Classification of Small Triorthogonal Codes

Abstract

Triorthogonal codes are a class of quantum error correcting codes used in magic state distillation protocols. We classify all triorthogonal codes with $n+k\le 38$, 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 show that every triorthogonal code with $n+k\le 38$ 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 classify the Reed-Muller polynomials of low weight using the results of Kasami, Tokura, and Azumi 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.

27 pages, 1 figure (v2) minor changes

References in corpus (4)

• Magic state distillation with low overhead
• Building a fault-tolerant quantum computer using concatenated cat codes
• Multilevel distillation of magic states for quantum computing
• The cost of universality: A comparative study of the overhead of state distillation and code switching with color codes

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