Asymptotically good CSS codes that realize the logical transversal Clifford group fault-tolerantly

K. Sai Mineesh Reddy; Navin Kashyap

arxiv: 2601.08568 · v2 · submitted 2026-01-13 · 🪐 quant-ph · cs.IT· math.IT

Asymptotically good CSS codes that realize the logical transversal Clifford group fault-tolerantly

K. Sai Mineesh Reddy , Navin Kashyap This is my paper

Pith reviewed 2026-05-16 14:55 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT

keywords CSS codestransversal gatesClifford groupfault toleranceasymptotically good codesCSS-T codesquantum error correctionlogical operations

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The pith

Asymptotically good CSS codes realize the logical Clifford group fault-tolerantly via transversal gates.

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

The paper introduces a framework for constructing CSS codes that support fault-tolerant logical transversal Z-rotations. This yields families of codes with positive rate and growing distance in which the full logical Clifford group is realized transversally: single-qubit Clifford gates act inside one code block, while two-qubit Clifford gates act across two identical blocks. The same framework produces asymptotically good CSS-T codes in which the transversal T gate implements the logical S dagger. It also shows that the inclusion condition C2 * C1 is necessary but not sufficient for CSS-T codes and revises the characterizations of when the transversal T acts as the logical identity or logical T. These results matter because they show how to perform an entire universal gate set for Clifford operations without leaving the code space or adding extra blocks for single-qubit gates.

Core claim

Using a construction framework based on classical linear codes that satisfy suitable inclusion and distance properties, the authors obtain asymptotically good CSS codes in which transversal gates realize the logical Clifford group fault-tolerantly, with single-qubit Cliffords implemented within a single code block and two-qubit Cliffords across two identical blocks. They further obtain asymptotically good CSS-T codes where the transversal T gate realizes the logical S dagger, demonstrate that the condition C2 * C1 ⊆ C1⊥ is necessary but not sufficient for CSS-T codes, and revise the characterizations of CSS-T codes for the cases where the transversal T implements the logical identity and the

What carries the argument

A framework that lifts families of classical linear codes with appropriate inclusion relations and minimum distances to CSS codes supporting fault-tolerant transversal Z-rotations, thereby enabling the full transversal Clifford group on the logical qubits.

If this is right

The resulting CSS codes achieve positive asymptotic encoding rate while supporting fault-tolerant implementation of the full Clifford group.
Transversal single-qubit Clifford gates are realized within one code block.
Transversal two-qubit Clifford gates are realized across two identical code blocks.
Asymptotically good CSS-T codes exist in which the transversal T gate implements the logical S dagger.
The condition C2 * C1 ⊆ C1⊥ is necessary but not sufficient for a CSS code to be CSS-T.

Where Pith is reading between the lines

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

These codes could lower the overhead of Clifford operations in large-scale quantum computation by removing the need for magic-state distillation or additional encoding layers for single-qubit gates.
Combining the framework with non-Clifford gate constructions might produce resource-efficient universal fault-tolerant computation.
The revised characterizations of CSS-T codes may guide searches for other transversal non-Clifford gates in similar code families.

Load-bearing premise

The existence of infinite families of classical linear codes satisfying the required inclusion and distance properties so that the resulting CSS codes have positive rate and growing minimum distance.

What would settle it

An explicit demonstration that no such infinite families of classical linear codes exist, or that any CSS codes built from them fail to maintain the claimed distances or to implement the logical Clifford operations fault-tolerantly under transversal gates.

read the original abstract

This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal $Z$-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal $T$ realizes the logical transversal $S^{\dagger}$; (b) show that the condition $C_2 \ast C_1 \subseteq C_1^{\perp}$ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal $T$ implements the logical identity and the logical transversal $T$, respectively.

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.

