Pith. sign in

REVIEW 3 major objections 4 minor 57 references

CSS-$T$ codes over Binary Extension Fields and their Physical Foundations

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read CSS-T codes over every binary extension field are characterized by a trace-code condition.

desk verdict The central characterization is false already for q=2, so the main existence result is unsupported, though the trace-code definition is worth salvaging. read the letter →

arxiv 2507.17611 v1 pith:HT3DLTKC submitted 2025-07-23 quant-ph

Jasper J. Postema , F. Conca , A. Ravagnani This is my paper
classification quant-ph MSC 81P7094B05 PACS 03.67.Pp
keywords CSS-TcodestransversalTgatebinaryextensionfieldstracestarproductLDPCquantumasymptoticallygooderrorcorrection
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 extends the CSS-T construction---quantum CSS codes on which the non-Clifford $T$ gate can be applied transversally---from binary qubits to qudits over every finite field $\mathbb{F}_{2^s}$. Its central claim is Theorem 4.6: a CSS pair $(C_1,C_2)$ over $\mathbb{F}_{2^s}$ is CSS-T if and only if $\operatorname{tr}(C_1)\star\operatorname{tr}(C_1)\subseteq \operatorname{tr}(C_2)^\perp$, where $\operatorname{tr}$ is the absolute trace to $\mathbb{F}_2$ and $\star$ is the coordinate-wise star product. Because the condition is purely about binary trace codes, the qudit problem is reduced to a binary coding problem. The paper further claims that length-doubling any asymptotically good family of CSS codes by the identity map produces asymptotically good LDPC CSS-T codes over every $\mathbb{F}_{2^s}$, so such families exist for all binary extension fields. A sympathetic reader would care because a transversal non-Clifford gate is the missing ingredient for low-overhead fault-tolerant universal computation, and larger alphabets offer more parameter choices than qubits.

What carries the argument

The machinery is the interaction of three objects: the absolute trace $\operatorname{tr}:\mathbb{F}_{2^s}\to\mathbb{F}_2$, the star product $a\star b=(a_1b_1,\ldots,a_nb_n)$, and the trace code $\operatorname{tr}(C)=\{(\operatorname{tr}(c_1),\ldots,\operatorname{tr}(c_n)):c\in C\}$. The q-ary $T$-gate is the diagonal operator with phase $\exp(i\pi\operatorname{tr}(\lambda x)/4)$, which is why the trace enters the definition. Theorem 4.6 is the load-bearing step: it transfers the q-ary CSS-T condition to the binary inclusion $\operatorname{tr}(C_1)\star\operatorname{tr}(C_1)\subseteq\operatorname{tr}(C_2)^\perp$, so the binary CSS-T toolbox, including length-doubling, can be reused. The trace-duality theorem and the Galois-invariance/trace-code equivalence are the supporting results that connect trace codes to duals and subfield subcodes.

What would settle it

Compute the matrix elements of $T X(\mu) T^\dagger$ directly from Eq. (3) on basis states: the phase picked up from the left and right $T$'s is $e^{i\pi(\operatorname{tr}(\mu x)-\operatorname{tr}(x))/4}$, so the result is a scalar multiple of $X(\mu)$ with no $X(\mu)Z(1)$ term. If this computation is correct, the spread term used in the proof is absent and the claimed equivalence needs another derivation; this single calculation settles whether the foundation of the paper's main theorem stands.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the physical condition 'the $T(\lambda)$-gate preserves the code space for every $\lambda\in\mathbb{F}_{2^s}$' is equivalent to a binary statement about trace codes: $(\operatorname{tr}(C_1),\operatorname{tr}(C_2))$ is a binary CSS-T pair. The proof conjugates the q-ary Pauli stabilisers by $T^{\otimes n}$ and reads off which transformed stabilisers must remain in the stabiliser group, obtaining $\operatorname{tr}(C_1)\star\operatorname{tr}(C_1)\subseteq \operatorname{tr}(C_2)^\perp$. From this equivalence the paper derives a chain of consequences: trace codes of CSS-T pairs are self-orthogonal; a transversal $T$ and a transversal Hadamard cannot coexist for distance greater than 2, matching the no-go theorem for transversal universal gate sets; and over generalized Reed-Muller codes with $s>1$ only the trivial pair $(\mathrm{GRM}_q(0,m),\mathrm{GRM}_q(1,m))$ works when $ms\ge 3$. The final claim is existential: applying the identity length-doubling map to the known asymptotically good LDPC CSS family turns it into an asymptotically good LDPC CSS-T family over every $\mathbb{F}_{2^s}$, with rate halved and distance at least halved.

