Building codes with transversal CCZ using projective geometry and SAT solvers

Bohan Lu; Kenneth R. Brown

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arXiv · 2610.09341

Building codes with transversal CCZ using projective geometry and SAT solvers

Bohan Lu·Kenneth R. Brown

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Abstract

We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical $T/T^\dagger$ gates and whose distance is at least 3. These codes start with 8-divisible $X$-stabilizers defined by projective geometry, which guarantees $d_Z \geq 3$. A SAT solver then finds three $X$-logical operators compatible with the transversal CCZ. We build codes at thirteen block lengths from $n=48$ to $n=496$, with $d_Z=3$ or 4, and illustrate the method with the [[48,3,3]] code $Q_{48}$. Additionally, we prove that no CSS code with three logical qubits, $Z$-distance at least 3, and a quasi-transversal CCZ gate (transversal $T$ with a diagonal Clifford correction) exists below $n=39$. The [[47,3,3]] code of Jacinto et al. attains the minimum length in this class, as established by the classification of Wills, Jain, and Singh.

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Bohan Lu, Kenneth R. Brown. 2026-10-16. Building codes with transversal CCZ using projective geometry and SAT solvers. https://arxiv.org/abs/2610.09341

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