Pith. sign in

REVIEW 2 major objections 5 minor 4 cited by

Quantum Codes with Addressable and Transversal Non-Clifford Gates

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read This paper constructs explicit qubit CSS codes in which any three logical qubits across one, two, or three code blocks can be addressed with a logical CCZ gate implemented by a depth-one circuit of physical CCZ gates, with…

desk verdict Strong Reed-Solomon construction with a genuinely new addressable-orthogonality framework; the asymptotically good AG instantiation is asserted by analogy and needs a real proof before acceptance. read the letter →

arxiv 2502.01864 v3 pith:UJCAOOYT submitted 2025-02-03 quant-ph cs.ITmath.IT

Zhiyang He , Vinod Vaikuntanathan , Adam Wills , Rachel Yun Zhang This is my paper · ORCID
classification quant-phcs.ITmath.IT MSC 81P7094B0594B27 PACS 03.67.Pp03.67.Lx
keywords quantumerrorcorrectionCSScodestransversalgatesaddressablelogicalCCZgatetriorthogonalityReed-Solomonalgebraicgeometry
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

Fault-tolerant quantum computers need error-correcting codes whose logical gates do not spread errors. Transversal gates satisfy this, but nearly all known constructions apply a logical gate to every logical qubit at once, which is far too coarse for high-rate codes that encode many qubits. This paper shows that fine-grained, addressable transversal gates are possible: it constructs an explicit family of qubit CSS codes whose rate and distance are within polylogarithmic factors of optimal, in which any triple of logical qubits across one, two, or three code blocks can be acted on by a logical CCZ gate through a depth-one circuit of physical CCZ gates. It also proves an asymptotically good version in which pre-designed, pairwise disjoint triples inside one code block are addressable. The unifying tool is a new framework, addressable orthogonality, that extends the triorthogonality framework and reduces the code construction to classical codes with multiplication properties.

What carries the argument

The load-bearing object is the addressable orthogonality property of a generator matrix $G$: for each logical index $A$ there is a weight vector $\Gamma^A$ such that weighted sums of products of rows vanish except when all rows coincide at index $A$, where the sum equals a nonzero value. When such a matrix is converted into a CSS code, this condition makes a transversal physical diagonal gate induce the intended logical gate on the addressed qubits. Reed-Solomon codes supply addressable strong $\ell$-orthogonality directly through their evaluation structure, while algebraic geometry codes supply strong $\ell$-multiplication, which yields addressable $(\ell-1)$-orthogonality through a general lemma. Qudit-to-qubit conversion then uses self-dual bases and multiplication-friendly embeddings, with $r = t^3$ qubits per qudit (or a duplicated version for depth-one intra-block gates), to turn the qudit CCZ gate into a depth-one circuit of qubit CCZ gates.

What would settle it

Search for the claimed family of algebraic geometry codes with the strong $\ell$-multiplication property for the needed $\ell$ (in particular $\ell = 7$) over a fixed field: either construct it explicitly from the cited tower of function fields and verify that $m$, $d$, and $d^\perp$ are all $\Theta(N)$, or exhibit an $\ell$ and a field where the simple generalization fails, which would disprove Theorem 5.8 and with it the asymptotically good qubit code of Theorem 1.2.

Watch

Extended reading notes

Core claim

The central discovery is an explicit qubit CSS code family with parameters $[[n, \Omega(n/\mathrm{polylog}(n)), \Omega(n/\mathrm{polylog}(n))]]_2$ that supports a transversal, addressable CCZ gate: given any three logical qubits lying in one, two, or three code blocks, a depth-one circuit of physical CCZ gates applies a logical CCZ gate to exactly those qubits. The paper further proves an asymptotically good family with parameters $[[n, \Theta(n), \Theta(n)]]_2$ in which the logical qubits of one block are partitioned into triples and each pre-designed triple can be addressed with a logical CCZ gate via a depth-one circuit of physical Z, CZ, and CCZ gates. These qubit codes are obtained by first building qudit codes from Reed-Solomon codes (for full addressability) and from algebraic geometry codes (for pre-designed addressability), then converting them to qubit codes while preserving addressability. The authors claim these are the first quantum codes with transversal, addressable non-Clifford gates in a strong sense, going beyond the weak single-index addressability that follows automatically from a transversal higher-level Clifford gate.

Load-bearing premise

The asymptotically good construction rests on the asserted existence of an explicit family of classical algebraic geometry codes over a fixed finite field with the strong $\ell$-multiplication property and with dimension, distance, and dual distance all linear in the block length; the paper cites a simple generalization of a known theorem but does not carry out that construction, so if such a family does not exist the asymptotically good claim collapses.

Editorial extensions

If this is right

  • Full addressability of the CCZ gate is achievable on qubit CSS codes with dimension and distance within polylogarithmic factors of optimal, so high-rate codes need not be limited to global-only transversal non-Clifford operations.
  • The same construction works for $C^{\ell-1}Z$ gates for any $\ell$ and for other diagonal gates with $\pm 1$ diagonal entries, making the addressability result a family of gates rather than a single gate.
  • The addressable orthogonality framework subsumes triorthogonality and its generalizations, giving a unified reduction: build a classical code with a multiplication property, derive an orthogonal matrix, and read off an addressable transversal gate.
  • An asymptotically good qubit code supports addressable CCZ on pre-designed disjoint intra-block triples via a constant-depth circuit of Z, CZ, and CCZ gates, so the asymptotically good regime is compatible with addressability in a weaker sense.
  • Up to Clifford corrections, the framework also yields addressable T gates through addressable triorthogonality, although the paper does not provide an instantiation.