Load-bearing premise

The proof of Theorem 4.6 assumes that conjugating $X(\mu)$ by the q-ary $T$ gate with $\operatorname{tr}(\mu)=1$ produces $\frac{1}{\sqrt{2}}(X(\mu)+iX(\mu)Z(1))$, whereas the paper's own definition of $T$ in Eq. (3) gives the pure phase $e^{i\pi\operatorname{tr}(\mu)/4}X(\mu)$; the derivation of $\operatorname{tr}(C_1)\star\operatorname{tr}(C_1)\subseteq\operatorname{tr}(C_2)^\perp$ needs a corrected conjugation rule to stand.

Editorial extensions

If this is right

  • Any CSS code over $\mathbb{F}_{2^s}$ whose trace codes satisfy $\operatorname{tr}(C_1)\star\operatorname{tr}(C_1)\subseteq\operatorname{tr}(C_2)^\perp$ admits a transversal $T(\lambda)$ for every $\lambda\in\mathbb{F}_{2^s}$.
  • Length-doubling a CSS code by the identity map always yields a CSS-T code over the same field, so every CSS code can be made CSS-T at the cost of doubling length and halving rate.
  • The known asymptotically good LDPC CSS families over finite fields yield asymptotically good LDPC CSS-T families over every $\mathbb{F}_{2^s}$, resolving the non-binary existence question.
  • For generalized Reed-Muller codes over $\mathbb{F}_{2^s}$ with $s>1$, the only nontrivial CSS-T pairs are $(\mathrm{GRM}_q(1,m),\mathrm{GRM}_q(0,m))$ with $ms\ge 3$, in contrast to the binary case.
  • No q-ary CSS-T code of distance greater than 2 can simultaneously admit transversal Hadamard and transversal $T$ gates, consistent with the no-go theorem for transversal universal gate sets.

Reading between the lines

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

  • Not stated in the paper but implied: because the characterization is entirely in terms of binary trace codes, any future binary CSS-T construction or bound can be lifted to $\mathbb{F}_{2^s}$ by choosing codes whose traces are the binary codes, potentially with better $q$-ary distances than the length-doubling construction.
  • If the conjugation rule in the proof of Theorem 4.6 is replaced by the scalar-phase rule from the paper's own Eq. (3), the spread term that produces the trace-star inclusion disappears; a corrected characterization might have a different form, while the structural conclusions may survive.
  • A concrete testable extension would be a small search over $\mathbb{F}_4$ and $\mathbb{F}_8$ for maps $\phi$ other than the identity in Theorem 6.3 that satisfy the parity condition, comparing the resulting rate/distance trade-offs against the halved-rate identity construction.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a q-ary generalization of CSS-T codes over binary extension fields F_{2^s}, defining them via the condition that the T(λ) gate preserves the CSS code space. Its central result, Theorem 4.6, characterizes q-ary CSS-T pairs by the trace-code star-product condition tr(C1)⋆tr(C1) ⊆ tr(C2)⊥. The paper then uses this characterization to prove Theorem 6.3 and Corollary 6.5, claiming the existence of asymptotically good LDPC CSS-T codes over every F_{2^s} via a length-doubling construction. It also compares the new definition with the earlier BCR definition and provides examples from Reed-Muller and cyclic codes.

Significance. If correct, the paper would extend the binary CSS-T theory to qudits over any binary extension field and would settle the existence of asymptotically good CSS-T codes in that setting. The use of trace codes is natural, and the paper is clearly structured. However, the central characterization Theorem 4.6 is false, as shown by a concrete q=2 counterexample, and the main existence result is built directly on this theorem. The paper also contains several proof gaps related to self-duality. These issues undermine the core claims.

