## Description

Qubit stabilizer code whose \(X\)-type generators form a \(k\)-orthogonal matrix (defined below) in the symplectic representation. In other words, the overlap between any \(k\) stabilizers (including the identity) is even. The original definition is for qubit CSS codes [1], but it can be extended to more general qubit stabilizer codes [3; Def. 1]. This entry is formulated for qubits, but an extension exists for modular qudits [1].

A matrix is \(k\)-orthogonal [3; Def. 4] if \begin{align} |x^1|&\equiv 0 \mod 2 \tag*{(1)}\\ |x^1\cdot x^2|&\equiv 0 \mod 2 \tag*{(2)}\\ |x^1\cdot x^2\cdot x^3|&\equiv 0 \mod 2 \tag*{(3)}\\ &\vdots \tag*{(4)}\\ |x^1\cdot x^2\cdot x^3\cdot\ldots\cdot x^k|&\equiv 0 \mod 2 \tag*{(5)}\\ \end{align} for all its rows \(x^j\), where the generalized dot-product notation means a sum of products of the respective coordinates of all vectors.

## Parent

## Children

- Quantum pin code — Quantum pin codes are \(\ell\)-orthogonal, i.e., the overlap between any \(\ell\) stabilizers is even [2].
- Triorthogonal code — \(k\)-orthogonal codes reduce to triorthogonal codes for \(k=3\).

## Cousins

- Modular-qudit color code — The notion of \(k\)-orthogonality can be extended to modular-qudit codes and is known as \(k^{\star}\)-orthogonality [1; Def. 2]. Modular-qudit color codes defined on lattices in \(D\) spatial dimension whose \(X\)-type stabilizers are placed on cells of dimension \(\nu \leq D\) are \(k^{\star}\)-orthogonal for all \(k \leq \nu\) [1; Lemma 5].
- \([[2^r-1, 1, 3]]\) simplex code — \([[2^r-1, 1, 3]]\) simplex codes are \((r-1)\)-orthogonal [3; Lemma 2].

## References

- [1]
- F. H. E. Watson et al., “Qudit color codes and gauge color codes in all spatial dimensions”, Physical Review A 92, (2015) arXiv:1503.08800 DOI
- [2]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
- [3]
- S. Koutsioumpas, D. Banfield, and A. Kay, “The Smallest Code with Transversal T”, (2022) arXiv:2210.14066

## Page edit log

- Victor V. Albert (2024-03-01) — most recent

## Cite as:

“\(k\)-orthogonal code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_k-orthogonal