Transversal Gates in Nonadditive Quantum Codes

Bei Zeng; Chao Zhang; Shilin Huang; Zipeng Wu

REVIEW 3 major objections 4 minor 39 references

Transversal Gates in Nonadditive Quantum Codes

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read This paper claims a family of nonadditive quantum codes whose transversal gate groups include a 6-qubit C10 code, 7-qubit codes with T and √T gates, and 8-qubit codes with T^{1/4}, shrinking the smallest known blocks for non-Clifford…

desk verdict The SS-LP method and most of the small examples look real, but the headline √T code is misprinted, so the paper needs revision before the central claim is credible. read the letter →

arxiv 2504.20847 v1 pith:FMHBSKH3 submitted 2025-04-29 quant-ph

Chao Zhang , Zipeng Wu , Shilin Huang , Bei Zeng This is my paper · ORCID

classification quant-ph MSC 81P7081P6894B60 PACS 03.67.Pp03.67.Lx

keywords nonadditivequantumcodestransversalgatesKnill-LaflammeconditionsbinarydihedralgroupsStiefelmanifoldsubset-sumlinearprogrammingicosahedralgroupfault-tolerantcomputation

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 seeks to show that nonadditive quantum codes can realize a wider variety of transversal logical gates than prior examples suggested, and to supply a systematic way to find them. It claims a new six-qubit distance-3 code whose transversal group is the cyclic group C10, the smallest known distance-3 code supporting a non-Clifford transversal gate, and a family of seven-qubit codes realizing binary dihedral groups BD_{2m} up to 2m=36, including transversal T and √T gates. The authors introduce a two-part strategy: direct search on the Stiefel manifold of code subspaces, and a Subset-Sum-Linear-Programming construction that reduces the search to integer partitions with linear constraints. If the claims hold, fault-tolerant designs gain smaller candidate code blocks for non-Clifford transversal operations.

What carries the argument

The SS-LP construction: assume the target logical diagonal gate is implemented transversally by local Z rotations with integer angles a_j modulo m; then each logical basis state can only be supported on computational basis strings whose weighted sum of bits is congruent to the corresponding logical eigenvalue b_k modulo m. This reduces the search to subset-sum/partition problems, filters candidates with a linear program enforcing Z-type Knill-Laflamme conditions, and finishes with block-separable amplitude optimization. The general search uses Stiefel manifold parameterization of orthonormal two-frames, combined with polar-decomposition updates and a composite loss L_KL + L_gate. The paper states that for BD_{2m} symmetry the X/Y/Z KL conditions simplify, so the LP reduces to a set of mean-zero constraints on supports.

What would settle it

Verify the displayed state in Sec. 6.3.2 by computing its squared norm and the angle sum ∑ a_j mod m. If the norm evaluates to 31 instead of 32, or the sum is -2 mod m instead of the required -1 for the BD32 example (and similarly for BD64 and BD84), then those printed examples do not realize the claimed transversal groups; a corrected example list would settle the SS-LP claim.

Watch

Extended reading notes

Core claim

The central claim is that the landscape of nonadditive ((n,2,3)) codes is much richer than stabilizer or permutation-invariant constructions alone would suggest: every requested finite subgroup of SU(2) that survives the search can be realized as the transversal group of a valid code. Specifically, the paper constructs a ((6,2,3)) code with transversal Z(2π/5), generating C10; several ((7,2,3)) codes with transversal 2I at signature norms λ* = 0 and √(3/4); and SS-LP-derived ((7,2,3)) codes with BD_{2m} for all 2m≤36, including the first 7-qubit examples with transversal T (BD16) and √T (BD32), which beat the previously smallest ((11,2,3)) and ((19,2,3)) constructions. For eight qubits it finds BD_{2m} for 2m>36, including a BD64 code supporting $T^{{1/4}}$. The authors argue these examples show a continuum of nonadditive codes parameterized by the signature norm and connected to algebraic constraints on transversal gate groups.