major comments (3)
  1. [Theorem 4.6, Eq. (5)] The claimed equivalence is false. Take q=2, C1=span{1111,0011}, C2=span{0011} in F2^4. Then C1⋆C1=C1 and C1⊆C2⊥, so condition (5) holds. The CSS code QS(C1,C2) is spanned by |0000>+|0011> and |1111>+|1100>. Applying T(1)⊗4 from Eq. (3) gives |0000>+i|0011> and −|1111>+i|1100>, neither of which lies in the code space. Hence Definition 4.1 fails while (5) holds, contradicting the 'if' direction of Theorem 4.6. The same failure occurs for the length-doubled code (C1^id,C2^id) used in Section 6: the condition of Theorem 6.3 is satisfied with ϕ=id, but T⊗8 maps |00000000>+|00110011> to |00000000>−|00110011>, outside the code space. Since Theorem 6.3 and Corollary 6.5 are derived from Theorem 4.6, the asymptotic existence claim is unsupported.
  2. [Proof of Theorem 4.6] The proof contains an unjustified step: the chain tr(C1)⊥ ⊆ Zj⊥ ⊆ Zj ⊆ tr(C1). From Zj ⊆ tr(C1) one obtains tr(C1)⊥ ⊆ Zj⊥, but Zj⊥ ⊆ Zj does not follow from the preceding argument, and in general for binary codes the dual of a subspace is larger, not smaller. Moreover, the subsequent use of a self-dual code Ctr(x) 'contained in the support of tr(x)' repeats the error in the proof of Theorem 3.5: a code of dimension wt(x)/2 supported on x is self-dual only when regarded as a code of length wt(x) on its support, not in the ambient space F2^n. Without a correct derivation, the proof does not establish the trace-condition characterization.
  3. [Proof of Theorem 3.5] The proof of the binary base case is flawed. The statement 'Since Cx⊥ = Cx ⊆ C1⊥, we have z∈Cx' assumes Cx is self-dual in the full ambient space F2^n, which is incompatible with the specified dimension wt(x)/2 when wt(x)<n. In the full space, dim(Cx⊥)=n−wt(x)/2, so the equality Cx⊥=Cx is false. The argument can be repaired by working with the punctured support of x, but as written it is invalid, and the same flaw propagates into the proof of Theorem 4.6.
minor comments (4)
  1. [Definition 2.5] The phrase 'self-dual code Cx ⊆ C1⊥ of dimension ωH(x)/2 and supported on x' is ambiguous: self-duality must be specified with respect to the punctured length-wt(x) space. Under the literal ambient-space reading, no such code exists when wt(x)<n, so the definition is inconsistent as stated.
  2. [Example 5.2] The text says 'at least one of the codewords in C2 is not even, e.g. x has length and Hamming weight 15', but the vector x is defined as an element of C1, not C2. This appears to be a typo or an error in the example.
  3. [Theorem 6.3, Eq. (6)] The displayed condition is missing a closing parenthesis in the second Hamming weight term: it should read ωH(tr(ϕ(x)) ⋆ tr(ϕ(y)) ⋆ tr(ϕ(z))) = 0 mod 2.
  4. [Proposition 6.1] The proof merely cites Theorem 3.9 of [23] and the inequality dim(tr(C)) ≥ dim(C). The inequality is true, but the transfer of the three bounds to the q-ary case is not shown, and given the failure of Theorem 4.6, any application of the trace-code characterization in this context would need revisiting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the q-ary CSS-T characterization is derived from trace-star conditions and external binary results, and the asymptotic existence proof uses an explicit length-doubling of external LDPC CSS codes.

full rationale

The paper's central result, Theorem 4.6, derives the condition tr(C1) ⋆ tr(C1) ⊆ tr(C2)⊥ from the conjugation action of the q-ary T-gate and from the binary CSS-T characterization of Theorem 3.5, which itself cites the external algebraic characterization in [34]. The trace code and subfield subcode tools (Delsarte's theorem, Galois invariance) are standard external results, not definitions tailored to the target claim. Corollary 6.5 does not fit the target result into an input: it takes Panteleev-Kalachev's external family of asymptotically good CSS codes, applies the explicit length-doubling construction with φ = id, and checks the resulting condition via Theorem 6.3. No fitted parameter is renamed as a prediction, and no definition of CSS-T is secretly assumed in the proof of its characterization. The only overlap with the authors' own prior work is the citation of [23] for Lemma 2.6, a criterion about existence of self-dual subcodes whose assumptions do not include the q-ary CSS-T property; this is supporting mathematical background rather than a load-bearing circular premise. The reader-identified conjugation-rule inconsistency and the skeptic's self-duality objection concern correctness of intermediate steps, not circularity: even if those steps were wrong, the claimed result would be unsupported rather than equivalent to its inputs.

