# Triorthogonal code[1]

## Description

A triorthogonal \(m \times n\) binary matrix is triorthogonal if its rows \(r_1, \ldots, r_m\) satisfy \(|r_i \cdot r_j| = 0\) and \(|r_i \cdot r_j \cdot r_k| = 0\), where addition and multiplication are done on \(\mathbb{Z}_2\). The triorthogonal code associated with the matrix is constructed by mapping non-zero entries in even-weight rows to \(X\) operators, and \(Z\) operators for each row in the orthogonal complement.

## Protection

Weight \(t\) Pauli errors, where \(t\) depends on the family. For example, Ref. [1] provides a family of distance \(2\) codes. It also presents a \([[49, 1, 5]]\) code.

## Magic

Depends on the matrix. Reference [1] gave a family of \(\frac{k}{3k+8}\) codes with magic-state distillation scaling exponent \(\gamma = \log_2 \frac{3k+8}{k}\).

## Encoding

Encoder for magic states for the code constructed in [1].

## Transversal Gates

## Gates

Triorthogonal codes can be used for high-quality magic-state distillation [1].

## Threshold

Approximately \(\frac{1}{3k + 1}\) [1].

## Notes

Reference [2] presents a classification of triorthogonal codes up to \(n + k \leq 38\) by associating each triorthogonal code with a Reed-Muller polynomial.

## Parent

## Child

- \([[15,1,3]]\) Reed-Muller code — The \([[15, 1, 3]]\) code is a triorthogonal code [2]

## Cousins

- Quantum Reed-Muller code — Classification of triorthongonal codes yields a connection to Reed-Muller polynomials [2].
- Color code — The 3D color code is triorthogonal.
- Quantum divisible code — Triorthogonal codes can be derived using a procedure that yields sufficient conditions for a CSS code to admit a given transversal diagonal logical gate. Quantum divisible codes are derived in a similar procedure, but one that yields necessary and sufficient conditions.

## Zoo code information

## References

- [1]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012). DOI; 1209.2426
- [2]
- Sepehr Nezami and Jeongwan Haah, “Classification of Small Triorthogonal Codes”. 2107.09684

## Cite as:

“Triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_triorthogonal