Reading between the lines

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

  • A natural next step, not claimed by the paper, is to bring the Reed-Solomon internal-structure argument to algebraic geometry codes; if that succeeds, the near-asymptotically good full-addressability result would likely become asymptotically good, resolving the paper's Open Problem 1.
  • The framework is strictly stronger than the folklore that a transversal gate at level $\ell+1$ gives an addressable gate at level $\ell$: for univariate gates it avoids a factor-of-two parameter loss, and for polynomial gates the saving is much larger, so the addressable orthogonality conditions should become the default reduction for diagonal Clifford-hierarchy gates.
  • The depth-one physical circuits closely match neutral-atom hardware, where three-atom CCZ interactions are native; a small demonstration of the inter-block addressable CCZ on the qubit code could be a concrete experimental testbed.
  • Combining the qudit-to-qubit conversion (self-dual bases plus multiplication-friendly embeddings) with homological or balanced-product LDPC constructions is a plausible route to addressable non-Clifford gates on low-weight stabilizer codes, which the current non-LDPC codes do not yet provide.
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

2 major / 5 minor

Summary. The paper introduces a framework of "addressable orthogonality" for CSS codes, generalizing the triorthogonality framework of Bravyi and Haah, and uses it to construct quantum codes supporting transversal, addressable non-Clifford gates. The primary result (Theorem 1.1) is an explicit qubit code family with parameters [[n, Ω(n/polylog(n)), Ω(n/polylog(n))]] that supports a logical CCZ gate on any triple of logical qubits across one, two, or three code blocks via a depth-one physical circuit of CCZ gates. This is obtained by combining a self-contained Reed-Solomon qudit construction (Section 3) with qudit-to-qubit conversion and multiplication-friendly embeddings from [Ngu24] (Sections 2.3 and 6.1). The paper also develops the addressable orthogonality framework (Section 4) and proves Theorem 1.2: an asymptotically good qubit code with CCZ gates on pre-designed, pairwise disjoint intra-block triples, based on an algebraic-geometry instantiation (Theorem 5.8) and a further conversion (Section 6.2). Additional results include generalizations to higher C^{ℓ-1}Z and polynomial gates, and an appendix on addressable T gates up to Clifford corrections.

Significance. If the proofs are completed, this is a substantial advance: it gives the first quantum code families with genuinely addressable transversal non-Clifford gates, a flexible framework that extends triorthogonality, and near-asymptotically optimal parameters for the strongest form of addressability. The Reed-Solomon construction and the qubit conversion are presented explicitly and in detail; the near-asymptotically good Theorem 1.1 is internally consistent up to imported lemmas that are standard in the area. However, the asymptotically good theorem rests on an unproved algebraic-geometry existence statement (Theorem 5.8), and Theorem 1.1 depends on the multiplication-friendly embedding lemma from [Ngu24]. Once these dependencies are made precise, the results would be a clear and useful contribution to quantum coding theory.

major comments (2)
  1. [Section 5.3, Theorem 5.8] Theorem 5.8 is load-bearing for Theorem 1.2, but its proof is a single sentence asserting that the result follows by replacing 37 with ℓ in Theorem 3.6 of [Ngu24]. This does not verify the required parameter balance. In particular, for an algebraic-geometry code C_L(D,G) with deg G = cN and genus g = εN, the ℓ-multiplication property requires roughly c < 1/ℓ, while the dual-distance condition d^⊥ - k = Θ(N) requires c to be sufficiently above 2ε plus the puncturing rate; the paper does not state the necessary field-size condition or prove that the Garcia-Stichtenoth tower supplies the required divisor. The statement also uses k in the conclusion m-k, d-k, d^⊥-k = Θ(N) without defining k. Finally, the proof does not establish that the code contains the all-ones vector, which Remark 5.2 needs to upgrade from regular to strong ℓ-multiplication. Because Corollary 5.9, Theorem 5.10, Section 6.2, and Theorem 1.2 all depend on this result, a complete proof or a precise statement of the imported theorem with all parameter conditions is needed.
  2. [Section 6.2, Eq. (190) and footnote 22] The conversion of the W gate to a logical CCZ gate in Theorem 6.10 requires the existence of β̂ satisfying tr(β̂ Σ_{cyc} α_a^4 α_b^2 α_c) = 1. The paper relegates this to a footnote claiming that the needed elements are "easily seen to be obtainable," without a proof. This is a small but load-bearing finite-field point: if no such β̂ exists, the logical action of Π^{β̂} would not reduce to CCZ after the gauge-fixing step. Please supply a short proof, or state explicitly the condition on q and the choice of elementary basis elements α_1, α_2, α_3.
minor comments (5)
  1. [Section 5.3, Theorem 5.8] The parameter k is used in the statement of Theorem 5.8 before it is defined; please introduce it explicitly, e.g., as the dimension of the punctured code used in Lemma 5.3.
  2. [Section 3.1, depth-4 argument] The scheduling of the n physical CCZ gates into a depth-4 circuit is correct but terse. The four triples associated with a given α are the four 3-subsets of a coset of the two-dimensional subspace spanned by Δ_AB and Δ_AC, and the depth-4 coloring can be described directly on these cosets. The current wording, which says there is exactly one gate with a given qudit in a given position, is easy to misread and could be clarified.
  3. [Section 6.1.2, Lemma 6.3] Theorem 1.1 depends crucially on the multiplication-friendly embedding lemma imported from [Ngu24] and on the qudit-to-qubit conversion lemma cited to [WHY24]. These dependencies should be listed explicitly in the proof of Theorem 6.1, so that the reader can see exactly which ingredients are proved here and which are imported.
  4. [Throughout] Several small typos and notation inconsistencies appear: in Eq. (37) a parenthesis is missing in θ(U)(|v⟩) = ψUψ^{-1}(|v⟩); in Section 4.4 the proof of Theorem 4.11 writes U^{βΛ_a}_7 with a lower-case 'a'; in Section 2.2 the phrase "for l > 3" uses 'l' instead of 'ℓ'. These should be corrected.
  5. [Section 5.2, Theorem 5.4] The proof uses α_A to denote an evaluation point, which can be confused with the self-dual basis elements α_i introduced later in Section 6.1.1. Please choose a different symbol for the evaluation points, or state the relation explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the addressable CCZ action is derived from polynomial/multiplication identities, not fitted or defined into existence.

