Quantum

Building codes with transversal CCZ using projective geometry and SAT solvers

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Briefings

This preprint establishes CSS codes with three logical qubits using projective geometry-based $X$-stabilizers and SAT solvers to implement transversal $T$-gates. The method achieves a logical distance of at least 3, with block lengths ranging from 48 to 496. It proves no such codes exist below $n=39$ with a quasi-transversal CCZ, leaving the interval $39 ext{ to } 46$ open for further research.

Key points

• The authors construct CSS codes with three logical qubits where the logical CCZ gate is implemented by transversal $T/T^\dagger$ gates.
• The method utilizes $X$-stabilizers defined by projective geometry to ensure a $Z$-distance of at least 3, verified by SAT solvers.

Context and limitations

The study covers block lengths from 48 to 496, with specific illustrations for $n=48$. The lower bound of 39 remains unverified due to the absence of a code in the 39-46 range.

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