Abstract
Triorthogonal codes are a class of quantum error-correcting codes used in magic state distillation protocols. We classify all triorthogonal codes with , where is the number of physical qubits and is the number of logical qubits of the code. We find 38 distinguished triorthogonal subspaces, and we show that every triorthogonal code with 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 , 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.
- Received 22 June 2022
- Accepted 11 July 2022
DOI:https://doi.org/10.1103/PhysRevA.106.012437
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