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arXiv:2609.08203

Asymptotically good binary triorthogonal codes and higher-level transversal gates

Rodrigo San-José

cs.ITquant-phSubmitted Sep 8, 2026• Announced Sep 9, 2026• 0 citations
v1 · Sep 8, 2026v2 · Sep 13, 2026
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Abstract

For each level of the Clifford hierarchy from the third level onward, we construct explicit asymptotically good binary CSS codes supporting diagonal transversal gates in that level. 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 no subsequent correction is required. Finally, adapting an error-correction-based distillation protocol, our binary $T$-gate families yield constant-overhead $T$-state block distillation under sufficiently weak input noise and ideal stabilizer operations.

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For each level of the Clifford hierarchy from the third level onward, we construct explicit asymptotically good binary CSS codes supporting diagonal transversal gates in that level. 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 no subsequent correction is required. Finally, adapting an error-correction-based distillation protocol, our binary TTT-gate families yield constant-overhead TTT-state block distillation under sufficiently weak input noise and ideal stabilizer operations.
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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.