuting operator framework

    Alexandre Conlon, Jason Crann, David W Kribs, and Rupert H Levene. Quantum teleportation in the commuting operator framework. Ann. Henri Poincaré, 24(5):1779–1821, 2023. doi:10.1007/ s00023-022-01255-0. 31

    2023

  15. [22]

    Private algebras in quantum information and infinite-dimensional complementarity.J

    Jason Crann, David W Kribs, Rupert H Levene, and Ivan G Todorov. Private algebras in quantum information and infinite-dimensional complementarity.J. Math. Phys., 57(1), 2016. doi:10.1063/1. 4935399

    2016 doi

  16. [23]

    State convertibility in the von Neumann algebra framework

    Jason Crann, David W Kribs, Rupert H Levene, and Ivan G Todorov. State convertibility in the von Neumann algebra framework. Commun. Math. Phys., 378(2):1123–1156, 2020. doi:10.1007/ s00220-020-03803-3

    2020

  17. [25]

    The capacity of a quantum channel for simultaneous transmission of classical and quantum information.Commun

    Igor Devetak and Peter W Shor. The capacity of a quantum channel for simultaneous transmission of classical and quantum information.Commun. Math. Phys., 256(2):287–303, 2005. doi:10.1007/ s00220-005-1317-6

    2005

  18. [26]

    Entanglement in the stabilizer formalism.arXiv preprint quant-ph/0406168, 2004

    David Fattal, Toby S Cubitt, Yoshihisa Yamamoto, Sergey Bravyi, and Isaac L Chuang. Entanglement in the stabilizer formalism.arXiv preprint quant-ph/0406168, 2004. doi:10.48550/arXiv.quant-ph/ 0406168

    2004 doi arXiv

  19. [27]

    Class of quantum error-correcting codes saturating the quantum Hamming bound

    Daniel Gottesman. Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A, 54(3):1862, 1996. doi:10.1103/PhysRevA.54.1862

    1996 doi

  20. [28]

    Stabilizer codes and quantum error correction

    Daniel Gottesman. Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, 1997.doi:10.48550/arXiv.quant-ph/9705052

    1997 doi arXiv

  21. [29]

    Encoding a qubit in an oscillator.Phys

    Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator.Phys. Rev. A, 64(1):012310, 2001. doi:/10.1103/PhysRevA.64.012310

    2001 doi

  22. [30]

    Codes for simultaneous transmission of quantum and classical information

    Markus Grassl, Sirui Lu, and Bei Zeng. Codes for simultaneous transmission of quantum and classical information. In2017 IEEE International Symposium on Information Theory (ISIT), pages 1718–1722,

  23. [31]

    Punctur- ing quantum stabilizer codes, 2024

    Jaron Skovsted Gundersen, René Bødker Christensen, Petar Popovski, and Rafał Wisniewski. Punctur- ing quantum stabilizer codes, 2024. URL:https://arxiv.org/abs/2410.17754, arXiv:2410.17754

    2024 arXiv

  24. [32]

    The Ryu–Takayanagi formula from quantum error correction.Commun

    Daniel Harlow. The Ryu–Takayanagi formula from quantum error correction.Commun. Math. Phys., 354(3):865–912, 2017. doi:10.1007/s00220-017-2904-z

    2017 doi

  25. [33]

    Hastings and Jeongwan Haah

    Matthew B. Hastings and Jeongwan Haah. Dynamically generated logical qubits.Quantum, 5:564,

  26. [34]

    Learning the alpha-bits of black holes.JHEP, 2019(12):7,

    Patrick Hayden and Geoffrey Penington. Learning the alpha-bits of black holes.JHEP, 2019(12):7,

    2019

  27. [35]

    Approximate quantum error correction revisited: Introducing the alpha-bit

    Patrick Hayden and Geoffrey Penington. Approximate quantum error correction revisited: Introducing the alpha-bit. Commun. Math. Phys., 374(2):369–432, 2020. doi:10.1007/s00220-020-03689-1

    2020 doi

  28. [36]

    doi:10.22331/q-2021-10-19-564

    2021 doi

  29. [37]

    Breuckmann

    Oscar Higgott and Nikolas P. Breuckmann. Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead.Phys. Rev. X, 11:031039, 2021. doi:10.1103/PhysRevX.11.031039