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

The paper relies on standard coding-theory theorems and on the existence of good q-ary CSS codes from prior work. No free parameters are fitted. The main unverified input is the incorrect conjugation rule, which is a proof error rather than an axiom.

assumptions (5)
  • standard math Delsarte's theorem relating trace codes and subfield subcodes
    Used in Lemma 4.5 and Remark 4.7. Standard theorem from [35].
  • standard math Lemma 4.4 (Galois invariance) from Giorgetti-Previtali
    Used to relate trace codes and subfield subcodes. External cited result from [36].
  • standard math Randriambololona lower bound on dimension of star product
    Used in Remark 4.7 to bound distance under simultaneous Hadamard and T transversality.
  • domain assumption Panteleev-Kalachev existence of asymptotically good q-ary LDPC CSS codes
    External theorem used in Corollary 6.5 to seed the length-doubling construction.
  • domain assumption The physical T gate for qudits is defined with the field trace
    This is the physical foundation of the new definition; taken from the literature on qudit gates.

how reviews work

0 comments
Cite this review

Pith. "Pith review of CSS-$T$ codes over Binary Extension Fields and their Physical Foundations." pith.science (2026). https://pith.science/paper/HT3DLTKC

@misc{pith2026250717611,
  author       = {Pith},
  title        = {Pith review of: CSS-$T$ codes over Binary Extension Fields and their Physical Foundations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HT3DLTKC}},
  note         = {Machine review of arXiv:2507.17611}
}
abstract

We investigate the class of CSS-$T$ codes, a family of quantum error-correcting codes that allows for a transversal $T$-gate. We extend the definition of a pair of linear codes $(C_1,C_2)$, $C_i\subseteq\mathbb{F}_q^n$, forming a $q$-ary CSS-$T$ code over binary extension fields, and demonstrate the existence of asymptotically good sequences of LDPC CSS-$T$ codes over any such field.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

57 extracted references · 46 canonical work pages

  1. [1]

    Let q = ps be a prime power and Fq the finite field with q elements

    QUANTUM COMPUTING This section contains a brief self-contained introduction to quantum computing and establishes the notation for the rest of the paper. Let q = ps be a prime power and Fq the finite field with q elements. Let C denote the complex field, and † Hermitian conjugation. A ket-vector |·⟩ denotes a column vector, while a bra-vector ⟨·| = |·⟩† de...

    work page

  2. [2]

    Calderbank and Shor, and indepen- dently Steane, introduced this class of codes in 1996

    THE CSS AND CSS-T CONSTRUCTIONS CSS codes, named after Calderbank, Shor, and Steane [12, 13], are a family of quantum error- correcting codes constructed from a pair of classical linear codes. Calderbank and Shor, and indepen- dently Steane, introduced this class of codes in 1996. Their constructions are different, but equivalent. 5 CSS codes have been mo...

    work page 1996

  3. [3]

    As we will see in Section 4, our characterisation has a natural extension to binary field extensions, while the original definition of CSS-T codes does not

    REVISITING CSS- T CODES FROM REED-MULLER CODES In this short section we establish a characterisation of CSS- T codes and show how it can be conve- niently used to derive one of the main results of [33] with a short proofs. As we will see in Section 4, our characterisation has a natural extension to binary field extensions, while the original definition of...

    work page

  4. [4]

    every codeword has an even Hamming weight) can be characterised as follows: a binary code C ⊆ Fn 2 is even if and only if Rn 2 ⊆ C ⊥

    The property of a binary code being even (i.e. every codeword has an even Hamming weight) can be characterised as follows: a binary code C ⊆ Fn 2 is even if and only if Rn 2 ⊆ C ⊥. The star product exhibits cyclical behaviour with respect to the standard inner product, in the following sense. Lemma 3.3 (Cyclicity of the star product) . For all a, b, c∈ Fq...

    work page

  5. [5]

    A generalization to arbitrary fields was proposed and studied in [23]

    q-AR Y CSS-T CODES The original definition of a CSS- T code was given only over the binary field; recall Definition 2.5. A generalization to arbitrary fields was proposed and studied in [23]. In this paper, we propose a new definition of CSS- T code over binary field extensions, which is different from the one proposed in [23] but which finds a precise ph...

