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arXiv:2609.08520

Quantum Matrix-Product Codes: CSS-T Characterization and Maximality

Delio Jaramillo-Velez, Alessandro Neri, Adway Patra, Diego Ruano, Flavio Salizzoni

Published Sep 8, 2026• 0 citations
arXivHTMLPDF

Abstract

CSS-T codes are quantum error-correcting codes that play an important role in fault-tolerant quantum computation, as they help mitigate the proliferation of errors. They admit a transversal T-gate and are defined from a pair of nested classical binary linear codes satisfying a specific algebraic condition expressed in terms of their Schur square. We extend this algebraic characterization to the propagation rule known as the $(u \mid u+v)$-construction, that is, the matrix-product code constructed from two constituent codes. Moreover, for cyclic constituent codes, we provide an explicit characterization in terms of the defining cyclotomic sets, which extends existing results for cyclic codes. This framework allows the construction of new and longer CSS-T codes.

DOI
10.48550/arxiv.2609.08520

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CSS-T codes are quantum error-correcting codes that play an important role in fault-tolerant quantum computation, as they help mitigate the proliferation of errors. They admit a transversal T-gate and are defined from a pair of nested classical binary linear codes satisfying a specific algebraic condition expressed in terms of their Schur square. We extend this algebraic characterization to the propagation rule known as the (u∣u+v)(u \mid u+v)(u∣u+v)-construction, that is, the matrix-product code constructed from two constituent codes. Moreover, for cyclic constituent codes, we provide an explicit characterization in terms of the defining cyclotomic sets, which extends existing results for cyclic codes. This framework allows the construction of new and longer CSS-T codes.
Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C1,C2)(C_1, C_2)(C1​,C2​) ( C 1 , C 2 ) such that C1C_1C1​ C 1 contains C2C_2C2​ C 2 , C2C_2C2​ C 2 is even, and the shortening of the dual of C1C_1C1​ C 1 with respect to the support of each codeword of C2C_2C2​ C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C1,C2)(C_1, C_2)(C1​,C2​) ( C 1 , C 2 ) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.
Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.\nWe prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual.
Designing quantum codes with both transversal non-Clifford gates and good error-correcting parameters is an important goal in fault-tolerant quantum computation. Here, we construct an explicit family of binary CSS codes with constant rate and linear distance that admit fully addressable transversal TTT gates. Specifically, any prescribed tensor product of logical powers of TTT is implemented by a tensor product of physical powers of TTT, without any subsequent correction. Our construction combines algebraic-geometry codes with suitable binary embedding to obtain generalized divisibility of binary codes. This divisibility then enables fully addressable transversal TTT gates. Furthermore, we formulate the minimum binary embedding length problem as a minimum-weight problem over an affine space and numerically improve the constants in the asymptotic rate and relative-distance bounds.
Realizing Logical Diagonal Gates via Transversal Physical ZZZ-Rotations in CSS Codes
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2⊆C1C_2 \subseteq C_1C2​⊆C1​, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C1,C2)(C_1, C_2)(C1​,C2​) whose resulting CSS codes realize a target logical diagonal gate via transversal physical ZZZ-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit ZZZ-rotations and multi-qubit controlled ZZZ-rotations via transversal physical ZZZ-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an [[n′,k′]][[n',k']][[n′,k′]] CSS code Q′Q'Q′ and a target logical ZZZ-rotation (single-qubit or multi-controlled) ULU_LUL​, and extends Q′Q'Q′ by systematically appending n′′n''n′′ physical qubits to obtain an [[n,k]][[n,k]][[n,k]] CSS code QQQ with n=n′+n′′n = n'+n''n=n′+n′′ and k=k′k=k'k=k′. The target logical gate ULU_LUL​ is realized in QQQ by applying a well-chosen physical transversal ZZZ-rotation to the n′′n''n′′ appended physical qubits. Moreover, any logical gate realized via transversal physical ZZZ-rotations in the input code Q′Q'Q′ remains transversally realizable in the extended code QQQ. The CSS code QQQ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code Q′Q'Q′ to obtain a CSS code QQQ that supports fault-tolerant implementations of multiple desired logical ZZZ-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled ZZZ-rotations.
Transversal $ T $ gates, which are seen as the most common non-Clifford gates, are an essential component of the universal set for fault tolerance. CSS-T codes are proposed to reduce the cost of implementing logical non-Clifford gates. So far, there have been limited studies on CSS-T codes. In this paper, we proposed new constructions of binary quantum CSS-T codes from four classes of special functions. Our constructions are based on a general construction of linear codes over the finite field $ \mathbb F_2 $, and we prove that such a pair of codes is actually a CSS-T pair under the given conditions. The resulting CSS-T codes are new compared with these codes in the literature.
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