    2021 doi

  30. [38]

    General entanglement-assisted quantum error- correcting codes

    Min-Hsiu Hsieh, Igor Devetak, and Todd Brun. General entanglement-assisted quantum error- correcting codes. Phys. Rev. A, 76:062313, 2007. doi:10.1103/PhysRevA.76.062313. 32

    2007 doi

  31. [39]

    Entanglement-assisted communication of classical and quantum information

    Min-Hsiu Hsieh and Mark M Wilde. Entanglement-assisted communication of classical and quantum information. IEEE Trans. Inf. Theory, 56(9):4682–4704, 2010. doi:10.1109/TIT.2010.2053903

    2010

  32. [40]

    Black holes as mirrors: quantum information in random subsystems

    Patrick Hayden and John Preskill. Black holes as mirrors: quantum information in random subsystems. JHEP, 2007(09):120, 2007. doi:10.1088/1126-6708/2007/09/120

    2007 doi

  33. [41]

    High performance entanglement-assisted quantum ldpc codes need little entanglement.IEEE Trans

    Min-Hsiu Hsieh, Wen-Tai Yen, and Li-Yi Hsu. High performance entanglement-assisted quantum ldpc codes need little entanglement.IEEE Trans. Inf. Theory, 57(3):1761–1769, 2011. doi:10.1109/TIT. 2011.2104590

    2011

  34. [42]

    The Ryu-Takayanagi formula from quantum error correction: an algebraic treatment of the boundary CFT.arXiv preprint arXiv:1912.02240, 2019

    Helia Kamal and Geoffrey Penington. The Ryu-Takayanagi formula from quantum error correction: an algebraic treatment of the boundary CFT.arXiv preprint arXiv:1912.02240, 2019. doi:10.48550/ arXiv.1912.02240

    1912 doi arXiv

  35. [43]

    Theory of quantum error-correcting codes.Phys

    Emanuel Knill and Raymond Laflamme. Theory of quantum error-correcting codes.Phys. Rev. A, 55(2):900, 1997. doi:10.1103/PhysRevA.55.900

    1997 doi

  36. [44]

    Trading classical communication, quantum communication, and entanglement in quantum Shannon theory.IEEE Trans

    Min-Hsiu Hsieh and Mark M Wilde. Trading classical communication, quantum communication, and entanglement in quantum Shannon theory.IEEE Trans. Inf. Theory, 56(9):4705–4730, 2010. doi: 10.1109/TIT.2010.2054532

    2010

  37. [45]

    Unified and generalized approach to quantum error correction

    David Kribs, Raymond Laflamme, and David Poulin. Unified and generalized approach to quantum error correction. Phys. Rev. Lett., 94:180501, 2005. doi:10.1103/PhysRevLett.94.180501

    2005 doi

  38. [46]

    Kribs, Raymond Laflamme, David Poulin, and Maia Lesosky

    David W. Kribs, Raymond Laflamme, David Poulin, and Maia Lesosky. Operator quantum error correction. Quantum Inf. Comput., 6:383–399, 2006. doi:10.26421/QIC6.4-5-6

    2006 doi

  39. [47]

    Single-shot quantum error correction with the three-dimensional subsystem toric code

    Aleksander Kubica and Michael Vasmer. Single-shot quantum error correction with the three-dimensional subsystem toric code. Nat. Commun., 13(1):6272, 2022. doi:10.1038/ s41467-022-33923-4

    2022

  40. [48]

    Classical enhancement of quantum-error-correcting codes

    Isaac Kremsky, Min-Hsiu Hsieh, and Todd A Brun. Classical enhancement of quantum-error-correcting codes. Phys. Rev. A, 78(1):012341, 2008. doi:10.1103/PhysRevA.78.012341

    2008 doi

  41. [49]

    Kuperberg

    G. Kuperberg. The capacity of hybrid quantum memory.IEEE Trans. Inf. Theory, 49(6):1465–1473,

  42. [50]

    Entanglement-assisted quantum error-correcting codes with imperfect ebits

    Ching-Yi Lai and Todd A Brun. Entanglement-assisted quantum error-correcting codes with imperfect ebits. Phys. Rev. A, 86(3):032319, 2012. doi:10.1103/PhysRevA.86.032319