    work page

  6. [6]

    C is Galois invariant),

    C = C ⋆2 (i.e. C is Galois invariant),

    work page

  7. [7]

    Property 1

    dim F2 (tr(C)) = dimFq (C). Property 1. in the previous lemma is often called Galois invariance ; see [36]. Duality and trace are related as follows. Lemma 4.5. Let C ⊆ Fn q be an Fq-linear code. We have tr(C)⊥Fp ⊆ tr(C ⊥Fq ). Proof. We omit the subscript Fp throughout this proof. First we prove that C ⊥|Fp ⊆ C|Fp ⊥ . (4) 10 Let x ∈ C ⊥|Fp . Then ⟨x, y⟩ =...

    work page

  8. [8]

    BCR” and to our definition as “CPR

    PROPER TIES AND APPLICA TIONS OF CSS-T CODES A. Comparisons As described earlier, in this paper we deviate from the definition of q-ary CSS-T codes given in [23]. A natural question is if the two definitions are related. We provide counterexamples showing that neither definition implies the other, but there also exist CSS- T codes that satisfy both. In th...

    work page

Show all 57 references
  1. [9]

    The defining set J = {j ∈ Zn | g(βj) = 0},

  2. [10]

    Note that I ∪ J = Zn

    The generating set I = {i ∈ Zn | g(βi) ̸= 0}. Note that I ∪ J = Zn. We also define −I = {−i ∈ Zn | i ∈ I}. 15 Lemma 5.9 (Cyclic trace codes) . We highlight two properties of trace codes of cyclic codes:

  3. [11]

    The trace code tr( C) of a cyclic code C is also cyclic

  4. [12]

    If C ⊆ Fq is a cyclic code with generator polynomial g(x) ∈ Fq[x], then tr(C) ⊆ Fp is a cyclic code with generator polynomial η(g(x)) ∈ Fp[x], where η(g(x)) is the unique largest-degree divisor of g(x) with all coefficients in Fp. Proof. We prove both statements separately:

  5. [13]

    , cn) = (c2,

    Let T be the cyclic shift operator on codewords c ∈ C, such that T (c1, c2, . . . , cn) = (c2, . . . , cn, c1). Then T ◦tr = tr ◦ Tis apparent from T (tr(c1, c2, . . . , cn)) = (tr(c2), . . . ,tr(cn), tr(c1)) = tr(c2, . . . , cn, c1) = tr(T (c1, . . . , cn)). Thus for all c ∈ ...

  6. [14]

    Example 5.10

    See Lemma 2.1 in [38]. Example 5.10. Let F4 = F2[α] with α2 + α + 1 = 0 . Let C ⊆ F9 4 be the [9 , 4, 4]-code generated by g(x) = (x3 + α)(x + α)(x + α + 1), with generator matrix G =   α α α 1 1 1 0 0 0 0 α α α 1 1 1 0 0 0 0 α α α 1 1 1 0 0 0 0 α α α 1 1 1   Then tr...

  7. [15]

    n /∈ I (2) 1 + I (2) 1 + I (2) 2 . Proof. (1) is a direct consequence of the CSS condition C2 ⊆ C1. For cyclic codes, this is equivalent to stating that g2(x) | g1(x), which is true if and only if I (q) 2 ⊆ I (q) 1 . For (2), we use the fact that tr(C(I (q))) = C(I (2)) ⊆ Fn 2...

  8. [16]

    More in detail, we show that asymptotically good sequences of CSS- T codes exist, and that they can be derived from sequences of CSS codes

    BOUNDS AND ASYMPTOTICALL Y GOOD CSS-T CODES OVER BINAR Y EXTENSION FIELDS In this section, we look at the asymptotic behaviour of CSS- T codes. More in detail, we show that asymptotically good sequences of CSS- T codes exist, and that they can be derived from sequences of CSS ...

  9. [17]

    If there exists a codeword x ∈ C2 with Hamming weight ωH(x) ≥ n + 1 − k2, then R + δ 2 ≤ 1 2

  10. [18]

    If there exists a codeword x ∈ C2 with Hamming weight ωH(x) ≥ n − d2, then R + δ ≤ 1 2 + 1 n

  11. [19]

    If there exists a codeword x ∈ C2 with Hamming weight ωH(x) ≥ n − d1, then R + 3 2 δ ≤ 1 2 + 2 n . Proof. These statements follow easily from Theorem 3.9 in Ref. [23], where the binary case is analyzed, using the fact that dim(tr( C)) ≥ dim(C). 17 To prove that an infinite num...

