# 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].

## Fault Tolerance

## Threshold

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

## Notes

Reference [4] presents a classification of triorthogonal codes up to \(n + k \leq 38\) by associating each triorthogonal code with a Reed-Muller polynomial.

## Parent

## Children

- \([[15,1,3]]\) quantum Reed-Muller code — The \([[15, 1, 3]]\) code is a triorthogonal code [4].
- Three-dimensional color code

## Cousins

- Quantum Reed-Muller code — Classification of triorthongonal codes yields a connection to Reed-Muller polynomials [4].
- Small-distance block quantum code — Ref. [1] provides a family of distance \(2\) triorthogonal codes. It also presents a \([[49, 1, 5]]\) code.
- 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.
- Quantum Reed-Solomon code — Triorthogonality can be generalized to qudit codes. Quantum RS codes achieve a magic-state distillation scaling exponent \(\gamma\) that is arbitrarily close to zero [5].
- Binary dihedral permutation-invariant code — The \(((27,2,5))\) binary dihedral permutation-invariant code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[49,1,5]]\) triorthogonal code.

## References

- [1]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
- [2]
- A. Paetznick and B. W. Reichardt, “Universal Fault-Tolerant Quantum Computation with Only Transversal Gates and Error Correction”, Physical Review Letters 111, (2013) arXiv:1304.3709 DOI
- [3]
- D. Banfield and A. Kay, “Implementing Logical Operators using Code Rewiring”, (2023) arXiv:2210.14074
- [4]
- S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
- [5]
- A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI

## Page edit log

- Victor V. Albert (2021-12-16) — most recent
- Benjamin Quiring (2021-12-16)

## Cite as:

“Triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/quantum_triorthogonal