Load-bearing premise

The load-bearing premise is that the printed codeword formulas, including normalization and the angle-sum condition ∑ a_j ≡ -1 (mod m) of Eq. (15), are accurate as written; if any displayed example fails these checks, its claimed transversal group is not actually realized.

Editorial extensions

If this is right

If the claims hold, a distance-3 quantum code with a non-Clifford transversal gate can be as small as six qubits, below the previous known block sizes.
Transversal T and √T gates, normally associated with larger stabilizer or permutation-invariant codes, can be realized in 7-qubit nonadditive codes, and T^{1/4} in an 8-qubit code.
The observed saturation at BD36 in 7-qubit codes suggests a finite limit on the order of diagonal transversal groups for a given block length, guiding future search strategies.
The signature-norm parametrization links continuous families of nonadditive codes to distinct transversal groups, providing new invariants for comparing code constructions.

Reading between the lines

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

If the examples are verified, the apparent saturation at 2m=36 for n=7 suggests a general relationship between physical qubit number and the maximal order of a diagonal transversal gate; a testable conjecture is that BD_{2m} at n=7 requires 2m to be bounded by a linear function of n, which the authors did not state.
The SS-LP reduction could plausibly extend to K>2 logical dimensions or higher distance d>3 by replacing the single-congruence condition with a system of congruences, though the paper only treats K=2, d=3.
The C10 ((6,2,3)) code implies that 5-fold rotations, not just Clifford-hierarchy gates, can be transversal, offering a possible route to non-Clifford fault-tolerant operations on very small code blocks.
A direct numerical check of the printed angle vectors for the BD32, BD64, and BD84 examples would settle whether those displayed states realize the claimed groups; if not, the SS-LP method may still be valid but those particular examples would require correction.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Ed:

The paper has a real idea and several apparently correct constructions, but the headline claim for the smallest transversal √T code rests on an example that fails basic arithmetic as printed. The BD32 state in §6.3.2 is not normalized (squared norm 31, not 32), and the displayed local Z rotations multiply to a logical phase of e^{-iπ/16} rather than the claimed e^{-iπ/8}. Moreover, the rotation angles include odd multiples of π/16, which cannot come from the SS-LP ansatz with integer a_j for m=16. So the claimed ((7,2,3)) sqrt(T) improvement over the ((19,2,3)) code is not supported by the manuscript.

That said, the paper does more than hand-wave. The SS-LP reduction to integer partitions subject to congruence constraints is a genuinely useful way to shrink the search for codes with transversal diagonal gates. The ((6,2,3)) C10 code in §4.1 checks out: the state is normalized, the phases satisfy the constraint equations, and the tensor product of Z rotations gives the claimed logical Z(-2π/5). The BD16 example in §6.3.1 also works — the angle vector sums to -1 mod 8 and the state has norm 16 on the 4|0L> scale. The 2I examples in §5 are plausible, though I did not verify every coefficient. The BD36 example also passes the same mechanical checks.

The reader's take flagged BD64 and BD84 as also broken, but that is not right: both use even angle numerators, giving integer a_j, and their sums satisfy ∑a_j ≡ -1 mod m. The problem is specific to BD32. That is still a serious problem because it is the distinguishing new example. The paper gives no verification code or certificates, and the search method is numerical, so the reader cannot easily separate transcription errors from actual failures. The weight enumerator formulas throughout could be checked symbolically, but without code that is laborious.

Who should read this: anyone working on nonadditive codes or transversal gate groups. The method and the correct examples are a step forward. But the paper as submitted is not trustworthy for the √T claim, and the lack of code makes it hard to verify the rest.

Recommendation: send to peer review, but require the authors to (i) correct or replace the BD32 example, (ii) add a script or certificate verifying the KL conditions for each claimed code, and (iii) reconcile the discrepancy with the abstract. With those fixes, the paper would be a solid contribution. If the BD32 example cannot be repaired, the stronger claim should be removed.