  12. [20]

    P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in Proceedings 35th Annual Symposium on Foundations of Computer Science(1994) pp. 124–134

    1994

  13. [21]

    P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing 26, 1484–1509 (1997)

    1997

  14. [22]

    L. K. Grover, A fast quantum mechanical algorithm for database search, in Proceedings of the Twenty- Eighth Annual ACM Symposium on Theory of Computing, STOC ’96 (Association for Computing Ma- chinery, New York, NY, USA, 1996) p. 212–219

    1996

  15. [23]

    Temme, S

    K. Temme, S. Bravyi, and J. M. Gambetta, Error mitigation for short-depth quantum circuits, Physical Review Letters 119, 10.1103/physrevlett.119.180509 (2017)

    2017 doi

  16. [24]

    de Keijzer, L

    R. de Keijzer, L. Visser, O. Tse, and S. Kokkelmans., Fidelity-enhanced variational quantum optimal control, Phys. Rev. A 111, 052625 (2025)

    2025

  17. [25]

    W. J. Huggins, S. McArdle, T. E. O’Brien, J. Lee, N. C. Rubin, S. Boixo, K. B. Whaley, R. Babbush, and J. R. McClean, Virtual distillation for quantum error mitigation, Phys. Rev. X 11, 041036 (2021)

    2021

  18. [26]

    Pelofske and V

    E. Pelofske and V. Russo, Digital zero-noise extrapolation with quantum circuit unoptimization (2025), arXiv:2503.06341 [quant-ph]

    2025 arXiv

  19. [27]

    Z. Cai, R. Babbush, S. C. Benjamin, S. Endo, W. J. Huggins, Y. Li, J. R. McClean, and T. E. O’Brien, Quantum error mitigation, Reviews of Modern Physics 95, 10.1103/revmodphys.95.045005 (2023)

    2023 doi

  20. [28]

    Gottesman, Stabilizer codes and quantum error correction (1997), arXiv:quant-ph/9705052 [quant-ph]

    D. Gottesman, Stabilizer codes and quantum error correction (1997), arXiv:quant-ph/9705052 [quant-ph]

    1997 arXiv

  21. [29]

    E. T. Campbell, Early fault-tolerant simulations of the Hubbard model, Quantum Science and Technology 7, 015007 (2021)

    2021

  22. [30]

    I. D. Kivlichan, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, W. Sun, Z. Jiang, N. Rubin, A. Fowler, A. Aspuru-Guzik, et al., Improved fault-tolerant quantum simulation of condensed-phase correlated elec- trons via trotterization, Quantum 4, 296 (2020)

    2020

  23. [31]

    A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Physical Review A 54, 1098–1105 (1996)

    1996

  24. [32]

    Steane, Multiple-particle interference and quantum error correction, Proceedings of the Royal Society of London

    A. Steane, Multiple-particle interference and quantum error correction, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551–2577 (1996)

    1996

  25. [33]

    D. J. C. MacKay, G. Mitchison, and P. L. McFadden, Sparse-graph codes for quantum error correction, IEEE Transactions on Information Theory 50, 2315 (2004)

    2004

  26. [34]

    A. A. Kovalev and L. P. Pryadko, Quantum Kronecker sum-product low-density parity-check codes with finite rate, Phys. Rev. A 88, 012311 (2013)

    2013

  27. [35]

    Bravyi, A

    S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, High-threshold and low- overhead fault-tolerant quantum memory, Nature 627, 778–782 (2024)

    2024

  28. [36]

    P. O. Boykin, T. Mor, M. Pulver, V. Roychowdhury, and F. Vatan, A new universal and fault-tolerant quantum basis, Information Processing Letters 75, 101 (2000)

    2000

  29. [37]

    Eastin and E

    B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Physical Review Letters 102, 10.1103/physrevlett.102.110502 (2009)

    2009 doi

  30. [38]

    Rengaswamy, R

    N. Rengaswamy, R. Calderbank, M. Newman, and H. D. Pfister, Classical coding problem from transversal T gates, in 2020 IEEE International Symposium on Information Theory (ISIT)(IEEE, 2020) p. 1891–1896