Desk Editor's Note private letter to a colleague

This paper supplies a framework for asymptotically good CSS codes that support fault-tolerant transversal Clifford gates, plus some clean revisions to the CSS-T conditions.

read the letter

The main point is that the authors give a way to build CSS codes from classical linear codes C1 and C2 that achieve good asymptotic rate and distance while letting transversal single-qubit Cliffords act inside one block and two-qubit Cliffords act across two identical blocks. They also produce families where transversal T implements logical S dagger, and they show that the star-product condition C2 * C1 ⊆ C1^perp is necessary but not sufficient for a code to be CSS-T. On top of that they revise the earlier characterizations of when transversal T acts as the identity or as logical T. Those revisions and the necessity result look like the clearest technical contributions. The asymptotic claims rest on the existence of classical codes with the right inclusion and distance properties, which is standard but still needs the framework to actually deliver the parameters without extra overhead. The work stays within established CSS machinery and does not appear to introduce hidden fitting or circular definitions. One soft spot is that the necessity proof and the revised characterizations will need line-by-line checking against small examples to confirm they cover all cases; the abstract alone does not show the full derivations. Another minor point is that the paper relies on known good classical code families rather than constructing new ones from scratch, so the novelty sits more in the gate-support conditions than in the underlying codes. This is aimed at people working on fault-tolerant quantum computation who care about explicit transversal gate sets and asymptotic overhead. It is worth sending to peer review because the claims are concrete, the framework is new relative to the cited literature, and the revisions to CSS-T could be useful if they hold up.

Referee Report

2 major / 2 minor

Summary. The paper introduces a framework for constructing CSS codes that support fault-tolerant logical transversal Z-rotations. Using this framework, it obtains asymptotically good CSS code families that realize the logical transversal Clifford group fault-tolerantly, with single-qubit Clifford gates realized transversally within one code block and two-qubit Clifford gates realized across two identical blocks. For CSS-T codes, it demonstrates asymptotically good families in which transversal T realizes the logical S†, proves that the condition C₂ ∗ C₁ ⊆ C₁⊥ is necessary but not sufficient, and revises the characterizations of CSS-T codes for the cases in which transversal T implements the logical identity or the logical T.

Significance. If the constructions and revised characterizations hold, the results are significant for quantum error correction: they supply the first explicit existence proofs of asymptotically good CSS codes supporting a full set of fault-tolerant transversal Clifford gates, which is a key requirement for scalable fault-tolerant quantum computation. The clarification that the star-product condition is necessary but insufficient, together with the revised CSS-T characterizations, refines the theoretical understanding of transversal gates in CSS constructions and may guide future code searches.

major comments (2)

[Framework section] Framework section: the asymptotic goodness claim rests on the existence of families of classical linear codes C₁ ⊇ C₂ satisfying the stated inclusion, star-product, and linear-distance conditions; the manuscript must supply explicit positive lower bounds on the rate and relative distance (or a concrete construction achieving them) rather than an existence argument alone, as this is load-bearing for the central claim.
[CSS-T codes section] CSS-T codes section: the demonstration that C₂ ∗ C₁ ⊆ C₁⊥ is necessary but not sufficient requires an explicit counterexample (a concrete pair of classical codes satisfying the inclusion yet failing to support the claimed transversal T action); without it the revised characterization cannot be verified and the necessity claim remains incomplete.

minor comments (2)

The notation for the star product ∗ should be defined explicitly at first use and distinguished from the standard Hadamard product or other code-theoretic operations to prevent reader confusion.
[Introduction] The introduction should include a short comparison table or paragraph contrasting the new framework with prior CSS constructions supporting transversal gates (e.g., those realizing only a subset of the Clifford group).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our manuscript. We are pleased that the significance of the results on asymptotically good CSS codes realizing the logical transversal Clifford group fault-tolerantly is recognized. We address each major comment below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

Referee: [Framework section] Framework section: the asymptotic goodness claim rests on the existence of families of classical linear codes C₁ ⊇ C₂ satisfying the stated inclusion, star-product, and linear-distance conditions; the manuscript must supply explicit positive lower bounds on the rate and relative distance (or a concrete construction achieving them) rather than an existence argument alone, as this is load-bearing for the central claim.

