Of the gates a fault-tolerant quantum computer needs, the T gate is the expensive one. Clifford gates are comparatively easy to protect, but they are not universal on their own, and the standard ways of supplying T, such as magic-state distillation or code switching, cost qubits and time. A code on which T can be applied transversally, one physical qubit at a time, is therefore valuable. A code on which T can be aimed at any single chosen logical qubit, while the code keeps constant rate and linear distance, has been an open construction target.
A new preprint by Tongyin Lin, Bujiao Wu and Bin Cheng, “Asymptotically Good Quantum Codes with Addressable Transversal T Gates” (arXiv:2609.35129v1), gives an explicit construction of such a family. This article was drafted with Claude and fact-checked against the cited primary sources.
What the paper does
The authors define fully addressable transversal T gates: for a fixed encoding of K logical qubits, every tensor product of logical powers of T (each exponent taken mod 8) must be implemented by a tensor product of single-qubit physical unitaries. Choosing the exponent vector to be a unit vector applies T to one logical qubit and the identity to all others. In their construction the physical gates are themselves powers of T, and no Clifford correction is needed afterwards.
The main result (Theorem 4.5, stated informally as Theorem 1.2) is an explicit family of binary CSS codes with parameters [[N, K = Θ(N), D = Θ(N)]] that has this property. The physical exponents depend linearly, over the integers mod 8, on the requested logical exponents.
The context matters. The Eastin–Knill theorem rules out a code with a universal set of transversal gates, so transversal T is a building block rather than a complete solution. Earlier work cited by the authors obtained asymptotically good binary codes with transversal CCZ (Golowich and Guruswami; Nguyen), addressable CCZ (He et al.), and asymptotically good CSS-T codes on which physical transversal T acts as the logical identity or as logical S†. Constructions that do realise logical T either had vanishing rate (Hastings and Haah; Haah) or constant rate with growing but not linear distance (Wills). The paper also cites independent, concurrent work by San-José that obtains asymptotically good codes with transversal diagonal rotations at every level of the Clifford hierarchy from the third upward, with addressable rotations via a hierarchy-lowering argument; Lin, Wu and Cheng take a more direct route to addressable T.
The key idea: divisibility as a phase dial
Applying Tbj to physical qubit j multiplies a computational basis state |w⟩ by eiπ/4 raised to the weighted weight Σj bjwj, taken mod 8. So if, for every codeword in a coset, that weighted weight equals the phase the logical T pattern should produce, the physical layer acts exactly as the desired logical gate.
The paper packages this as weighted 8-divisibility: the set of coefficient vectors a in (ℤ8)n for which every codeword of a binary code has a-weighted weight divisible by 8. Haah studied this with odd coefficient vectors; the new ingredient is allowing even and zero coefficients too, so that each logical qubit’s phase can be prescribed independently. Lemma 3.1 shows that if such coefficient vectors exist for every information coordinate, then shortening and puncturing the classical code on those coordinates (a Krishna–Tillich style construction) produces a CSS code with fully addressable transversal T.
How the construction works
Parity lifting (Lemmas 4.1–4.2). If a binary code has the 4-multiplication property (the component-wise product of any five codewords has even weight), a coefficient vector known only mod 2 can be lifted to mod 4 and then mod 8 by solving two binary linear systems. Inverting a matrix over ℤ8 then yields the prescribed per-qubit coefficients. All of this is polynomial time.
Algebraic-geometry codes (Lemma 4.3). From an optimal Garcia–Stichtenoth function-field tower over the field with 1024 elements, the authors build AG codes with the 15-multiplication property (products of sixteen codewords sum to zero). Their rate tends to 17/496, with relative distance at least 463/496 and dual relative distance at least 1/496.
Binary embedding (Lemma 4.4). A fixed linear map σ from the 1024-element field into L bits, built from a trace of a product over odd-sized subsets of five inputs, turns the sixteenfold field identity into vanishing fivefold binary overlaps, i.e. the binary 4-multiplication property. The generic bound is L ≤ 637.
Selecting logical coordinates (Theorem 4.5). K field coordinates are restricted to a one-dimensional binary subspace generated by a trace-one element and compressed to single bits, with K = ⌊n/992⌋. The resulting CSS codes satisfy K/N → 1/(991L), relative X-distance at least 925/(991L), and relative Z-distance at least 1/(991L).
Shorter embeddings (Section 5, Appendix D). The minimum embedding length is recast as a minimum-weight problem in a binary affine space. The triangular construction gives length 353; a randomized local search found a length-244 embedding whose image is a binary [244, 10, 100] code. Plugging this in (Corollary D.1) gives rate tending to 1/241804, relative Z-distance at least 1/241804, and relative X-distance at least 92500/241804.
Why it matters
What is demonstrated here is a proof, not an experiment: an explicit family of binary codes that is simultaneously asymptotically good and supports T on any chosen subset of logical qubits, implemented only by single-qubit physical T powers on a fixed encoding. The authors point to magic-state distillation and code switching as settings where codes with transversal T are used, and pose the space-time overhead of universal computation with such codes as an open question.
Technical perspective (interpretation)
The following is my interpretation, not a claim from the paper. The appealing move is treating divisibility coefficients as a programmable dial: once the 4-multiplication property holds, choosing which logical qubits get which T power becomes linear algebra over ℤ8. That separates the hard part (building codes with strong multiplication structure) from the addressability part.
The constants make clear that this is an existence-and-explicitness result rather than a hardware proposal. A rate of 1/241804 means roughly a quarter of a million physical qubits per logical qubit in the limit, and the guaranteed relative distance is of the same order; the bounds are also only asserted for sufficiently large tower levels. AG-code-based CSS codes are not low-density parity-check codes, so their checks are high weight. Those gaps are exactly where follow-up work on better constants, LDPC versions and decoders would have to land before the idea could shape an architecture.
Limitations and open questions
The paper is purely theoretical (27 pages, no figures): no noise model, decoder, threshold or circuit-level simulation is given.
The explicit constants are very small, and the stabilizers are not bounded-weight.
Transversal T alone is not universal (Eastin–Knill); complementary gates are still required.
The authors list open questions: general structural principles for fully addressable transversal non-Clifford gates; whether asymptotically good quantum LDPC codes can have fully addressable transversal T; and the optimal space-time tradeoff once noisy syndrome extraction, ancilla preparation, complementary gates and classical processing are included.
The paper includes a statement that AI models were used to assist with brainstorming, exploring proof strategies, writing proofs and preparing the manuscript, with the authors taking full responsibility for its claims. It is a v1 preprint and has not been peer reviewed.
Paper information
Title: Asymptotically Good Quantum Codes with Addressable Transversal T Gates
Authors: Tongyin Lin, Bujiao Wu (International Quantum Academy, Shenzhen); Bin Cheng (Centre for Quantum Technologies, National University of Singapore)
arXiv: 2609.35129v1 [quant-ph], submitted 28 September 2026
Code/data: none listed (theory paper); the explicit length-244 embedding is given in hexadecimal in Appendix D.