    2012 doi

  43. [51]

    Error correction schemes for fully correlated quantum channels protecting both quantum and classical information.Quantum Inf

    Chi-Kwong Li, Seth Lyles, and Yiu-Tung Poon. Error correction schemes for fully correlated quantum channels protecting both quantum and classical information.Quantum Inf. Process., 19(5):1–17, 2020. doi:10.1007/s11128-020-02639-z

    2020 doi

  44. [52]

    The encoding and decoding complexities of entanglement-assisted quantum stabilizer codes

    Kao-Yueh Kuo and Ching-Yi Lai. The encoding and decoding complexities of entanglement-assisted quantum stabilizer codes. In 2019 IEEE International Symposium on Information Theory (ISIT), pages 2893–2897, 2019. doi:10.1109/ISIT.2019.8849380

    2019

  45. [53]

    Non-binary entanglement-assisted stabilizer codes

    Priya J Nadkarni and Shayan Srinivasa Garani. Non-binary entanglement-assisted stabilizer codes. Quantum Inf. Process, 20(8):256, 2021. doi:10.1007/s11128-021-03174-1

    2021 doi

  46. [54]

    Nadkarni and Shayan Srinivasa Garani

    Priya J. Nadkarni and Shayan Srinivasa Garani. Quantum error correction architecture for qudit stabilizer codes. Phys. Rev. A, 103:042420, 2021. doi:10.1103/PhysRevA.103.042420

    2021 doi

  47. [55]

    Quantum data-syndrome codes: Subsystem and impure code constructions.Quantum Inf

    Andrew Nemec. Quantum data-syndrome codes: Subsystem and impure code constructions.Quantum Inf. Process., 22(11):408, 2023. doi:10.1142/S0219749921500416. 33

    2023 doi

  48. [56]

    Infinite families of quantum-classical hybrid codes.IEEE Trans

    Andrew Nemec and Andreas Klappenecker. Infinite families of quantum-classical hybrid codes.IEEE Trans. Inf. Theory, 67(5):2847–2856, 2021. doi:10.1109/TIT.2021.3051037

    2021

  49. [57]

    Nadkarni and Shayan Srinivasa Garani

    Priya J. Nadkarni and Shayan Srinivasa Garani. Encoding of nonbinary entanglement-unassisted and assisted stabilizer codes.IEEE Trans. Quantum Eng., 2:1–22, 2021.doi:10.1109/TQE.2021.3050848

    2021

  50. [58]

    Entanglement wedge reconstruction and the information paradox

    Geoffrey Penington. Entanglement wedge reconstruction and the information paradox. JHEP, 2020(9):2, 2020. doi:10.1007/JHEP09(2020)002

    2020 doi

  51. [59]

    Stabilizer formalism for operator quantum error correction.Phys

    David Poulin. Stabilizer formalism for operator quantum error correction.Phys. Rev. Lett., 95:230504,

  52. [60]

    E.M. Rains. Nonbinary quantum codes.IEEE Transactions on Information Theory, 45(6):1827–1832,

  53. [61]

    Encoding subsystem codes with and without noisy gauge qubits

    Pradeep Kiran Sarvepalli and Andreas Klappenecker. Encoding subsystem codes with and without noisy gauge qubits. In2009 Third International Conference on Quantum, Nano and Micro Technolo- gies, pages 48–53, 2009.doi:10.1109/ICQNM.2009.10

    2009 doi

  54. [62]

    Reichardt

    Adam Paetznick and Ben W. Reichardt. Universal fault-tolerant quantum computation with only transversal gates and error correction.Phys. Rev. Lett., 111:090505, 2013.doi:10.1103/PhysRevLett. 111.090505

    2013 doi

  55. [63]

    Fault-tolerantquantumcomputation

    PeterW.Shor. Fault-tolerantquantumcomputation. In Proceedings of 37th Conference on Foundations of Computer Science, pages 56–65, 1996.doi:10.1109/SFCS.1996.548464

    1996

  56. [64]