    2020

  31. [39]

    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)

    2020

  32. [40]

    Panteleev and G

    P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical LDPC codes (2022), arXiv:2111.03654 [cs.IT]

    2022 arXiv

  33. [41]

    Berardini, R

    E. Berardini, R. Dastbasteh, J. E. Martinez, S. Jain, and O. S. Larrarte, Asymptotically good CSS- T codes exist, arXiv preprint arXiv:2412.08586 (2024). 19

    2024 arXiv

  34. [42]

    Berardini, A

    E. Berardini, A. Caminata, and A. Ravagnani, Structure of CSS and CSS- T quantum codes, Designs, Codes and Cryptography 92, 2801–2823 (2024)

    2024

  35. [43]

    Gottesman, The Heisenberg representation of quantum computers, in 22nd International Colloquium on Group Theoretical Methods in Physics(1998) arXiv:quant-ph/9807006 [quant-ph]

    D. Gottesman, The Heisenberg representation of quantum computers, in 22nd International Colloquium on Group Theoretical Methods in Physics(1998) arXiv:quant-ph/9807006 [quant-ph]

    1998 arXiv

  36. [44]

    A. Y. Kitaev, Quantum computations: algorithms and error correction, Russian Mathematical Surveys 52, 1191 (1997)

    1997

  37. [45]

    Aharonov, A simple proof that Toffoli and Hadamard are quantum universal (2003), arXiv:quant- ph/0301040 [quant-ph]

    D. Aharonov, A simple proof that Toffoli and Hadamard are quantum universal (2003), arXiv:quant- ph/0301040 [quant-ph]

    2003

  38. [46]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

    2010

  39. [47]

    Litinski, A game of surface codes: Large-scale quantum computing with lattice surgery, Quantum 3, 128 (2019)

    D. Litinski, A game of surface codes: Large-scale quantum computing with lattice surgery, Quantum 3, 128 (2019)

    2019

  40. [48]

    Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2–30 (2003)

    A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2–30 (2003)

    2003

  41. [49]

    A. J. Landahl, J. T. Anderson, and P. R. Rice, Fault-tolerant quantum computing with color codes (2011), arXiv:1108.5738 [quant-ph]

    2011 arXiv

  42. [50]

    Bravyi and A

    S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Physical Review A 71, 10.1103/physreva.71.022316 (2005)

    2005 doi

  43. [51]

    Bravyi and J

    S. Bravyi and J. Haah, Magic-state distillation with low overhead, Physical Review A 86, 10.1103/phys- reva.86.052329 (2012)

    2012 doi

  44. [52]

    Andrade, J

    E. Andrade, J. Bolkema, T. Dexter, H. Eggers, V. Luongo, F. Manganiello, and L. Szramowski, CSS- T codes from Reed Muller codes (2025), arXiv:2305.06423 [cs.IT]

    2025 arXiv

  45. [53]

    Camps-Moreno, H

    E. Camps-Moreno, H. H. L´ opez, G. L. Matthews, D. Ruano, R. San-Jos´ e, and I. Soprunov, An algebraic characterization of binary CSS- T codes and cyclic CSS- T codes for quantum fault tolerance, Quantum Information Processing 23, 10.1007/s11128-024-04427-5 (2024)

    2024 doi

  46. [54]

    Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Transactions on Information Theory 21, 575 (2003)

    P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Transactions on Information Theory 21, 575 (2003)

    2003

  47. [55]

    Giorgetti and A

    M. Giorgetti and A. Previtali, Galois invariance, trace codes and subfield subcodes, Finite Fields and Their Applications 16, 96 (2010)

    2010

  48. [56]

    Randriambololona, An upper bound of Singleton type for componentwise products of linear codes, IEEE Transactions on Information Theory 59, 7936–7939 (2013)

    H. Randriambololona, An upper bound of Singleton type for componentwise products of linear codes, IEEE Transactions on Information Theory 59, 7936–7939 (2013)

    2013

  49. [57]

    Gao and F.-W

    Z.-H. Gao and F.-W. Fu, Linear recurring sequences and subfield subcodes, in Proceedings of the Fifth International Workshop on Signal Design and Its Applications in Communications(2011) pp. 142–145

    2011

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.