# 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 \(GF(2)\). The triorthogonal CSS 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.

See [2; Appx. D][3; Sec. 2.3] for generalized versions of triorthogonality.

## Protection

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

## Encoding

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

## Transversal Gates

Admits transversal \(T\) gates [1] and controlled-controlled-\(Z\) gate [4]. The \(T\) gates are realized via physical \(T\) gates on each qubit; this is an if-and-only-if condition [5]. The CCZ gates are realized via physical CCZ gates on three code blocks.Triorthogonality is necessary for physical \(T\) gates on each qubit to realize the identity logical gate [5; Thm. 12].

## Fault Tolerance

## Notes

Reference [7] presents a classification of triorthogonal codes up to \(n + k \leq 38\) by associating each triorthogonal code with a Reed-Muller polynomial.

## Parent

- \(k\)-orthogonal code — \(k\)-orthogonal codes reduce to triorthogonal codes for \(k=3\).

## Children

- \([[3k + 8, k, 2]]\) triorthogonal code
- \([[49,1,5]]\) triorthogonal code
- 3D color code
- \([[15,1,3]]\) quantum Reed-Muller code — The \([[15, 1, 3]]\) code is a triorthogonal code [7].

## Cousins

- Quantum Reed-Muller code — Classification of triorthogonal codes yields a connection to Reed-Muller polynomials [7].
- Generalized quantum divisible code — Triorthogonal codes are stabilizer codes, while generalized quantum divisible codes are CSS codes. Every level-three generalized divisible code is a triorthogonal code, but whether the converse is true or false is not known [8; Sec. VI.C].
- Quantum Golay code — A \([[95,1,7]]\) triorthogonal code with a transversal \(T\) gate can be obtained from the qubit Golay code via the doubling transformation [9].
- Prime-qudit RS code — Triorthogonality can be generalized to qudit codes. Prime-qudit RS codes achieve a magic-state distillation scaling exponent \(\gamma\) that is arbitrarily close to zero [10].

## References

- [1]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
- [2]
- E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
- [3]
- J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 DOI
- [4]
- 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
- [5]
- N. Rengaswamy et al., “On Optimality of CSS Codes for Transversal T”, IEEE Journal on Selected Areas in Information Theory 1, 499 (2020) arXiv:1910.09333 DOI
- [6]
- D. Banfield and A. Kay, “Implementing Logical Operators using Code Rewiring”, (2023) arXiv:2210.14074
- [7]
- S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
- [8]
- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
- [9]
- M. Sullivan, “Code conversion with the quantum Golay code for a universal transversal gate set”, Physical Review A 109, (2024) arXiv:2307.14425 DOI
- [10]
- 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