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arXiv:2609.35129

Asymptotically Good Quantum Codes with Addressable Transversal T Gates

Tongyin Lin, Bujiao Wu, Bin Cheng

quant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

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 $T$ gates. Specifically, any prescribed tensor product of logical powers of $T$ is implemented by a tensor product of physical powers of $T$, 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 $T$ 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.

Comments
27 pages, no figures. Comments are welcome

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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.
In this work, we study quantum CSS codes with transversal TTT gates. Here, TTT gate transversality is meant in the strongest sense; the application of physical TTT to every physical qubit yields logical TTT on every logical qubit, without Clifford corrections. Despite the importance of the TTT gate in fault-tolerant quantum computation, the parameters of asymptotic families of such codes have not been improved since the work of Hastings and Haah in 2017, and Haah in 2018. In this work, we significantly broaden the achievable parameters of quantum code families with transversal TTT gates, both expanding the regime of achievable polynomial rate and distance, and constructing such codes with constant rate and growing distance; this is the first time the latter has been achieved, even when allowing Clifford corrections after the transversal TTT gate. These are also the first codes achieving γ→0γ\to 0γ→0 for a code with a transversal TTT gate, where γγγ is the overhead exponent of magic state distillation. To do this, we develop a framework of divisible decreasing monomial codes, punctured at a downward-closed set on the Boolean hypercube to create logical qubits. We prove a closed-form expression for the distance of such a code punctured at such a set, which may be of independent interest. We first instantiate this with an explicit construction based on weighted Reed-Muller codes, puncturing at low Hamming-weight points, and then with a randomised construction, where a small random set of points is protected from the puncturing to save quantum code distance, achieving improved parameters.
Recent work by Wills constructed codes with constant rate and transversal TTT. Extending this line of work, we construct asymptotically good CSS codes with \textbf{fully addressable} transversal TTT, permitting selective control over which logical qubits receive a TTT gate. We also give Pauli-LDPC codes with transversal TTT and growing distance at vanishing rate; and a protected-subsystem construction with fully addressable transversal TTT, constant protected rate, distance Ω(n)Ω(\sqrt n)Ω(n​), and sparse (non-Pauli) checks. More generally, we obtain asymptotically good binary codes with full transversal addressability for every fixed finite-order one-qubit gate, retaining CSS realizations for dyadic phase gates such as T\sqrt TT​ and using generally nonadditive codes for rotations such as Rz(π/7)R_z(π/7)Rz​(π/7). The remaining challenge is to construct asymptotically good Pauli-LDPC codes with fully addressable transversal non-Clifford one-qubit gates.
Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.
Constructing quantum codes with good parameters and useful transversal gates is a central problem in quantum error correction. In this paper, we continue our work in arXiv:2502.01864 and construct the first family of asymptotically good quantum codes (over qubits) supporting transversally addressable non-Clifford gates. More precisely, given any three logical qubits across one, two, or three codeblocks, the logical CCZ\mathsf{CCZ}CCZ gate can be executed on those three logical qubits via a depth-one physical circuit of CCZ\mathsf{CCZ}CCZ gates. This construction is based on the transitive, iso-orthogonal algebraic geometry codes constructed by Stichtenoth (IEEE Trans. Inf. Theory, 2006). This improves upon our construction from arXiv:2502.01864, which also supports transversally addressable CCZ\mathsf{CCZ}CCZ gates and has inverse-polylogarithmic rate and relative distance.
It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford Cr−1ZC^{r-1}ZCr−1Z gate, for every constant r≥3r\geq 3r≥3. In particular, we obtain length-nnn quantum LDPC codes (with constant-weight stabilizers) of polynomial distance d≥n(1−ε)/rd\geq n^{(1-ε)/r}d≥n(1−ε)/r supporting transversal Cr−1ZC^{r-1}ZCr−1Z gates on a close-to-linear number k≥n1−εk\geq n^{1-ε}k≥n1−ε of disjoint tuples of logical qubits, for arbitrarily small ε>0ε>0ε>0. Our construction is the first with constant-weight stabilizers that obtains dk≫ndk\gg ndk≫n, and as a consequence achieves arbitrarily small magic state overhead exponent γ=log⁡(n/k)/log⁡(d)>0γ=\log(n/k)/\log(d)>0γ=log(n/k)/log(d)>0. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many k=Ω(n)k=Ω(n)k=Ω(n) logical Cr−1ZC^{r-1}ZCr−1Z gates, though with stabilizer weight and physical circuit depth nεn^εnε. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits. To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal Cr−1ZC^{r-1}ZCr−1Z. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.
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.