full rationale

The central constructions are not circular. In Section 3, the logical CCZ action is derived by evaluating a physical product of CCZ gates on Reed–Solomon codewords; the interpolation identity (50) converts the resulting phase into u_A u_B u_C, and the logical gate is a consequence rather than an input. In Section 4, addressable orthogonality is an algebraic sufficient condition stated independently of any gate; Theorems 4.6, 4.7 and 4.11 prove that the physical phase operators induce the claimed logical phases via equations (81) and (109). The classical multiplication-property instantiations in Section 5 translate these conditions through Lemma 5.3 and the RS construction of Theorem 5.4, neither of which renames the target gate as the hypothesis. The qubit conversions in Section 6 use the external multiplication-friendly embedding lemma from [Ngu24] and a gauge-fixing step; no fitted parameter is later called a prediction. The only self-citation is to [WHY24], which overlaps with an author, but it is used for background material such as Lemma 2.5 and the Clifford-hierarchy level of U_{q,7}, and the paper explicitly states that the qudit-to-qubit conversions of [WHY24] are too weak and that it exclusively uses [Ngu24, GG24]. The unproved assertion of Theorem 5.8 by analogy to [Ngu24] is a correctness/completeness risk, not a circularity, since [Ngu24] is external and the claimed generalization is not equivalent to the paper's own inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central Reed-Solomon construction is largely self-contained, but the asymptotically good qubit claims inherit two external existence results, namely AG ℓ-multiplication and multiplication-friendly embeddings, plus one unproved beta-hat choice. The free parameters are construction choices rather than empirical fits.

free parameters (3)
  • n, a power of two = power of two, with q = n^2
    Length of the Reed-Solomon code; chosen so that F_q contains a subfield K of order n. This is a construction parameter, not a data fit.
  • m, the Reed-Solomon dimension = ceil(n/3) in the asymptotic choice
    The degree bound m < n/3 + 1 ensures 3m - 3 < n so the dual-RS quadrature applies to the product polynomial. Chosen by hand to balance rate and distance.
  • k, number of logical qudits = m/2 in the asymptotic choice
    Must satisfy k <= m and k < d(C), d(C^⊥); chosen as m/2 to make the code asymptotically good.
assumptions (5)
  • standard math Finite field facts: trace is F_2-linear, self-dual bases exist, duals of RS codes are GRS codes, and Fact 2.11 quadrature formula holds.
    Used throughout Sections 2 and 3; these are standard results with cited proofs.
  • standard math CSS code construction from punctured and shortened classical codes, with the distance bounds in Lemma 2.7 and Fact 2.8.
    Used in Section 2.4 to convert a classical generator matrix into a quantum code with the claimed distance.
  • domain assumption There exists an explicit family of AG codes over a fixed F_q with the strong ℓ-multiplication property and linear m, d, d^⊥.
    Theorem 5.8 is load-bearing for the asymptotically good qubit code in Theorem 1.2, but its proof is deferred to a simple generalization of [Ngu24].
  • domain assumption A degree-3 multiplication-friendly embedding of F_{2^t} into F_2^{t^3} exists (Lemma 6.3, from [Ngu24]).
    Needed for the depth-one physical CCZ implementation on qubits after conversion; cited, not proved.
  • domain assumption The gate U_{q,7} is in exactly the third level of the Clifford hierarchy for sufficiently large t.
    Used to justify that the W gate in Section 4.4 is non-Clifford; cited from [WHY24].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Codes with Addressable and Transversal Non-Clifford Gates." pith.science (2026). https://pith.science/paper/UJCAOOYT

@misc{pith2026250201864,
  author       = {Pith},
  title        = {Pith review of: Quantum Codes with Addressable and Transversal Non-Clifford Gates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UJCAOOYT}},
  note         = {Machine review of arXiv:2502.01864}
}
abstract

