Root code for the Group Kingdom

## Description

Encodes a logical Hilbert space, finite- or infinite-dimensional, into a physical Hilbert space of \(\ell^2\)-normalizable functions on a second-countable unimodular group. For \(K\)-dimensional logical subspace and for groups \(G^{n}\), can be denoted as \(((n,K))_G\). When the logical subspace is the Hilbert space of \(\ell^2\)-normalizable functions on \(G^{ k}\), can be denoted as \([[n,k]]_G\). Ideal codewords may not be normalizable, depending on whether \(G\) is continuous and/or noncompact, so approximate versions have to be constructed in practice.

## Parent

## Children

- Tenfold code
- Group GKP code
- Rotor code — Group quantum codes whose physical spaces are constructed using either the group of the integers \(\mathbb{Z}\) or the circle group \(U(1)\) are rotor codes.
- Twisted quantum double (TQD) code
- Bosonic code — Group quantum codes whose physical spaces are constructed using the group of the reals \(\mathbb{R}\) under addition are bosonic codes.
- Modular-qudit code — Group quantum codes whose physical spaces are constructed using modular-integer groups \(\mathbb{Z}_q\) are modular-qudit codes.
- Galois-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [1]; see Sec. 5.3 of Ref. [2]. Interpreted this way, Galois-qudit codes are group quantum codes whose physical spaces are constructed using Galois fields \(GF(q)\) as groups.

## Cousins

- Category-based quantum code — Category quantum codes whose physical spaces are constructed using a finite group as the category are group codes.
- Group-alphabet code

## References

- [1]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [2]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074

## Page edit log

- Victor V. Albert (2021-12-03) — most recent

## Cite as:

“Group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2021. https://errorcorrectionzoo.org/c/group_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/group_quantum.yml.