Group-based quantum code
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 \(G^{\times n}\), can be denoted as \(((n,K))_G\). When the logical subspace is the Hilbert space of \(\ell^2\)-normalizable functions on \(G^{\times k}\), 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
Child
Cousins
- Qubit code — Group quantum codes whose physical spaces are constructed using the group \(\mathbb{Z}_2\) are qubit codes.
- Modular-qudit code — Group quantum codes whose physical spaces are constructed using modular-integer groups \(\mathbb{Z}_q\) are modular-qudit codes.
- Bosonic code — Group quantum codes whose physical spaces are constructed using the group of the reals \(\mathbb{R}\) under addition are bosonic codes.
- Group-based code
Zoo code information
Cite as:
“Group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group_quantum
Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/groups/group_quantum.yml.