Classification of small triorthogonal codes

Sepehr Nezami and Jeongwan Haah
Phys. Rev. A 106, 012437 – Published 28 July 2022

Abstract

Triorthogonal codes are a class of quantum error-correcting codes used in magic state distillation protocols. We classify all triorthogonal codes with n+k38, 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+k38 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.

  • Figure
  • Received 22 June 2022
  • Accepted 11 July 2022

DOI:https://doi.org/10.1103/PhysRevA.106.012437

©2022 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Authors & Affiliations

Sepehr Nezami1 and Jeongwan Haah2

  • 1Institute for Quantum Information and Matter and Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
  • 2Microsoft Quantum, Redmond, Washington 98052, USA

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Vol. 106, Iss. 1 — July 2022

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