The development of quantum codes with good error correction parameters and useful sets of transversal gates is a problem of major interest in quantum error-correction. Abundant prior works have studied transversal gates which are restricted to acting on all logical qubits simultaneously. In this work, we study codes that support transversal gates which induce $\textit{addressable}$ logical gates, i.e., the logical gates act on logical qubits of our choice. As we consider scaling to high-rate codes, the study and design of low-overhead, addressable logical operations presents an important problem for both theoretical and practical purposes. Our primary result is the construction of an explicit qubit code for which $\textit{any}$ triple of logical qubits across one, two, or three codeblocks can be addressed with a logical $\mathsf{CCZ}$ gate via a depth-one circuit of physical $\mathsf{CCZ}$ gates, and whose parameters are asymptotically good, up to polylogarithmic factors. The result naturally generalizes to other gates including the $\mathsf{C}^{\ell} Z$ gates for $\ell \neq 2$. Going beyond this, we develop a formalism for constructing quantum codes with $\textit{addressable and transversal}$ gates. Our framework, called $\textit{addressable orthogonality}$, encompasses the original triorthogonality framework of Bravyi and Haah (Phys. Rev. A 2012), and extends this and other frameworks to study addressable gates. We demonstrate the power of this framework with the construction of an asymptotically good qubit code for which $\textit{pre-designed}$, pairwise disjoint triples of logical qubits within a single codeblock may be addressed with a logical $\mathsf{CCZ}$ gate via a physical depth-one circuit of $\mathsf{Z}$, $\mathsf{CZ}$ and $\mathsf{CCZ}$ gates. In an appendix, we show that our framework extends to addressable and transversal $T$ gates, up to Clifford corrections.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Transversal non-Clifford gates on almost-good quantum LDPC and quantum locally testable codes

    quant-ph 2026-04 conditional novelty 7.5 of 10

    Almost-good qLDPC and qLTC codes admit nontrivial transversal logical multi-controlled-Z gates via cohomological cup products and two-way product-expanding punctured Reed–Solomon local codes.

  2. Construction of the full logical Clifford group for high-rate quantum Reed-Muller codes using only transversal and fold-transversal gates

    quant-ph 2026-02 accept novelty 7.0 of 10

    High-rate self-dual quantum Reed–Muller codes admit ancilla-free addressable Clifford gates generated by transversal H and fold-transversal phase gates.

  3. Asymptotically Good Quantum Codes with Addressable and Transversal Non-Clifford Gates

    quant-ph 2025-07 reject novelty 7.0 of 10

    An asymptotically good qubit CSS code family with constant rate and distance and transversally addressable logical CCZ gates is claimed, built from Stichtenoth's transitive iso-orthogonal algebraic geometry codes.

  4. No-Go Theorem on Fault Tolerant Gadgets for Multiple Logical Qubits

    quant-ph 2026-02 reject novelty 6.0 of 10

    No stabilizer code can implement the full logical Clifford group on multiple logical qubits using transversal gates, fold-transversal gates beyond two qubits, or code automorphisms.

Reference graph

Works this paper leans on

97 extracted references · 44 canonical work pages · cited by 4 Pith papers

  1. [1]

    Quantum error correction below the surface code threshold , 2024

    Rajeev Acharya, Laleh Aghababaie-Beni, Igor Aleiner, Trond I Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Nikita Astrakhantsev, Juan Atalaya, et al. Quantum error correction below the surface code threshold , 2024

    2024

  2. [2]

    Fault-tolerant quantum computation with constant error

    Dorit Aharonov and Michael Ben-Or. Fault-tolerant quantum computation with constant error . In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing , pages 176--188, 1997

    1997

  3. [3]

    Suppressing quantum errors by scaling a surface code logical qubit

    Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit . Nature , 614(7949):676--681, 2023

    2023

  4. [4]

    Breuckmann and Simon Burton

    Nikolas P. Breuckmann and Simon Burton. Fold- T ransversal C lifford G ates for Q uantum C odes . Quantum , 8:1372, jun 2024

    2024

  5. [5]

    High-threshold and low-overhead fault-tolerant quantum memory

    Sergey Bravyi, Andrew W Cross, Jay M Gambetta, Dmitri Maslov, Patrick Rall, and Theodore J Yoder. High-threshold and low-overhead fault-tolerant quantum memory . Nature , 627(8005):778--782, 2024

    2024

  6. [6]

    Cups and Gates I : Cohomology invariants and logical quantum operations

    Nikolas P Breuckmann, Margarita Davydova, Jens N Eberhardt, and Nathanan Tantivasadakarn. Cups and Gates I : Cohomology invariants and logical quantum operations . arXiv preprint arXiv:2410.16250 , 2024

    arXiv 2024

  7. [7]

    Balanced product quantum codes

    Nikolas P Breuckmann and Jens N Eberhardt. Balanced product quantum codes . IEEE Transactions on Information Theory , 67(10):6653--6674, 2021

    2021

  8. [8]

    Quantum low-density parity-check codes

    Nikolas P Breuckmann and Jens Niklas Eberhardt. Quantum low-density parity-check codes . PRX Quantum , 2(4):040101, 2021

    2021

Show all 97 references
  1. [9]

    Logical quantum processor based on reconfigurable atom arrays

    Dolev Bluvstein, Simon J Evered, Alexandra A Geim, Sophie H Li, Hengyun Zhou, Tom Manovitz, Sepehr Ebadi, Madelyn Cain, Marcin Kalinowski, Dominik Hangleiter, et al. Logical quantum processor based on reconfigurable atom arrays . Nature , 626(7997):58--65, 2024

    2024

  2. [10]

    Magic-state distillation with low overhead

    Sergey Bravyi and Jeongwan Haah. Magic-state distillation with low overhead . Physical Review A , 86(5):052329, 2012

    2012

  3. [11]

    Universal quantum computation with ideal Clifford gates and noisy ancillas

    Sergey Bravyi and Alexei Kitaev. Universal quantum computation with ideal Clifford gates and noisy ancillas . Physical Review A , 71(2):022316, 2005

    2005

  4. [12]

    Bombin and M

    H. Bombin and M. A. Martin-Delgado. Topological Quantum Distillation . Physical Review Letters , 97(18), oct 2006

    2006

  5. [13]

