Classification of small triorthogonal codes
Sepehr Nezami, Jeongwan Haah · Physical Review A · 2022
Triorthogonal codes are a class of quantum error-correcting codes used in magic state distillation protocols. We classify all triorthogonal codes with $n+k\ensuremath{\le}38$, where $n$ is the number of physical qubits and $k$ is the number of logical qubits of the code. We find 38 distinguished triorthogonal subspaces, and we show that every triorthogonal code with $n+k\ensuremath{\le}38$ descends from one of these subspaces through elementary operations such as puncturing and deleting qubits. Specifically, we associate each triorthogonal code with a Reed-Muller polynomial of weight $n+k$, and we classify the Reed-Muller polynomials of low weight using the results of Kasami, Tokura, and Azumi [IEEE Trans. Inf. Theory 16, 752 (1970); Inf. Contr. 30, 380 (1976)] and an extensive computerized search. In an Appendix independent of the main text, we improve a magic state distillation protocol by reducing the time variance due to stochastic Clifford corrections.