Building codes with transversal CCZ using projective geometry and SAT solvers

Abstract

We construct CSS codes with three logical qubits whose logical CCZ gate is implemented by transversal physical $T/T^{†}$ gates and whose distance is at least 3. These codes start with 8-divisible $X$-stabilizers defined by projective geometry, which guarantees $d_Z\ge 3$. 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.

arXiv comment40 pages, 3 figures. Code and data: https://github.com/jerrylvx/gate2code-public