Referee Report

3 major / 4 minor

Summary. The paper introduces two computational methods for constructing quantum codes with prescribed transversal gate groups: a Stiefel-manifold optimization with composite Knill–Laflamme/gate losses, and a Subset-Sum-Linear-Programming (SS-LP) construction for codes with transversal diagonal gates. The authors report a ((6,2,3)) code with transversal group C10, new ((7,2,3)) codes with binary icosahedral group 2I, a family of ((7,2,3)) codes with binary dihedral groups BD_{2m} for 2m≤36 (including claimed first ((7,2,3)) codes with transversal T and sqrt(T) gates), and ((8,2,3)) codes for larger BD_{2m}, including a T^{1/4} example.

Significance. If the explicit constructions are correct, the paper would be significant: it provides a systematic search framework, substantially reduces the known qubit counts for transversal non-Clifford gates, and expands the known landscape of nonadditive codes. The paper's strengths include explicit logical states, explicit tensor-product gate implementations, and weight-enumerator data for many examples, connected to the signature-norm invariant. However, the central BD32 example fails elementary arithmetic checks as printed, and no verification code or certificate is supplied, so the headline 'smallest sqrt(T) code' claim is currently unsubstantiated.

major comments (3)

[Sec. 6.3.2] The displayed BD32 logical state is not normalized: the coefficients in the formula for sqrt(32)|0L> have squared moduli 1 + (1+32θ^2) + (6−32θ^2) + 4 + 3 + 7 + 5 + (4−32θ^2) + 32θ^2 = 31, with the θ^2 contributions cancelling. Thus the state as printed does not define a valid code subspace.
[Sec. 6.3.2] The displayed BD32 transversal gate is inconsistent with the SS-LP convention. For m=16 the local diagonal gates must be diag(1, exp(2π i a_j/16)) = e^{iπ a_j/8} Z(a_jπ/8) with integer a_j, so the local rotation angles must be even multiples of π/16. The printed gate uses Z(3π/16), Z(5π/16), and Z(7π/16), which are odd multiples and cannot arise from integer a_j. Moreover the product of the displayed rotations is Z(31π/16) = −Z(−π/16), whose relative phase is e^{−iπ/16}, not the claimed Z(−2π/16) = Z(−π/8) relative phase e^{−iπ/8}. The angle-sum condition (15) also fails for the implied half-integer a_j. The sqrt(T) claim is therefore unsubstantiated as printed.
[Secs. 3-6 and Appendices D-E] Because the only evidence for existence of the claimed codes is the printed logical states and gates, and because the BD32 example fails both normalization and gate-action checks, the authors should supply a machine-checkable verification (for example, a script that checks normalization, the Knill-Laflamme conditions for all weight-1 and weight-2 Pauli errors, and equality of the tensor-product gate with the claimed logical gate) for every explicit code in the main text and appendices. Without such verification, the existence claims for the headline examples cannot be accepted.

minor comments (4)

[Sec. 6.3.2] The text reads 'He we present one example'; this should be 'Here we present one example'.
[Sec. 5.1] The B weight enumerator for the λ* = sqrt(3/4) 2I code appears garbled in the printed expression ('99/2 096t2'); please correct and re-check this formula.
[Throughout] There are several typographical slips such as 'stablizer' (Sec. 6.3.1), 'T able' in table captions, and inconsistent spacing in 'a nd', 'y et', and related words; these should be cleaned up.
[Table 3] The λ* intervals in Table 3 are described as 'numerically estimated'; the paper should state explicitly which entries are proven exact values and which are only numerical evidence from the optimization.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: target gates are imposed by construction; KL conditions are checked independently.

