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arXiv:2412.08586 · IEEE Journal on Selected Areas in Information Theory

Asymptotically Good CSS-T Codes and a New Construction of Triorthogonal Codes

Elena Berardini, Reza Dastbasteh, Josu Etxezarreta Martínez, Shreyas Jain, Olatz Sanz Larrarte

Published Jan 1, 2025• 2 citations
arXivHTMLPDF

Abstract

We propose a new systematic construction of CSS-T codes from any given CSS code using a map ϕ. When ϕ is the identity map I, we retrieve the construction of hu2021mitigating and use it to prove the existence of asymptotically good binary CSS-T codes, resolving a previously open problem in the literature, and of asymptotically good quantum LDPC CSS-T codes. We analyze the structure of the logical operators corresponding to certain non-Clifford gates supported by the quantum codes obtained from this construction (ϕ=I), concluding that they always result in the logical identity. An immediate application of these codes in dealing with coherent noise is discussed. We then develop a new doubling transformation for obtaining triorthogonal codes, which generalizes the doubling construction presented in jain2024. Our approach permits using self-orthogonal codes, instead of only doubly-even codes, as building blocks for triorthogonal codes. This broadens the range of codes available for magic state distillation.

DOI
10.1109/jsait.2025.3582156

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We propose a new systematic construction of CSS-T codes from any given CSS code using a map ϕ. When ϕ is the identity map I, we retrieve the construction of hu2021mitigating and use it to prove the existence of asymptotically good binary CSS-T codes, resolving a previously open problem in the literature, and of asymptotically good quantum LDPC CSS-T codes. We analyze the structure of the logical operators corresponding to certain non-Clifford gates supported by the quantum codes obtained from this construction (ϕ=I), concluding that they always result in the logical identity. An immediate application of these codes in dealing with coherent noise is discussed. We then develop a new doubling transformation for obtaining triorthogonal codes, which generalizes the doubling construction presented in jain2024. Our approach permits using self-orthogonal codes, instead of only doubly-even codes, as building blocks for triorthogonal codes. This broadens the range of codes available for magic state distillation.
For each level of the Clifford hierarchy from the third level onward, we construct explicit asymptotically good binary CSS codes for which a strongly transversal diagonal rotation, followed by a correction from the preceding level, implements the corresponding logical rotation. We achieve this using algebraic geometry codes over binary extension fields and performing an alphabet reduction that translates orthogonality conditions over the extension field to binary overlap conditions. In particular, we obtain the first asymptotically good family of binary triorthogonal codes. We also give a stronger construction with worse parameters for which the strongly transversal physical rotation implements the desired logical rotation, with no subsequent correction. Finally, adapting an error-correction-based distillation protocol, our binary TTT-gate families yield direct constant-overhead TTT-state block distillation from noisy TTT-state inputs under sufficiently weak input noise and ideal stabilizer operations.
Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical T\mathsf{T}T gates. As a non-Clifford operation, the T\mathsf{T}T gate enables universal fault-tolerant quantum computation when combined with Clifford operations. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate) that are not necessarily constructed from triply even binary linear codes. By imposing automorphism-group constraints, the formulation also recovers the known [[49,1,5]] and [[95,1,7]] triorthogonal codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding (AED) and guessing random additive-noise decoding (GRAND) are promising options.
This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal ZZZ-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 TTT realizes the logical transversal S†S^{\dagger}S†; (b) show that the condition C2∗C1⊆C1⊥C_2 \ast C_1 \subseteq C_1^{\perp}C2​∗C1​⊆C1⊥​ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal TTT implements the logical identity and the logical transversal TTT, respectively.
We give an asymptotically good family of quantum CSS codes on qubits with a transversal CCZ gate, meaning that the parallel logical CCZ on all logical qubits is performed by parallel physical CCZs on (a subset of) physical qubits. The construction is based on the observation that any classical code satisfying a multiplication property can be used to construct a quantum CSS code with transversal (qudit) CCZ. To obtain a constant-rate and linear-distance family, we then instantiate this construction with a classical good family of algebraic-geometry codes on a non-binary, but constant-sized, alphabet. Finally, we use a technique from the arithmetic secret sharing literature to reduce the alphabet to binary. As a corollary, the constructed code family provides a magic state distillation scheme with constant space overhead.
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Qudits offer the potential for low-overhead magic state distillation, although previous results for asymptotically good codes have required qudit dimension q≫100q\gg 100q≫100 or code length N≫100\mathcal{N}\gg 100N≫100. These parameters far exceed experimental demonstrations of qudit platforms, and thus motivate the search for better codes. Using a novel lifting procedure, we construct the first family of good triorthogonal codes on the F22m\mathbb{F}_{2^{2m}}F22m​ alphabet with m≥3m \geq 3m≥3 that lies above the Tsfasman-Vladut-Zink bound. These codes yield a family of asymptotically good quantum codes with transversal CCZ gates, enabling constant space overhead magic state distillation with qudit dimension as small as q=64q=64q=64. Further, we identify a promising code with parameters [[42,14,6]]64[[42,14,6]]_{64}[[42,14,6]]64​. Finally, we show that a distilled ∣CCZ⟩22m|{CCZ}\rangle_{2^{2m}}∣CCZ⟩22m​ can be reduced to a ∣CCZ⟩2n|{CCZ}\rangle_{2^n}∣CCZ⟩2n​ state for arbitrary nnn with a constant-depth Clifford circuit of at most 9 computational basis measurements, 12 single-qudit and 9 two-qudit Clifford gates.