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Triorthogonal Codes and Self-dual Codes

Minjia Shi, Haodong Lu, Jon-Lark Kim, Patrick Sole

TL;DR

This work addresses the construction of triorthogonal matrices and their associated codes, motivated by quantum distillation of magic states. It develops two parallel strategies: (i) adapting classical shortening, extending, and propagation techniques to build larger triorthogonal matrices from known ones, and (ii) leveraging binary self-dual codes to generate longer triorthogonal codes and subcodes, with explicit parameter control. A key contribution is an algorithm to locate the largest triorthogonal subspaces inside a given self-dual code, together with bounds on their sizes and constructive methods to produce new codes with diverse $[[n,k,d_Z]]$ parameters. The results enrich the catalog of triorthogonal codes, enable flexible code design for quantum error correction, and point toward potential infinite families and applications in magic-state distillation.

Abstract

Triorthogonal matrices were introduced in Quantum Information Theory in connection with distillation of magic states (Bravyi and Haah (2012)). We give an algorithm to construct binary triorthogonal matrices from binary self-dual codes. Further, we generalize to this setting the classical coding techniques of shortening and extending. We also give some simple propagation rules.

Triorthogonal Codes and Self-dual Codes

TL;DR

This work addresses the construction of triorthogonal matrices and their associated codes, motivated by quantum distillation of magic states. It develops two parallel strategies: (i) adapting classical shortening, extending, and propagation techniques to build larger triorthogonal matrices from known ones, and (ii) leveraging binary self-dual codes to generate longer triorthogonal codes and subcodes, with explicit parameter control. A key contribution is an algorithm to locate the largest triorthogonal subspaces inside a given self-dual code, together with bounds on their sizes and constructive methods to produce new codes with diverse parameters. The results enrich the catalog of triorthogonal codes, enable flexible code design for quantum error correction, and point toward potential infinite families and applications in magic-state distillation.

Abstract

Triorthogonal matrices were introduced in Quantum Information Theory in connection with distillation of magic states (Bravyi and Haah (2012)). We give an algorithm to construct binary triorthogonal matrices from binary self-dual codes. Further, we generalize to this setting the classical coding techniques of shortening and extending. We also give some simple propagation rules.
Paper Structure (17 sections, 33 theorems, 76 equations, 2 tables)

This paper contains 17 sections, 33 theorems, 76 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical error correcting codes
  4. Triorthogonality: Matrices and codes
  5. Triorthogonal spaces and triorthogonal matrices
  6. Triorthogonal codes from matrices
  7. Shortening and extending
  8. Propagation rules
  9. New triorthogonal matrices from known ones
  10. Triorthogonal codes from self-dual codes
  11. Background material
  12. Longer triorthogonal codes from self-dual codes
  13. Largest triorthogonal subcodes from self-dual codes
  14. Applications and examples
  15. Conclusion
  16. ...and 2 more sections

Key Result

Proposition 1

If $\mathbf{x},\mathbf{y},\mathbf{z}$ are vectors in $\mathbb{F}_2^c$, then $\mathbf{x}\wedge \mathbf{x} = \mathbf{x}$, $\mathbf{x}\wedge \mathbf{1}_c=\mathbf{x}$, $\mathbf{x}\wedge(\mathbf{y}+\mathbf{z}) = (\mathbf{x}\wedge \mathbf{y}) + (\mathbf{x}\wedge \mathbf{z})$, where $\mathbf{1}_c$ is the a

Theorems & Definitions (66)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Definition 1: Triorthogonal matrices in basis paper 1
  • Definition 2: Triorthogonal codes in basis paper 1
  • Definition 3: data paper
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • ...and 56 more