Authors: We agree that making the asymptotic goodness fully explicit strengthens the central claim. In the revised manuscript we will add explicit positive lower bounds on both the rate and relative distance for the families of classical codes C₁ ⊇ C₂. These bounds follow from applying the Gilbert-Varshamov bound to linear codes satisfying the required inclusion C₁ ⊇ C₂, the star-product condition C₂ ∗ C₁ ⊆ C₁^⊥, and the linear-distance conditions; the resulting constants are strictly positive and can be stated directly in the framework section. revision: yes
Referee: [CSS-T codes section] CSS-T codes section: the demonstration that C₂ ∗ C₁ ⊆ C₁⊥ is necessary but not sufficient for CSS-T codes requires an explicit counterexample (a concrete pair of classical codes satisfying the inclusion yet failing to support the claimed transversal T action); without it the revised characterization cannot be verified and the necessity claim remains incomplete.

Authors: We thank the referee for highlighting this point. In the revised version we will supply an explicit counterexample: a concrete pair of binary linear codes C₁ and C₂ of modest length that satisfy C₁ ⊇ C₂ and C₂ ∗ C₁ ⊆ C₁^⊥, yet for which the transversal T gate on the associated CSS code fails to implement the logical S†. This example will be presented with the explicit generator matrices and verification that the star-product condition holds while the logical action does not, thereby completing the demonstration that the condition is necessary but not sufficient. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces a new framework for CSS code constructions supporting fault-tolerant transversal Z-rotations and derives asymptotically good families realizing the transversal Clifford group from classical linear codes C1 ⊇ C2 satisfying inclusion, distance, and star-product conditions. These conditions are standard in CSS theory and are not defined in terms of the target logical gates or fitted to the output; the existence claim is an explicit construction output rather than a renaming or self-referential fit. Revisions to CSS-T characterizations are presented as analytical corrections to prior conditions (e.g., necessity but insufficiency of C2 * C1 ⊆ C1⊥), without load-bearing self-citations or by-construction reductions. The central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on a new framework whose correctness depends on standard CSS code properties and the existence of suitable classical codes; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

standard math CSS codes are constructed from a pair of classical linear codes C1 and C2 with C2 subset C1perp
Standard background fact invoked throughout the abstract.
domain assumption Existence of infinite families of classical codes with the distance and inclusion properties needed for asymptotic goodness and transversal fault tolerance
Required for the main existence claim but not proven in the abstract.

pith-pipeline@v0.9.0 · 5460 in / 1307 out tokens · 75743 ms · 2026-05-16T14:55:06.482724+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/reality_from_one_distinction reality_from_one_distinction (8-tick period) echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

CSS code CSS(C1,C2,sZ) is a CSS-T code iff for all x∈C2, y∈C1, wH(x)−2wH(x∗(y⊕sZ))≡0(mod 8)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

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E. Berardini, R. Dastbasteh, J.E. Martinez, S. Jain, and O.S. Larrarte. Asymptotically good CSS-T codes and a new construction of triorthogonal codes.IEEE Journal on Selected Areas in Information Theory, 6:189–198, 2025

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Krishna and J.P

A. Krishna and J.P. Tillich. Towards low overhead magic state distillation. Phys. Rev. Lett., 123, 2019

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Quynh T. Nguyen. Good binary quantum codes with transversal CCZ gate. InProceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC ’25, page 697–706. Association for Computing Machinery, 2025

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C.M. Perez and W. Wolfgang. Self-dual doubly even 2-quasi-cyclic transitive codes are asymptotically good.IEEE Transactions on Information Theory, 53:4302–4308, 2007

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Rengaswamy, R

N. Rengaswamy, R. Calderbank, M. Newman, and H.D. Pfister. On optimality of CSS codes for transversal T .IEEE Journal on Selected Areas in Information Theory, 1:499–514, 2020

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Wills, MH

A. Wills, MH. Hsieh, and H Yamasaki. Constant-overhead magic state distillation.Nat. Phys., 21:1842–1846, 2025. VI. APPENDIX A. Proof of Lemma 1, Lemma 4, and Lemma 7 The proposition below summarizes the properties of the matrixArequired for subsequent proofs. Proposition 21.The following statements hold: 1)For every integerpsuch that2≤p≤min{m, k}, any se...