full rationale

The paper's central claims are existence claims for explicit nonadditive code states. The SS-LP construction (Sec. 6) fixes an angle vector a and defines support subsets S_k by the congruence sum_j a_j s_j ≡ b_k (mod m); this makes the tensor product of local Z rotations act as the desired diagonal logical gate by construction. The nontrivial content is the Knill–Laflamme condition, which is not implied by that congruence and is instead enforced through the LP filter and the subsequent amplitude optimization (Sec. 6.1, Steps 2 and 3). No fitted parameter is later relabeled as a prediction, and no equation has the claimed result as the value of a fit. The use of the authors' earlier invariant λ* [11] is background and independently checkable; the new code states are displayed explicitly and can be verified directly. The paper honestly notes numerical limitations (possible local minima in Sec. 4; deferral of X/Y KL verification in Sec. 6.1), which are completeness caveats rather than circular reductions. One displayed example, BD32 in Sec. 6.3.2, appears internally inconsistent as printed (squared norm 31 rather than 32, and local angles with odd numerators k=3,5,7 outside the SS-LP integer-a_j convention); this is a correctness risk in a specific example, not a circularity of the derivation. Overall, the construction is self-contained against the explicit states, and the claimed results do not reduce to their inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The constructions are explicit and no new physical entities are introduced. The main ledger items are standard QEC axioms plus the unsupported assumption that the numerical search and printed formulas exactly certify the KL conditions. That assumption is contradicted by several displayed formulas in the BD family.

free parameters (2)

SS-LP angle vector a_j = Various integer vectors, e.g., (1,3,4,4,5,6,7) for BD32 and (8,14,20,24,34,46,48,60) for BD64
The diagonal transversal gate is parameterized by integer angles that are enumerated, not fit to data. The claimed group realization depends on these vectors satisfying Eq. (15), which the printed vectors do not.
Continuous phase and sign parameters in code families = Free ranges or +/-1 signs, e.g., theta in [0, sqrt(1/8)] for BD32
Many advertised codes are families parameterized by continuous phases and sign choices. The KL validity is asserted over these ranges without a complete proof.

assumptions (4)

standard math Finite subgroups of SU(2) classification and the restriction that transversal gates of an ((n,2,d>1)) code form a finite subgroup of SU(2).
Invoked in Secs. 1 and 2.3; standard group theory, cited from [12] and [39].
domain assumption A distance-3 code is verified by the Knill-Laflamme conditions for all Pauli errors of weight at most 2.
Used throughout Secs. 3 and 6; this is the standard definition of quantum code distance.
domain assumption The LP filter on Z-type KL conditions followed by the nonlinear KL optimization is sufficient to certify the full KL conditions.
Sec. 6.1 states this two-stage procedure; no completeness theorem, tolerance, or certificate is supplied.
ad hoc to paper The numerical search on the Stiefel manifold and SU(2) parameterizations converges to exact solutions.
Sec. 3.4 and Table 3 rely on optimizer outputs, but no code, convergence thresholds, or verification scripts are provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Transversal Gates in Nonadditive Quantum Codes." pith.science (2026). https://pith.science/paper/FMHBSKH3

@misc{pith2026250420847,
  author       = {Pith},
  title        = {Pith review of: Transversal Gates in Nonadditive Quantum Codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMHBSKH3}},
  note         = {Machine review of arXiv:2504.20847}
}

abstract

Transversal gates play a crucial role in suppressing error propagation in fault-tolerant quantum computation, yet they are intrinsically constrained: any nontrivial code encoding a single logical qubit admits only a finite subgroup of $\mathrm{SU}(2)$ as its transversal operations. We introduce a systematic framework for searching codes with specified transversal groups by parametrizing their logical subspaces on the Stiefel manifold and minimizing a composite loss that enforces both the Knill-Laflamme conditions and a target transversal-group structure. Applying this method, we uncover a new $((6,2,3))$ code admitting a transversal $Z\bigl(\tfrac{2\pi}{5}\bigr)$ gate (transversal group $\mathrm{C}_{10}$), the smallest known distance $3$ code supporting non-Clifford transversal gates, as well as several new $((7,2,3))$ codes realizing the binary icosahedral group $2I$. We further propose the \emph{Subset-Sum-Linear-Programming} (SS-LP) construction for codes with transversal \emph{diagonal} gates, which dramatically shrinks the search space by reducing to integer partitions subject to linear constraints. In a more constrained form, the method also applies directly to the binary-dihedral groups $\mathrm{BD}_{2m}$. Specializing to $n=7$, the SS-LP method yields codes for all $\mathrm{BD}_{2m}$ with $2m\le 36$, including the first $((7,2,3))$ examples supporting transversal $T$ gate ($\mathrm{BD}_{16}$) and $\sqrt{T}$ gate ($\mathrm{BD}_{32}$), improving on the previous smallest examples $((11,2,3))$ and $((19,2,3))$. Extending the SS-LP approach to $((8,2,3))$, we construct new codes for $2m>36$, including one supporting a transversal $T^{1/4}$ gate ($\mathrm{BD}_{64}$). These results reveal a far richer landscape of nonadditive codes than previously recognized and underscore a deeper connection between quantum error correction and the algebraic constraints on transversal gate groups.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 35 canonical work pages