    Synchronizable hybrid subsystem codes.arXiv preprint arXiv:2409.11312, 2024

    Theerapat Tansuwannont and Andrew Nemec. Synchronizable hybrid subsystem codes.arXiv preprint arXiv:2409.11312, 2024. doi:10.48550/arXiv.2409.11312

    2024 doi arXiv

  57. [65]

    Black hole entanglement and quantum error correction.JHEP, 2013(10):107, 2013

    Erik Verlinde and Herman Verlinde. Black hole entanglement and quantum error correction.JHEP, 2013(10):107, 2013. doi:10.1007/JHEP10(2013)107

    2013 doi

  58. [66]

    Wilde, Hari Krovi, and Todd A

    Mark M. Wilde, Hari Krovi, and Todd A. Brun. Convolutional entanglement distillation. In2010 IEEE International Symposium on Information Theory, pages 2657–2661, 2010. doi:10.1109/ISIT. 2010.5513666

    2010

  59. [67]

    Recovery algorithms for clifford Hayden-Preskill problem

    Beni Yoshida. Recovery algorithms for clifford Hayden-Preskill problem. arXiv preprint arXiv:2106.15628, 2021. doi:10.48550/arXiv.2106.15628. 34 A 15-qubit subsystem color code We consider a [ [15, 1, 3; 6, 0, 1] ]subsystem color code [13,57] with distance 3 whose stabilizer a...

    2021 doi arXiv

  60. [69]

    Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A, 52:R2493–R2496, 1995. doi:10.1103/PhysRevA.52.R2493

    1995 doi

  61. [75]

    Let f = |EQ|

    Let EQ denote the set of qubit indices such that no Pauli operator whose support is a subset ofEQ belongs to Z(S). Let f = |EQ|. For obtaining an EAOAQEC code with f ebits with1 ≤ f <min wt(Z(S)), EQ can be chosen to be anyf qubit indices

  62. [76]

    ConstructH ′ from H by permuting the columns ofH1 and H2 corresponding to the qubit indices in EQ to the first2f columns

    Obtain the check matrixH = [H1|H2] of the stabilizer code by stacking the binary representation of stabilizer generators. ConstructH ′ from H by permuting the columns ofH1 and H2 corresponding to the qubit indices in EQ to the first2f columns. Perform partial Gaussian eliminat...

  63. [77]

    Consider the 2f stabilizer generators Sj corresponding to the pivotal rows in the Gaussian elimination procedure to be the extended symplectic pairs and the rest stabilizer generators to correspond to the extended isotropic operators. The qubits with indices inEQ correspond to...

  64. [78]

    Multiply each elementT of T0 with the corresponding stabilizer generators Sj that have non-identity operators on the ebits to obtain the productT ′ to have identity on the ebits.T (n−f ) 0 ’ is obtained to be the restriction ofT ′ to the qubit indices not inEQ. return H and T ...

  65. [79]

    Obtain the check matrixH = H 0 0 H of the CSS code

  66. [80]

    Let H (G) be the matrix obtained after Gaussian elimination procedure

    Using Gaussian elimination, convert theH into row reduced echelon form. Let H (G) be the matrix obtained after Gaussian elimination procedure. Let Sjs be the stabilizer generators corresponding to the check matrix H = H (G) 0 0 H (G) . Let f = rank(H)

  67. [81]

    The qubits corresponding to the pivotal columns in the Gaussian elimination procedure are the ebits while the rest are Alice’s qubits.H is obtained from the restriction of elements ofSj to Alice’s qubit indices

  68. [82]

    Multiply each elementT of T0 with the corresponding stabilizer generatorsSj that have non-identity operators on the ebits to obtain the productT ′ to have identity on the ebits.T ′(n−f ) 0 is obtained to be the restriction ofT ′ to Alice’s qubit indices. return H and T ′(n−f )...

  69. [1999]

    doi:10.1109/18.782103

    doi

  70. [2003]

    doi:10.1109/TIT.2003.811917

    2003

  71. [2005]

    doi:10.1103/PhysRevLett.95.230504

    doi

  72. [2017]

    doi:10.1109/ISIT.2017.8006823

    2017

  73. [2019]

    doi:10.1007/JHEP12(2019)007

    2019 doi

  74. [2021]

    doi:10.1080/03081087.2020.1748852

    2020

Pith tools

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