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[25]

SinceCis a2 m-divisible code (andm≥2), applying (2) yields: 0 (mod 2 m−p+1) =w H (er1 ∗

Since A is a submatrix of the systematic generator matrix GC, each row ri of A extends to a codeword eri = (e i, ri)∈C , wheree i denotes the standard basis vector. SinceCis a2 m-divisible code (andm≥2), applying (2) yields: 0 (mod 2 m−p+1) =w H (er1 ∗. . .∗erp) =w H (e1 ∗. . .∗e p) +w H (r1 ∗. . .∗r p) =w H (r1 ∗. . .∗r p) where the last equality follows...

work page
[26]

SinceCis a2 m-divisible code, for anyi∈[k], we have: 0 (mod 2 m) =w H (eri) =w H (ei) +w H (ri) = 1 +w H (ri)

work page
[27]

From Item 2, the rows of A have odd weights

From Item 1 (with p= 2 ), distinct rows have pairwise even overlap. From Item 2, the rows of A have odd weights. These two conditions imply that the rows ofAare linearly independent overF 2. The next proposition summarizes the properties of the codesC 1 andC 2 required for subsequent proofs. Proposition 22.The following statements hold: 1)The codeC 2 is2 ...

work page
[28]

SinceCis a2 m-divisible code,C 2 inherits this property

For anyx∈C 2, the vector(0 t, x)lies inCby construction. SinceCis a2 m-divisible code,C 2 inherits this property

work page
[29]

Applying (2), we get: 0 (mod 2 m−1) =w H (0t, x)∗(y ′, y) =w H (x∗y)

For anyx∈C 2,y∈C 1, there exists ay ′ ∈F t 2 such that(0 t, x),(y ′, y)∈C. Applying (2), we get: 0 (mod 2 m−1) =w H (0t, x)∗(y ′, y) =w H (x∗y)

work page
[30]

, yt correspond to the firsttrows of the matrixA

Observe thaty 1, . . . , yt correspond to the firsttrows of the matrixA. wH (ya) = tX i=1 (−2)i−1 X {j1,...,ji} ⊆[t] iY p=1 ajp wH (yj1 ∗. . .∗y ji ) ! (20) = tX j=1 aj wH (yj) (mod 2 m)(21) =−w H (a) (mod 2 m)(22) where Eqs. (20), (21) and (22) follow from (1), (18) and (19), respectively. Proof of Lemma 1

work page
[31]

(7)), dim(C2) =k−t

By construction (Eq. (7)), dim(C2) =k−t . Furthermore, since Proposition 21 establishes that A has full row rank, the generator matrixG C1 has linearly independent rows, implying thatdim(C 1) =k

work page
[32]

For any x∈C 1, by the puncturing construction, there exists x′ ∈F t 2 such that (x′, x)∈C . Since wH (x′)≤t and dmin(C) =d, we obtain: dmin(C1)≥d−t≥t Regarding C ⊥ 2 , observe that the generator matrix of C ⊥ 2 (GC⊥ 2 ) forms a submatrix of the generator matrix of C ⊥ (GC⊥ ): GC⊥ 2 = h AT 2 In−k i GC⊥ = h AT In−k i = h AT 1 GC⊥ 2 i Thus, anyz∈C ⊥ 2 extend...

work page
[33]

Sincew H (z′)≤tandd min(C ⊥) =d ⊥, we obtain: dmin(C ⊥ 2 )≥d ⊥ −t≥t

work page
[34]

If C is self-dual thenk= n 2

Proposition 22 (Item 2) implies that the vectors in C2 and C1 are mutually orthogonal; hence, C2 ⊆C ⊥ 1 . If C is self-dual thenk= n 2 . Consequently,dim(C 2) = n 2 −tanddim(C ⊥ 1 ) =n−t−k= n 2 −t, which implies thatC 2 =C ⊥ 1 . Proof of Lemma 4

work page
[35]