[1]

Boolean functi ons, projection operators, and quantum error correcting codes

V aneet Aggarwal and A Robert Calderbank. Boolean functi ons, projection operators, and quantum error correcting codes. IEEE T ransactions on Information Theory, 54(4):1700–1707, 2008

work page 2008
[2]

Classiﬁcat ion of transversal gates in qubit stabilizer codes

Jonas T Anderson and T omas Jochym-O’Connor. Classiﬁcat ion of transversal gates in qubit stabilizer codes. Quantum Information & Computation , 16(9-10):771–802, 2016

work page 2016
[3]

Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group

Vikraman Arvind, Piyush P Kurur, and KR Parthasarathy . N onstabilizer quantum codes from abelian subgroups of the error group. arXiv preprint quant-ph/0210097 , 2002. 27

work page Pith review arXiv 2002
[4]

Calderbank, E.M

A.R. Calderbank, E.M. Rains, P .M. Shor, and N.J.A. Sloan e. Quantum error correction via codes over gf(4). IEEE T ransactions on Information Theory, 44(4):1369–1387, 1998

work page 1998
[5]

Quantum variational learning for quantum error-correcting codes

Chenfeng Cao, Chao Zhang, Zipeng Wu, Markus Grassl, and B ei Zeng. Quantum variational learning for quantum error-correcting codes. Quantum, 6:828, October 2022

work page 2022
[6]

Codeword stabilized quantum codes: Algo- rithm and structure

Isaac Chuang, Andrew Cross, Graeme Smith, John Smolin, a nd Bei Zeng. Codeword stabilized quantum codes: Algo- rithm and structure. Journal of Mathematical Physics , 50(4), 2009

work page 2009
[7]

C odeword stabilized quantum codes

Andrew Cross, Graeme Smith, John A Smolin, and Bei Zeng. C odeword stabilized quantum codes. IEEE transactions on information theory , 55(1):433–438, 2009

work page 2009
[8]

Small binary stabilizer s ubsystem codes

Andrew Cross and Drew V andeth. Small binary stabilizer s ubsystem codes. arXiv preprint arXiv:2501.17447 , 2025

arXiv 2025

Show all 39 references

[9]

R. H. Dicke. Coherence in spontaneous radiation process es. Phys. Rev., 93:99–110, Jan 1954

1954
[10]

Computing irreducible representations o f groups

John D Dixon. Computing irreducible representations o f groups. Mathematics of Computation , 24(111):707–712, 1970

1970
[11]

Cha racterizing quantum codes via the coefﬁcients in knill- laﬂamme conditions

Mengxin Du, Chao Zhang, Yiu-Tung Poon, and Bei Zeng. Cha racterizing quantum codes via the coefﬁcients in knill- laﬂamme conditions. arXiv preprint arXiv:2410.07983 , 2024

2024 arXiv
[12]

Restrictions on transv ersal encoded quantum gate sets

Bryan Eastin and Emanuel Knill. Restrictions on transv ersal encoded quantum gate sets. Physical review letters , 102(11):110502, 2009

2009
[13]

Stabilizer codes and quantum error c orrection

Daniel Gottesman. Stabilizer codes and quantum error c orrection. arXiv: Quantum Physics , 1997

1997
[14]

A note on non-additive qu antum codes

Markus Grassl and Thomas Beth. A note on non-additive qu antum codes. arXiv preprint quant-ph/9703016 , 1997