    Quantum measurements and gates by code deformation

    H \'e ctor Bomb \' n and Miguel Angel Martin-Delgado. Quantum measurements and gates by code deformation . Journal of Physics A: Mathematical and Theoretical , 42(9):095302, 2009

    2009

  6. [14]

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

    H \'e ctor Bomb \' n. Gauge color codes: optimal transversal gates and gauge fixing in topological stabilizer codes . New Journal of Physics , 17(8):083002, 2015

    2015

  7. [15]

    Dimensional jump in quantum error correction

    H \'e ctor Bomb \' n. Dimensional jump in quantum error correction . New Journal of Physics , 18(4):043038, 2016

    2016

  8. [16]

    Hyperbolic and semi-hyperbolic surface codes for quantum storage

    Nikolas P Breuckmann, Christophe Vuillot, Earl Campbell, Anirudh Krishna, and Barbara M Terhal. Hyperbolic and semi-hyperbolic surface codes for quantum storage . Quantum Science and Technology , 2(3):035007, 2017

    2017

  9. [17]

    Magic-state distillation in all prime dimensions using quantum reed-muller codes

    Earl T Campbell, Hussain Anwar, and Dan E Browne. Magic-state distillation in all prime dimensions using quantum reed-muller codes . Physical Review X , 2(4):041021, 2012

    2012

  10. [18]

    Diagonal gates in the Clifford hierarchy

    Shawn X Cui, Daniel Gottesman, and Anirudh Krishna. Diagonal gates in the Clifford hierarchy . Physical Review A , 95(1):012329, 2017

    2017

  11. [19]

    Improved Q LDPC Surgery: Logical Measurements and Bridging Codes

    Andrew Cross, Zhiyang He, Patrick Rall, and Theodore Yoder. Improved Q LDPC Surgery: Logical Measurements and Bridging Codes . arXiv preprint arXiv:2407.18393 , 2024

    2024

  12. [20]

    Parallel Logical Measurements via Quantum Code Surgery

    Alexander Cowtan, Zhiyang He, Dominic J Williamson, and Theodore J Yoder. Parallel Logical Measurements via Quantum Code Surgery . arXiv preprint arXiv:2503.05003 , 2025

    2025 arXiv

  13. [21]

    Low-overhead fault-tolerant quantum computing using long-range connectivity

    Lawrence Z Cohen, Isaac H Kim, Stephen D Bartlett, and Benjamin J Brown. Low-overhead fault-tolerant quantum computing using long-range connectivity . Science Advances , 8(20):eabn1717, 2022

    2022

  14. [22]

    A. Cowtan. SSIP: automated surgery with quantum LDPC codes . arXiv:2407.09423 , 2024

    2024 arXiv

  15. [23]

    Quantum error correction via codes over gf (4)

    A Robert Calderbank, Eric M Rains, Peter M Shor, and Neil JA Sloane. Quantum error correction via codes over gf (4). IEEE Transactions on Information Theory , 44(4):1369--1387, 1998

    1998

  16. [24]

    Good quantum error-correcting codes exist

    A Robert Calderbank and Peter W Shor. Good quantum error-correcting codes exist . Physical Review A , 54(2):1098, 1996

    1996

  17. [25]

    Good Quantum LDPC Codes with Linear Time Decoders

    Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick. Good Quantum LDPC Codes with Linear Time Decoders . In Proceedings of the 55th Annual ACM Symposium on Theory of Computing , STOC 2023, page 905–918, New York, NY, USA, 2023. Association for Computing Machinery

    2023

  18. [26]

    High-fidelity parallel entangling gates on a neutral-atom quantum computer

    Simon J Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H Li, Alexandra A Geim, Tout T Wang, Nishad Maskara, et al. High-fidelity parallel entangling gates on a neutral-atom quantum computer . Nature , 622(7982):268--272, 2023

    2023

  19. [27]

    In Victor V

    Triorthogonal code . In Victor V. Albert and Philippe Faist, editors, The Error Correction Zoo . 2024

    2024

  20. [28]

    Restrictions on Transversal Encoded Quantum Gate Sets

    Bryan Eastin and Emanuel Knill. Restrictions on Transversal Encoded Quantum Gate Sets . Physical Review Letters , 102:110502, Mar 2009

    2009

  21. [29]

    Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes

    Jens Niklas Eberhardt and Vincent Steffan. Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes . arXiv preprint arXiv:2407.03973 , 2024

    2024 arXiv

  22. [30]

    Constant overhead quantum fault tolerance with quantum expander codes

    Omar Fawzi, Antoine Grospellier, and Anthony Leverrier. Constant overhead quantum fault tolerance with quantum expander codes . Communications of the ACM , 64(1):106--114, 2020

    2020

  23. [31]

    Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations

    Daniel Gottesman and Isaac L Chuang. Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations . Nature , 402(6760):390--393, 1999

    1999

  24. [32]

    Asymptotically Good Quantum Codes with Transversal Non- Clifford Gates , 2024

    Louis Golowich and Venkatesan Guruswami. Asymptotically Good Quantum Codes with Transversal Non- Clifford Gates , 2024

    2024

  25. [33]

    Halving the cost of quantum addition

    Craig Gidney. Halving the cost of quantum addition . Quantum , 2:74, jun 2018

    2018

  26. [34]

    Quantum LDPC Codes with Transversal Non- Clifford Gates via Products of Algebraic Codes

    Louis Golowich and Ting-Chun Lin. Quantum LDPC Codes with Transversal Non- Clifford Gates via Products of Algebraic Codes . arXiv preprint arXiv:2410.14662 , 2024