2)d (p) X = 2 pdX follows directly from the repetition construction

The2 p-fold repetition preserves code dimensions; hence,dim(C (p) i ) = dim(C i)fori∈ {1,2}. 2)d (p) X = 2 pdX follows directly from the repetition construction. It remains to showd (p) Z =d Z: a)d (p) Z ≥d Z: Let(z 1, . . . , z2p )be a minimum weight vector in(C (p) 2 ) ⊥ \(C (p) 1 ) ⊥ then, i) for any(x, . . . , x)∈C (p) 2 , we have: 0 (mod 2) =w H (z1,...

work page
[36]

Lemma 24 implies: T ⊗n 0k L =e ι π 4 wH (sZ ) 0k L =⇒γ T =e ι π 4 wH (sZ ) Now consider any y∈C 1/C2

First, consider the zero logical state 0k (corresponding toy a = 0). Lemma 24 implies: T ⊗n 0k L =e ι π 4 wH (sZ ) 0k L =⇒γ T =e ι π 4 wH (sZ ) Now consider any y∈C 1/C2. Let |a⟩ be the logical state associated with y (i.e., ya =y ). Equating the action from Lemma 24 withγ T |a⟩L: γT eι π 4 (wH (y)−2wH (y∗sZ )) |a⟩L =γ T |a⟩L Since|a⟩ L ̸= 0, we obtain: w...

work page
[37]

Lemma 24 implies: T ⊗n 0k L =e ι π 4 wH (sZ ) 0k L =⇒γ T =e ι π 4 wH (sZ ) Now consider any a∈F k 2 and the corresponding coset representative ya ∈C 1/C2

First, consider the zero logical state 0k (corresponding toy a = 0). Lemma 24 implies: T ⊗n 0k L =e ι π 4 wH (sZ ) 0k L =⇒γ T =e ι π 4 wH (sZ ) Now consider any a∈F k 2 and the corresponding coset representative ya ∈C 1/C2. Equating the action from Lemma 24 with γT eι π 4 wH (a) |a⟩L: γT eι π 4 (wH (ya)−2wH (ya∗sZ )) |a⟩L =γ T eι π 4 wH (a) |a⟩L Since|a⟩ ...

work page

[1] [1]

Berardini, R

E. Berardini, R. Dastbasteh, J.E. Martinez, S. Jain, and O.S. Larrarte. Asymptotically good CSS-T codes and a new construction of triorthogonal codes.IEEE Journal on Selected Areas in Information Theory, 6:189–198, 2025

work page 2025

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Betsumiya and A

K. Betsumiya and A. Munemasa. On triply even binary codes.Journal of the London Mathematical Society, 86, 2012

work page 2012

[3] [3]

Magic-state distillation with low overhead.Phys

Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead.Phys. Rev. A, 86:052329, Nov 2012

work page 2012

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E. T. Campbell and C. Vuillot. Roads towards fault-tolerant universal quantum computation.Nature, 549:172–179, 2017

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Camps-Moreno, H.H

E. Camps-Moreno, H.H. López, G.L. Matthews, D. Ruano, R. San-José, and I. Soprunov. An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance.Quantum Information Processing, 23, 2024

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Restrictions on transversal encoded quantum gate sets.Phys

Bryan Eastin and Emanuel Knill. Restrictions on transversal encoded quantum gate sets.Phys. Rev. Lett., 102:110502, Mar 2009

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Asymptotically good quantum codes with transversal non-clifford gates

Louis Golowich and Venkatesan Guruswami. Asymptotically good quantum codes with transversal non-clifford gates. InProceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC ’25, page 707–717. Association for Computing Machinery, 2025

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Hastings

Jeongwan Haah and Matthew B. Hastings. Codes and protocols for distillingT, controlled-S, and Toffoli gates.Quantum, 2:71, 2018

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Hastings and Jeongwan Haah

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[11] [11]

J. Hu, Q. Liang, and R. Calderbank. Designing the quantum channels induced by diagonal gates.Quantum, 6:802, 2022

work page 2022

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Jain and V .V

S.P. Jain and V .V . Albert. Transversal Clifford andT -gate codes of short length and high distance.IEEE Journal on Selected Areas in Information Theory, 6:127–137, 2025

work page 2025

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Krishna and J.P

A. Krishna and J.P. Tillich. Towards low overhead magic state distillation. Phys. Rev. Lett., 123, 2019

work page 2019

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Quynh T. Nguyen. Good binary quantum codes with transversal CCZ gate. InProceedings of the 57th Annual ACM Symposium on Theory of Computing, STOC ’25, page 697–706. Association for Computing Machinery, 2025