1997 arXiv
[15]

Non-additive quant um codes from goethals and preparata codes

Markus Grassl and Martin Rotteler. Non-additive quant um codes from goethals and preparata codes. In 2008 IEEE Information Theory Workshop, pages 396–400. IEEE, 2008

2008
[16]

Theory of quantum er ror-correcting codes

Emanuel Knill and Raymond Laﬂamme. Theory of quantum er ror-correcting codes. Phys. Rev. A, 55:900–911, Feb 1997

1997
[17]

Quantum Error Correction with T ransversal non-Clifford Ga tes

Stergios Koutsioumpas. Quantum Error Correction with T ransversal non-Clifford Ga tes. PhD thesis, Royal Holloway University of London, 2024

2024
[18]

Family of quantum code s with exotic transversal gates

Eric Kubischta and Ian T eixeira. Family of quantum code s with exotic transversal gates. Physical Review Letters , 131(24):240601, 2023

2023
[19]

Permutation-invaria nt quantum codes with transversal generalized phase gates, 2024

Eric Kubischta and Ian T eixeira. Permutation-invaria nt quantum codes with transversal generalized phase gates, 2024

2024
[20]

Trivializations for gradient-b ased optimization on manifolds

Mario Lezcano Casado. Trivializations for gradient-b ased optimization on manifolds. Advances in Neural Information Processing Systems, 32, 2019

2019
[21]

D. A. Lidar, I. L. Chuang, and K. B. Whaley . Decoherence- free subspaces for quantum computation. Phys. Rev. Lett. , 81:2594–2597, Sep 1998

1998
[22]

Lidar and K

Daniel A. Lidar and K. Birgitta Whaley . Decoherence-fr ee subspaces and subsystems. In Fabio Benatti and Roberto Floreanini, editors, Irreversible Quantum Dynamics, pages 83–120. Springer Berlin Heidelberg, Berlin, Heidel berg, 2003

2003
[23]

Experimental measurem ent and a physical interpretation of quantum shadow enumerators

Daniel Miller, Kyano Levi, Lukas Postler, Alex Steiner , Lennart Bittel, Gregory AL White, Yifan T ang, Eric J Kuehnk e, Antonio A Mele, Sumeet Khatri, et al. Experimental measurem ent and a physical interpretation of quantum shadow enumerators. arXiv preprint arXiv:2408.16914 , 2024

2024 arXiv
[24]

A non-commuting stabilizer formalism

Xiaotong Ni, Oliver Buerschaper, and Maarten V an den Ne st. A non-commuting stabilizer formalism. Journal of Mathematical Physics, 56(5), 2015

2015
[25]

Permutation-invariant quantum codes

Yingkai Ouyang. Permutation-invariant quantum codes . Physical Review A , 90(6):062317, 2014

2014
[26]

Permutation-invariant quantum codin g for quantum deletion channels

Yingkai Ouyang. Permutation-invariant quantum codin g for quantum deletion channels. In 2021 IEEE International Symposium on Information Theory (ISIT) , pages 1499–1503. IEEE, 2021

2021
[27]

Permutational ly invariant codes for quantum error correction

Harriet Pollatsek and Mary Beth Ruskai. Permutational ly invariant codes for quantum error correction. Linear Algebra and its Applications , 392:255–288, 2004

2004
[28]

E.M. Rains. Quantum weight enumerators. IEEE T ransactions on Information Theory, 44(4):1388–1394, 1998

1998
[29]

E.M. Rains. Quantum shadow enumerators. IEEE T ransactions on Information Theory, 45(7):2361–2366, 1999. 28

1999
[30]

Eric M. Rains. Quantum codes of minimum distance two. IEEE T ransactions on Information theory, 45(1):266–271, 1999

1999
[31]

Rains, R

Eric M. Rains, R. H. Hardin, Peter W . Shor, and N. J. A. Slo ane. A nonadditive quantum code. Phys. Rev. Lett., 79:953– 954, Aug 1997

1997
[32]

