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 dZ ≥ 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 dZ=3 or 4, and illustrate the method with the [[48,3,3]] code Q48. 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.
Related
- Asymptotically good binary triorthogonal codes and higher-level transversal gates
- Parallelizable and addressable transversal non-Clifford gates on good quantum LDPC codes
- Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes
- Error-Transparent Code Switching Between Permutation-Invariant Codes with Transversal Clifford and Icosahedral Gates
- Realizing Logical Diagonal Gates via Transversal Physical Z-Rotations in CSS Codes
BibTeX
@misc{lu2026building,
title = {{Building codes with transversal CCZ using projective geometry and SAT solvers}},
author = {Bohan Lu and Kenneth R. Brown},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09341},
eprint = {2610.09341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09341},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Building codes with transversal CCZ using projective geometry and SAT solvers
AU - Bohan Lu
AU - Kenneth R. Brown
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.09341
UR - https://arxiv.org/abs/2610.09341
AB - 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 ≥ 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.
ER -