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Nielsen and Isaac L

Michael A. Nielsen and Isaac L. Chuang.Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, USA, 10th edition, 2011

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Perez and W

C.M. Perez and W. Wolfgang. Self-dual doubly even 2-quasi-cyclic transitive codes are asymptotically good.IEEE Transactions on Information Theory, 53:4302–4308, 2007

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Rengaswamy, R

N. Rengaswamy, R. Calderbank, M. Newman, and H.D. Pfister. On optimality of CSS codes for transversal T .IEEE Journal on Selected Areas in Information Theory, 1:499–514, 2020

work page 2020

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Narayanan Rengaswamy, Robert Calderbank, Michael Newman, and Henry D. Pfister. Classical coding problem from transversal T gates. In 2020 IEEE International Symposium on Information Theory (ISIT), page 1891–1896. IEEE Press, 2020

work page 2020

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Sloane, F.J

N.J.A. Sloane, F.J. MacWilliams, and J.G. Thompson. Good self dual codes exist.Discrete Mathematics, 3:153–162, 1972

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Tansuwannont, Y

T. Tansuwannont, Y . Takada, and K. Fujii. Clifford gates with logical transversality for self-dual CSS codes, 2025. arXiv:2503.19790

work page arXiv 2025

[24] [24]

Wills, MH

A. Wills, MH. Hsieh, and H Yamasaki. Constant-overhead magic state distillation.Nat. Phys., 21:1842–1846, 2025. VI. APPENDIX A. Proof of Lemma 1, Lemma 4, and Lemma 7 The proposition below summarizes the properties of the matrixArequired for subsequent proofs. Proposition 21.The following statements hold: 1)For every integerpsuch that2≤p≤min{m, k}, any se...

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[25] [25]

SinceCis a2 m-divisible code (andm≥2), applying (2) yields: 0 (mod 2 m−p+1) =w H (er1 ∗

Since A is a submatrix of the systematic generator matrix GC, each row ri of A extends to a codeword eri = (e i, ri)∈C , wheree i denotes the standard basis vector. SinceCis a2 m-divisible code (andm≥2), applying (2) yields: 0 (mod 2 m−p+1) =w H (er1 ∗. . .∗erp) =w H (e1 ∗. . .∗e p) +w H (r1 ∗. . .∗r p) =w H (r1 ∗. . .∗r p) where the last equality follows...

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[26] [26]

SinceCis a2 m-divisible code, for anyi∈[k], we have: 0 (mod 2 m) =w H (eri) =w H (ei) +w H (ri) = 1 +w H (ri)

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[27] [27]

From Item 2, the rows of A have odd weights

From Item 1 (with p= 2 ), distinct rows have pairwise even overlap. From Item 2, the rows of A have odd weights. These two conditions imply that the rows ofAare linearly independent overF 2. The next proposition summarizes the properties of the codesC 1 andC 2 required for subsequent proofs. Proposition 22.The following statements hold: 1)The codeC 2 is2 ...

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[28] [28]

SinceCis a2 m-divisible code,C 2 inherits this property

For anyx∈C 2, the vector(0 t, x)lies inCby construction. SinceCis a2 m-divisible code,C 2 inherits this property

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[29] [29]

Applying (2), we get: 0 (mod 2 m−1) =w H (0t, x)∗(y ′, y) =w H (x∗y)

For anyx∈C 2,y∈C 1, there exists ay ′ ∈F t 2 such that(0 t, x),(y ′, y)∈C. Applying (2), we get: 0 (mod 2 m−1) =w H (0t, x)∗(y ′, y) =w H (x∗y)

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[30] [30]

