# Triorthogonal code[1]

## Description

Qubit CSS code whose \(X\)-type logicals and stabilizer generators form a triorthogonal matrix (defined below) in the symplectic representation.

An \(m \times n\) binary matrix is triorthogonal if its rows \(r_1, \ldots, r_m\) satisfy \(|r_i \cdot r_j| = 0\) and \(|r_i \cdot r_j \cdot r_k| = 0\) with binary addition and multiplication. 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

## Encoding

## Transversal Gates

## Fault Tolerance

## Notes

## Parents

- CSS-T code — CSS-T codes reduce to triorthogonal codes when the logical action of the physical transversal \(T\) gate is a logical \(T\) gate on all encoded qubits. Triorthogonality is necessary for physical transversal \(T\) gates on each qubit to realize the identity logical gate [4; Thm. 12]. The \(X\)-type stabilizer generator matrix for a CSS-T code is always triorthogonal [8; Corr. 5].
- \(k\)-orthogonal code — \(k\)-orthogonal codes reduce to triorthogonal codes for \(k=3\).
- Prime-qudit triorthogonal code — Prime-qudit triorthogonal codes reduce to triorthogonal codes when \(p=2\).

## 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].
- Self-dual linear code — Self-dual binary codes can be used to construct triorthogonal codes [9].
- Binary dihedral PI code — There exist binary dihedral PI codes that have distance 5 (7, 9, 11, 13) and encode in 27 (49, 73, 107, 147) qubits, all realizing transversal \(T\) gates.
- 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 [10; Sec. VI.C].
- Quantum rainbow code — Hypergraph products of color codes yield quantum rainbow codes with growing distance and transversal gates in the Clifford hierarchy. In particular, utilizing this construction for quasi-hyperbolic color codes [11] yields an \([[n,O(n),O(\log n)]]\) triorthogonal code family with magic-state yield parameter \(\gamma\to 0\) [12].
- 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 [13].
- Quantum quadratic-residue (QR) code — Qubit quantum QR codes admit transversal implementations of the single-qubit Clifford group [14]. They yield a family of high-distance triorthogonal codes via the doubling transformation [14]; such codes admit transversal implementations of the \(T\) gate.
- Quantum AG code — By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the Clifford hierarchy and attains a zero magic-state yield parameter, \(\gamma = 0\) [15]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see [16,18][17; Sec. 5.3]. Two other such asymptotically good families exist [19,20], admitting a different diagonal gate at the third level of the Clifford hierarchy.

## 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]
- 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
- [5]
- 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
- [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]
- E. Camps-Moreno et al., “An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance”, Quantum Information Processing 23, (2024) arXiv:2312.17518 DOI
- [9]
- M. Shi et al., “Triorthogonal Codes and Self-dual Codes”, (2024) arXiv:2408.09685
- [10]
- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
- [11]
- G. Zhu et al., “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2023) arXiv:2310.16982
- [12]
- T. R. Scruby, A. Pesah, and M. Webster, “Quantum Rainbow Codes”, (2024) arXiv:2408.13130
- [13]
- M. Sullivan, “Code conversion with the quantum Golay code for a universal transversal gate set”, Physical Review A 109, (2024) arXiv:2307.14425 DOI
- [14]
- S. P. Jain and V. V. Albert, “High-distance codes with transversal Clifford and \(T\)-gates”, (2024) arXiv:2408.12752
- [15]
- A. Wills, M.-H. Hsieh, and H. Yamasaki, “Constant-Overhead Magic State Distillation”, (2024) arXiv:2408.07764
- [16]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [17]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [18]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [19]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
- [20]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140

## Page edit log

- Victor V. Albert (2024-07-29) — most recent
- Shubham P. Jain (2024-07-29)
- Narayanan Rengaswamy (2024-07-29)
- Victor V. Albert (2021-12-16)
- Benjamin Quiring (2021-12-16)

## Cite as:

“Triorthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_triorthogonal