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 \(GF(2)\). 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.


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


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

Universal fault-tolerant gates can be performed without magic-state distillation [4,6].


Reference [7] 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 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].


S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
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
J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 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
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
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
J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
M. Sullivan, “Code conversion with the quantum Golay code for a universal transversal gate set”, Physical Review A 109, (2024) arXiv:2307.14425 DOI
A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI
Page edit log

Your contribution is welcome!

on (edit & pull request)— see instructions

edit on this site

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={} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:

Cite as:

“Triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021.