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.

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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

Internal code ID: group_quantum

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Zoo Code ID: group_quantum

Cite as:
“Group-based quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group_quantum
BibTeX:
@incollection{eczoo_group_quantum, title={Group-based quantum code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/group_quantum} }
Permanent link:
https://errorcorrectionzoo.org/c/group_quantum

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.