    2024 arXiv

  27. [35]

    Algebraico-geometric codes

    Valerii Denisovich Goppa. Algebraico-geometric codes . Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya , 46(4):762--781, 1982

    1982

  28. [36]

    Stabilizer codes and quantum error correction

    Daniel Gottesman. Stabilizer codes and quantum error correction . PhD thesis, California Institute of Technology, 1997

    1997

  29. [37]

    Fault-tolerant quantum computation with constant overhead

    Daniel Gottesman. Fault-tolerant quantum computation with constant overhead . arXiv preprint arXiv:1310.2984 , 2013

    2013 arXiv

  30. [38]

    Surviving as a quantum computer in a classical world

    Daniel Gottesman. Surviving as a quantum computer in a classical world . 2024

    2024

  31. [39]

    Leveraging automorphisms of quantum codes for fault-tolerant quantum computation

    Markus Grassl and Martin Roetteler. Leveraging automorphisms of quantum codes for fault-tolerant quantum computation. In 2013 IEEE International Symposium on Information Theory , pages 534--538. IEEE, 2013

    2013

  32. [40]

    A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound

    Arnaldo Garcia and Henning Stichtenoth. A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound . Inventiones mathematicae , 121(1):211--222, 1995

    1995

  33. [41]

    Notes on coding theory

    Jonathan I Hall. Notes on coding theory . Michigan State University, 2003

    2003

  34. [42]

    Extractors: QLDPC Architectures for Efficient Pauli-Based Computation

    Zhiyang He, Alexander Cowtan, Dominic J Williamson, and Theodore J Yoder. Extractors: QLDPC Architectures for Efficient Pauli-Based Computation . arXiv preprint arXiv:2503.10390 , 2025

    2025

  35. [43]

    Distillation with sublogarithmic overhead

    Matthew B Hastings and Jeongwan Haah. Distillation with sublogarithmic overhead . Physical review letters , 120(5):050504, 2018

    2018

  36. [44]

    Fiber bundle codes: breaking the N^ 1/2 polylog (N) barrier for quantum LDPC codes

    Matthew B Hastings, Jeongwan Haah, and Ryan O'Donnell. Fiber bundle codes: breaking the N^ 1/2 polylog (N) barrier for quantum LDPC codes . In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing , pages 1276--1288, 2021

    2021

  37. [45]

    Homomorphic logical measurements

    Shilin Huang, Tomas Jochym-O’Connor, and Theodore J Yoder. Homomorphic logical measurements . PRX Quantum , 4(3):030301, 2023

    2023

  38. [46]

    Classifying Logical Gates in Quantum Codes via Cohomology Operations and Symmetry

    Po-Shen Hsin, Ryohei Kobayashi, and Guanyu Zhu. Classifying Logical Gates in Quantum Codes via Cohomology Operations and Symmetry . arXiv preprint arXiv:2411.15848 , 2024

    2024 arXiv

  39. [47]

    Climbing the diagonal clifford hierarchy

    Jingzhen Hu, Qingzhong Liang, and Robert Calderbank. Climbing the diagonal clifford hierarchy. arXiv preprint arXiv:2110.11923 , 2021

    2021 arXiv

  40. [48]

    Designing the Q uantum C hannels I nduced by D iagonal G ates

    Jingzhen Hu, Qingzhong Liang, and Robert Calderbank. Designing the Q uantum C hannels I nduced by D iagonal G ates . Quantum , 6:802, sep 2022

    2022

  41. [49]

    Cary Huffman and Vera Pless

    W. Cary Huffman and Vera Pless. Fundamentals of Error-Correcting Codes . Cambridge University Press, jun 2003

    2003

  42. [50]

    Decomposition of diagonal hermitian quantum gates using multiple-controlled pauli Z gates

    Mahboobeh Houshmand, Morteza Saheb Zamani, Mehdi Sedighi, and Mona Arabzadeh. Decomposition of diagonal hermitian quantum gates using multiple-controlled pauli Z gates . ACM Journal on Emerging Technologies in Computing Systems (JETC) , 11(3):1--10, 2014

    2014

  43. [51]

    Fault-tolerant logical measurements via homological measurement

    Benjamin Ide, Manoj G Gowda, Priya J Nadkarni, and Guillaume Dauphinais. Fault-tolerant logical measurements via homological measurement . arXiv preprint arXiv:2410.02753 , 2024

    2024 arXiv

  44. [52]

    High-distance codes with transversal Clifford and T -gates

    Shubham P Jain and Victor V Albert. High-distance codes with transversal Clifford and T -gates . arXiv preprint arXiv:2408.12752 , 2024

    2024 arXiv

  45. [53]

    Universal transversal gates with color codes: A simplified approach

    Aleksander Kubica and Michael E Beverland. Universal transversal gates with color codes: A simplified approach . Physical Review A , 91(3):032330, 2015

    2015

  46. [54]

    E. Knill. Scalable quantum computing in the presence of large detected-error rates . Physical Review A , 71(4), apr 2005

    2005

  47. [55]

    Fault-tolerant gates on hypergraph product codes

    Anirudh Krishna and David Poulin. Fault-tolerant gates on hypergraph product codes . Physical Review X , 11(1):011023, 2021

    2021

  48. [56]

    Magic state distillation with punctured polar codes

    Anirudh Krishna and Jean-Pierre Tillich. Magic state distillation with punctured polar codes . arXiv preprint arXiv:1811.03112 , 2018

