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Advances in Mathematics of Communications

Constructions of binary Quantum CSS-T codes from some special functions

Bin Zheng, Shixin Zhu

Published Dec 5, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.3934/amc.2026003

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Transversal TTT 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 F2\mathbb F_2F2​, 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.
Universal quantum computation requires the implementation of a logical non-Clifford gate. In this paper, we characterize all stabilizer codes whose code subspaces are preserved under physical T and T†gates. For example, this could enable magic state distillation with non-CSS codes and, thus, provide better parameters than CSS-based protocols. However, among non-degenerate stabilizer codes that support transversal T, we prove that CSS codes are optimal. We also show that triorthogonal codes are, essentially, the only family of CSS codes that realize logical transversal T via physical transversal T. Using our algebraic approach, we reveal new purely-classical coding problems that are intimately related to the realization of logical operations via transversal T. Decreasing monomial codes are also used to construct a code that realizes logical CCZ. Finally, we use Ax's theorem to characterize the logical operation realized on a family of quantum Reed-Muller codes. This result is generalized to finer angle Z-rotations in https://arxiv.org/abs/1910.09333.
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.
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.
The challenge of quantum computing is to combine error resilience with universal computation. There are many finite sets of gates that are universal, and a standard choice is to augment the set of Clifford gates by a non-Clifford unitary such as the T gate. Given a CSS code, we introduce a method of synthesizing all possible diagonal physical gates that preserve the codespace and induce a target logical gate. We denote an 〚n, k = k1− k2, d〛 CSS code C\mathcal{C}C by CSS⁡(X,C2;Z,C1⊥)\operatorname{CSS} \left( {X,{\mathcal{C}_2};Z,\mathcal{C}_1^ \bot } \right)CSS(X,C2​;Z,C1⊥​), where the [n, k2] binary code C2{\mathcal{C}_2}C2​ determines the X-stabilizers in C\mathcal{C}C, and the [n, n−k1] binary code $\mathcal{C}_1^ \bot $ determines the Z-stabilizers in C\mathcal{C}C. The diagonal entries of a diagonal physical gate are indexed by binary vectors in F2n\mathbb{F}_2^nF2n​. We show that a diagonal physical gate preserves the CSS codespace if and only if entries from the same coset of C2{\mathcal{C}_2}C2​ in C1{\mathcal{C}_1}C1​ (same X-logical) are identical. We also show that the target logical operator only specifies 2k1{2^{{k_1}}}2k1​ out of 2ndiagonal entries of the diagonal physical gate. The remaining degrees of freedom can be used to optimize implementation of the physical gate within a particular quantum computing infrastructure. This encompasses optimization with respect to locality of the physical gate, a criterion that is essential to fault tolerance. When the target logical operator is the identity, the physical gates that preserve the CSS code represent noise operators to which the codespace is oblivious. We illustrate our method by providing several examples of code-gate pairs for which the target logical gate is a non-Clifford unitary. The framework is extended to stabilizer codes in https://arxiv.org/abs/2109.13481.
Finding efficient schemes for non-Clifford logic or magic state preparation is one of the central challenges on the way to fault-tolerant quantum computation. Many of the proposed schemes rely on diagonal non-Clifford logical gates acting on CSS codes in space or decorating CSS-type syndrome-extraction circuits in spacetime. Here we propose and implement efficient algorithms to find all (spacetime) logical gates of a given CSS code (circuit) composed from a prescribed set of ansatz gates. Depending on the choice of ansatz gates, this means finding transversal gates, more general locality-preserving logical circuits, folding gates, or similar. While we focus on qubit diagonal gates in the Clifford hierarchy, we also discuss the generalization to arbitrary diagonal non-hierarchy gates, certain non-diagonal gates, as well as prime and composite-dimensional qudits. Our method works by rephrasing code-space preserving gates as the kernel of the ``pullback'' of the XXX check matrix onto phase functions, which maps between finite abelian 2-groups. We implement a fast ``filtration'' method to find this kernel. The runtime for finding fault-tolerant logical gates in a qLDPC code with O(n)O(n)O(n) qubits or a circuit with O(n)O(n)O(n) gates in a naive dense implementation is O(n3)O(n^3)O(n3), with potential for improvement making use of sparsity.
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.