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 [2].


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

Fault Tolerance

Universal fault-tolerant gates can be performed without magic-state distillation [2,3].


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


Reference [4] 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 [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.


S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
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
D. Banfield and A. Kay, “Implementing Logical Operators using Code Rewiring”, (2023) arXiv:2210.14074
S. Nezami and J. Haah, “Classification of small triorthogonal codes”, Physical Review A 106, (2022) arXiv:2107.09684 DOI
A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI
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Zoo Code ID: quantum_triorthogonal

Cite as:
“Triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.
@incollection{eczoo_quantum_triorthogonal, title={Triorthogonal code}, booktitle={The Error Correction Zoo}, year={2021}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.