    2018 arXiv

  49. [57]

    Towards low overhead magic state distillation

    Anirudh Krishna and Jean-Pierre Tillich. Towards low overhead magic state distillation . Physical review letters , 123(7):070507, 2019

    2019

  50. [58]

    Unfolding the color code

    Aleksander Kubica, Beni Yoshida, and Fernando Pastawski. Unfolding the color code . New Journal of Physics , 17(8):083026, aug 2015

    2015

  51. [59]

    Scaling and logic in the color code on a superconducting quantum processor

    Nathan Lacroix, Alexandre Bourassa, Francisco JH Heras, Lei M Zhang, Johannes Bausch, Andrew W Senior, Thomas Edlich, Noah Shutty, Volodymyr Sivak, Andreas Bengtsson, et al. Scaling and logic in the color code on a superconducting quantum processor . arXiv preprint arXiv:2412....

    2024

  52. [60]

    Transversal non- Clifford gates for quantum LDPC codes on sheaves

    Ting-Chun Lin. Transversal non- Clifford gates for quantum LDPC codes on sheaves . arXiv preprint arXiv:2410.14631 , 2024

    2024 arXiv

  53. [61]

    Parallel implementation of high-fidelity multiqubit gates with neutral atoms

    Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Tout T Wang, Sepehr Ebadi, Hannes Bernien, Markus Greiner, Vladan Vuleti \'c , Hannes Pichler, et al. Parallel implementation of high-fidelity multiqubit gates with neutral atoms . Physical review letters , 123(1...

    2019

  54. [62]

    Finite Fields

    Rudolf Lidl and Harald Niederreiter. Finite Fields . Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2 edition, 1996

    1996

  55. [63]

    Quantum two-block group algebra codes

    Hsiang-Ku Lin and Leonid P Pryadko. Quantum two-block group algebra codes . Physical Review A , 109(2):022407, 2024

    2024

  56. [64]

    Quantum expander codes

    Anthony Leverrier, Jean-Pierre Tillich, and Gilles Z \'e mor. Quantum expander codes . In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science , pages 810--824. IEEE, 2015

    2015

  57. [65]

    Leverrier and G

    A. Leverrier and G. Zemor. Quantum Tanner codes . In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) , pages 872--883, Los Alamitos, CA, USA, nov 2022. IEEE Computer Society

    2022

  58. [66]

    Quynh T. Nguyen. Good binary quantum codes with transversal CCZ gate , 2024

    2024

  59. [67]

    Quantum fault tolerance with constant-space and logarithmic-time overheads

    Quynh T Nguyen and Christopher A Pattison. Quantum fault tolerance with constant-space and logarithmic-time overheads . arXiv preprint arXiv:2411.03632 , 2024

    2024 arXiv

  60. [68]

    Degenerate quantum LDPC codes with good finite length performance

    Pavel Panteleev and Gleb Kalachev. Degenerate quantum LDPC codes with good finite length performance . Quantum , 5:585, 2021

    2021

  61. [69]

    Quantum LDPC codes with almost linear minimum distance

    Pavel Panteleev and Gleb Kalachev. Quantum LDPC codes with almost linear minimum distance . IEEE Transactions on Information Theory , 68(1):213--229, 2021

    2021

  62. [70]

    Asymptotically good Quantum and locally testable classical LDPC codes

    Pavel Panteleev and Gleb Kalachev. Asymptotically good Quantum and locally testable classical LDPC codes . In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing , STOC 2022, page 375–388, New York, NY, USA, 2022. Association for Computing Machinery

    2022

  63. [71]

    Reichardt

    Adam Paetznick and Ben W. Reichardt. Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction . Physical Review Letters , 111(9), aug 2013

    2013

  64. [72]

    Quintavalle, Paul Webster, and Michael Vasmer

    Armanda O. Quintavalle, Paul Webster, and Michael Vasmer. Partitioning qubits in hypergraph product codes to implement logical gates . Quantum , 7:1153, oct 2023

    2023

  65. [73]

    On products and powers of linear codes under componentwise multiplication

    Hugues Randriambololona. On products and powers of linear codes under componentwise multiplication . Algorithmic arithmetic, geometry, and coding theory , 637:3--78, 2015

    2015

  66. [74]

    On optimality of CSS codes for transversal T

    Narayanan Rengaswamy, Robert Calderbank, Michael Newman, and Henry D Pfister. On optimality of CSS codes for transversal T . IEEE Journal on Selected Areas in Information Theory , 1(2):499--514, 2020

    2020

  67. [75]

    Unifying the clifford hierarchy via symmetric matrices over rings

    Narayanan Rengaswamy, Robert Calderbank, and Henry D Pfister. Unifying the clifford hierarchy via symmetric matrices over rings. Physical Review A , 100(2):022304, 2019

    2019

  68. [76]

    Experimental Demonstration of Logical Magic State Distillation

    Pedro Sales Rodriguez, John M Robinson, Paul Niklas Jepsen, Zhiyang He, Casey Duckering, Chen Zhao, Kai-Hsin Wu, Joseph Campo, Kevin Bagnall, Minho Kwon, et al. Experimental Demonstration of Logical Magic State Distillation . arXiv preprint arXiv:2412.15165 , 2024

    2024

  69. [77]

    Polynomial codes over certain finite fields

    Irving S Reed and Gustave Solomon. Polynomial codes over certain finite fields . Journal of the society for industrial and applied mathematics , 8(2):300--304, 1960

    1960

  70. [78]

