# Generalized quantum divisible code[1]

## Description

A level-\(\nu\) generalized quantum divisible code is a CSS code whose \(X\)-type stabilizers, in the symplectic representation, have zero norm and form a \((\nu,t)\)-null matrix (defined below) with respect to some odd-integer vector \(t\) [1; Def. V.1]. Such codes admit gates at the \(\nu\)th level of the Clifford hierarchy. Such codes can also be level-lifted [1; Theorem V.6], \(\nu\to\nu+1\), which recursively yields towers of generalized divisible codes from a particular ground code.

Given an odd-integer coefficient length-\(n\) vector \(t\), two vectors \(v,w\) are \((\nu,t)\)-orthogonal if \begin{align} \sum_i v_i t_i w_i \equiv 0 \mod 2^{\nu-1}~. \tag*{(1)}\end{align} A matrix whose rows make up such vectors is called \((\nu,t)\)-orthogonal.

## Transversal Gates

## Parent

- Qubit CSS code — Generalized quantum divisible codes are CSS codes. Any self-dual CSS code yields a level-three generalized quantum divisible code when level-lifted [1; Sec. VI.C].

## Children

- Quantum divisible code — Generalized level-\(\nu\) quantum divisible codes reduce to quantum level-\(\nu\) divisible codes when \(t\) is a vector with \(\pm 1\) entries. The classical code formed by their \(X\)-type stabilizer generator matrix is \(\nu\)-even [1; pg. 10]. Both types of codes realize transversal gates outside of the Clifford group.
- \([[k+4,k,2]]\) H code — H codes are level-two generalized divisible codes [1; Sec. VI.C].

## Cousins

- Triorthogonal 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 [1; Sec. VI.C].
- Random stabilizer code — Random CSS codes [2] can be used to construct families of \([[O(d^{\nu−1}), \Omega(d), d]]\) level-\(\nu\) generalized quantum divisible codes [1; Sec. VI.A].

## References

- [1]
- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
- [2]
- A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes exist”, Physical Review A 54, 1098 (1996) arXiv:quant-ph/9512032 DOI

## Page edit log

- Connor Clayton (2024-03-15) — most recent
- Victor V. Albert (2024-02-28)

## Cite as:

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