, yt correspond to the firsttrows of the matrixA

Observe thaty 1, . . . , yt correspond to the firsttrows of the matrixA. wH (ya) = tX i=1 (−2)i−1 X {j1,...,ji} ⊆[t] iY p=1 ajp wH (yj1 ∗. . .∗y ji ) ! (20) = tX j=1 aj wH (yj) (mod 2 m)(21) =−w H (a) (mod 2 m)(22) where Eqs. (20), (21) and (22) follow from (1), (18) and (19), respectively. Proof of Lemma 1

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[31] [31]

(7)), dim(C2) =k−t

By construction (Eq. (7)), dim(C2) =k−t . Furthermore, since Proposition 21 establishes that A has full row rank, the generator matrixG C1 has linearly independent rows, implying thatdim(C 1) =k

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[32] [32]

For any x∈C 1, by the puncturing construction, there exists x′ ∈F t 2 such that (x′, x)∈C . Since wH (x′)≤t and dmin(C) =d, we obtain: dmin(C1)≥d−t≥t Regarding C ⊥ 2 , observe that the generator matrix of C ⊥ 2 (GC⊥ 2 ) forms a submatrix of the generator matrix of C ⊥ (GC⊥ ): GC⊥ 2 = h AT 2 In−k i GC⊥ = h AT In−k i = h AT 1 GC⊥ 2 i Thus, anyz∈C ⊥ 2 extend...

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[33] [33]

Sincew H (z′)≤tandd min(C ⊥) =d ⊥, we obtain: dmin(C ⊥ 2 )≥d ⊥ −t≥t

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[34] [34]

If C is self-dual thenk= n 2

Proposition 22 (Item 2) implies that the vectors in C2 and C1 are mutually orthogonal; hence, C2 ⊆C ⊥ 1 . If C is self-dual thenk= n 2 . Consequently,dim(C 2) = n 2 −tanddim(C ⊥ 1 ) =n−t−k= n 2 −t, which implies thatC 2 =C ⊥ 1 . Proof of Lemma 4

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[35] [35]

2)d (p) X = 2 pdX follows directly from the repetition construction

The2 p-fold repetition preserves code dimensions; hence,dim(C (p) i ) = dim(C i)fori∈ {1,2}. 2)d (p) X = 2 pdX follows directly from the repetition construction. It remains to showd (p) Z =d Z: a)d (p) Z ≥d Z: Let(z 1, . . . , z2p )be a minimum weight vector in(C (p) 2 ) ⊥ \(C (p) 1 ) ⊥ then, i) for any(x, . . . , x)∈C (p) 2 , we have: 0 (mod 2) =w H (z1,...

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[36] [36]

Lemma 24 implies: T ⊗n 0k L =e ι π 4 wH (sZ ) 0k L =⇒γ T =e ι π 4 wH (sZ ) Now consider any y∈C 1/C2

First, consider the zero logical state 0k (corresponding toy a = 0). Lemma 24 implies: T ⊗n 0k L =e ι π 4 wH (sZ ) 0k L =⇒γ T =e ι π 4 wH (sZ ) Now consider any y∈C 1/C2. Let |a⟩ be the logical state associated with y (i.e., ya =y ). Equating the action from Lemma 24 withγ T |a⟩L: γT eι π 4 (wH (y)−2wH (y∗sZ )) |a⟩L =γ T |a⟩L Since|a⟩ L ̸= 0, we obtain: w...

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[37] [37]

Lemma 24 implies: T ⊗n 0k L =e ι π 4 wH (sZ ) 0k L =⇒γ T =e ι π 4 wH (sZ ) Now consider any a∈F k 2 and the corresponding coset representative ya ∈C 1/C2

First, consider the zero logical state 0k (corresponding toy a = 0). Lemma 24 implies: T ⊗n 0k L =e ι π 4 wH (sZ ) 0k L =⇒γ T =e ι π 4 wH (sZ ) Now consider any a∈F k 2 and the corresponding coset representative ya ∈C 1/C2. Equating the action from Lemma 24 with γT eι π 4 wH (a) |a⟩L: γT eι π 4 (wH (ya)−2wH (ya∗sZ )) |a⟩L =γ T eι π 4 wH (a) |a⟩L Since|a⟩ ...

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