On the structur e of additive quantum codes and the existence of nonad- ditive codes

Vwani P Roychowdhury and Farrokh V atan. On the structur e of additive quantum codes and the existence of nonad- ditive codes. arXiv preprint quant-ph/9710031 , 1997

1997 arXiv
[33]

Quantum analog of the ma cwilliams identities for classical coding theory

Peter Shor and Raymond Laﬂamme. Quantum analog of the ma cwilliams identities for classical coding theory . Phys. Rev. Lett., 78:1600–1602, Feb 1997

1997
[34]

A. M. Steane. Error correcting codes in quantum theory . Phys. Rev. Lett., 77:793–797, Jul 1996

1996
[35]

A. M. Steane. Simple quantum error-correcting codes. Phys. Rev. A, 54:4741–4751, Dec 1996

1996
[36]

The xp stabiliser formalism: a generalisation of the pauli stabiliser formalism with arbitrary phases

Mark A W ebster, Benjamin J Brown, and Stephen D Bartlett . The xp stabiliser formalism: a generalisation of the pauli stabiliser formalism with arbitrary phases. Quantum, 6:815, 2022

2022
[37]

Nonadditive quant um error-correcting code

Sixia Yu, Qing Chen, CH Lai, and CH Oh. Nonadditive quant um error-correcting code. Physical review letters , 101(9):090501, 2008

2008
[38]

Graphical quantum error-correcting codes

Sixia Yu, Qing Chen, and Choo Hiap Oh. Graphical quantum error-correcting codes. arXiv preprint arXiv:0709.1780 , 2007

2007 arXiv
[39]

Transversal ity versus universality for additive quantum codes

Bei Zeng, Andrew Cross, and Isaac L Chuang. Transversal ity versus universality for additive quantum codes. IEEE T ransactions on Information Theory, 57(9):6272–6284, 2011. 29

2011

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Boolean functi ons, projection operators, and quantum error correcting codes

V aneet Aggarwal and A Robert Calderbank. Boolean functi ons, projection operators, and quantum error correcting codes. IEEE T ransactions on Information Theory, 54(4):1700–1707, 2008

work page 2008

[2] [2]

Classiﬁcat ion of transversal gates in qubit stabilizer codes

Jonas T Anderson and T omas Jochym-O’Connor. Classiﬁcat ion of transversal gates in qubit stabilizer codes. Quantum Information & Computation , 16(9-10):771–802, 2016

work page 2016

[3] [3]

Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group

Vikraman Arvind, Piyush P Kurur, and KR Parthasarathy . N onstabilizer quantum codes from abelian subgroups of the error group. arXiv preprint quant-ph/0210097 , 2002. 27

work page Pith review arXiv 2002

[4] [4]

Calderbank, E.M

A.R. Calderbank, E.M. Rains, P .M. Shor, and N.J.A. Sloan e. Quantum error correction via codes over gf(4). IEEE T ransactions on Information Theory, 44(4):1369–1387, 1998

work page 1998

[5] [5]

Quantum variational learning for quantum error-correcting codes

Chenfeng Cao, Chao Zhang, Zipeng Wu, Markus Grassl, and B ei Zeng. Quantum variational learning for quantum error-correcting codes. Quantum, 6:828, October 2022

work page 2022

[6] [6]

Codeword stabilized quantum codes: Algo- rithm and structure

Isaac Chuang, Andrew Cross, Graeme Smith, John Smolin, a nd Bei Zeng. Codeword stabilized quantum codes: Algo- rithm and structure. Journal of Mathematical Physics , 50(4), 2009

work page 2009

[7] [7]

C odeword stabilized quantum codes

Andrew Cross, Graeme Smith, John A Smolin, and Bei Zeng. C odeword stabilized quantum codes. IEEE transactions on information theory , 55(1):433–438, 2009

work page 2009

[8] [8]

Small binary stabilizer s ubsystem codes

Andrew Cross and Drew V andeth. Small binary stabilizer s ubsystem codes. arXiv preprint arXiv:2501.17447 , 2025

arXiv 2025