    Scheme for reducing decoherence in quantum computer memory

    Peter W Shor. Scheme for reducing decoherence in quantum computer memory . Physical review A , 52(4):R2493, 1995

    1995

  71. [79]

    Universal adapters between quantum LDPC codes

    Esha Swaroop, Tomas Jochym-O'Connor, and Theodore J Yoder. Universal adapters between quantum LDPC codes . arXiv preprint arXiv:2410.03628 , 2024

    2024

  72. [80]

    Fault-Tolerant Logical Clifford Gates from Code Automorphisms

    Hasan Sayginel, Stergios Koutsioumpas, Mark Webster, Abhishek Rajput, and Dan E Browne. Fault-Tolerant Logical Clifford Gates from Code Automorphisms . arXiv preprint arXiv:2409.18175 , 2024

    2024 arXiv

  73. [81]

    Factorization of symmetric matrices and trace-orthogonal bases in finite fields

    Gadiel Seroussi and Abraham Lempel. Factorization of symmetric matrices and trace-orthogonal bases in finite fields . SIAM Journal on Computing , 9(4):758--767, 1980

    1980

  74. [82]

    Quantum rainbow codes

    Thomas R Scruby, Arthur Pesah, and Mark Webster. Quantum rainbow codes . arXiv preprint arXiv:2408.13130 , 2024

    2024

  75. [83]

    Multiple-particle interference and quantum error correction

    Andrew Steane. Multiple-particle interference and quantum error correction . Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences , 452(1954):2551--2577, 1996

    1954

  76. [84]

    Algebraic function fields and codes , volume 254

    Henning Stichtenoth. Algebraic function fields and codes , volume 254. Springer Science & Business Media, 2009

    2009

  77. [85]

    Polylog-time-and constant-space-overhead fault-tolerant quantum computation with quantum low-density parity-check codes

    Shiro Tamiya, Masato Koashi, and Hayata Yamasaki. Polylog-time-and constant-space-overhead fault-tolerant quantum computation with quantum low-density parity-check codes . arXiv preprint arXiv:2411.03683 , 2024

    2024 arXiv

  78. [86]

    Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength

    Jean-Pierre Tillich and Gilles Z \'e mor. Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength . IEEE Transactions on Information Theory , 60(2):1193--1202, 2013

    2013

  79. [87]

    Three-dimensional surface codes: Transversal gates and fault-tolerant architectures

    Michael Vasmer and Dan E Browne. Three-dimensional surface codes: Transversal gates and fault-tolerant architectures . Physical Review A , 100(1):012312, 2019

    2019

  80. [88]

    Quantum pin codes

    Christophe Vuillot and Nikolas P Breuckmann. Quantum pin codes . IEEE Transactions on Information Theory , 68(9):5955--5974, 2022

    2022

  81. [89]

    Constant-overhead magic state distillation

    Adam Wills, Min-Hsiu Hsieh, and Hayata Yamasaki. Constant-overhead magic state distillation . arXiv preprint arXiv:2408.07764 , 2024

    2024 arXiv

  82. [90]

    Transversal diagonal logical operators for stabiliser codes

    Mark A Webster, Armanda O Quintavalle, and Stephen D Bartlett. Transversal diagonal logical operators for stabiliser codes. New Journal of Physics , 25(10):103018, October 2023

    2023

  83. [91]

    Low-overhead fault-tolerant quantum computation by gauging logical operators

    Dominic J Williamson and Theodore J Yoder. Low-overhead fault-tolerant quantum computation by gauging logical operators . arXiv preprint arXiv:2410.02213 , 2024

    2024 arXiv

  84. [92]

    Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays

    Qian Xu, J Pablo Bonilla Ataides, Christopher A Pattison, Nithin Raveendran, Dolev Bluvstein, Jonathan Wurtz, Bane Vasi \'c , Mikhail D Lukin, Liang Jiang, and Hengyun Zhou. Constant-overhead fault-tolerant quantum computation with reconfigurable atom arrays . Nature Physics ,...

    2024

  85. [93]

    Fast and parallelizable logical computation with homological product codes

    Qian Xu, Hengyun Zhou, Guo Zheng, Dolev Bluvstein, J Ataides, Mikhail D Lukin, and Liang Jiang. Fast and parallelizable logical computation with homological product codes. arXiv preprint arXiv:2407.18490 , 2024

    2024 arXiv

  86. [94]

    Universal fault-tolerant gates on concatenated stabilizer codes

    Theodore J Yoder, Ryuji Takagi, and Isaac L Chuang. Universal fault-tolerant gates on concatenated stabilizer codes . Physical Review X , 6(3):031039, 2016

    2016

  87. [95]

    A topological theory for qLDPC : non- Clifford gates and magic state fountain on homological product codes with constant rate and beyond the N^ 1/3 distance barrier

    Guanyu Zhu. A topological theory for qLDPC : non- Clifford gates and magic state fountain on homological product codes with constant rate and beyond the N^ 1/3 distance barrier . arXiv preprint arxiv:2501.19375 , 2025

    2025

  88. [96]

    Time-efficient logical operations on quantum LDPC codes

    Guo Zhang and Ying Li. Time-efficient logical operations on quantum LDPC codes . arXiv preprint arXiv:2408.01339 , 2024

    2024 arXiv

  89. [97]

    Non- Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries

    Guanyu Zhu, Shehryar Sikander, Elia Portnoy, Andrew W Cross, and Benjamin J Brown. Non- Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries . arXiv preprint arXiv:2310.16982 , 2023

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.