Triorthogonal code[1]


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.


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.


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


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

Transversal Gates

Admits transversal \(T\) gates [1] and the controlled-controlled-\(Z\) gate.


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


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


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




  • 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

Internal code ID: quantum_triorthogonal

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Zoo Code ID: quantum_triorthogonal

Cite as:
“Triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_quantum_triorthogonal, title={Triorthogonal code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Permanent link:


S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012). DOI; 1209.2426
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.