Galois-qudit CWS code 

Description

A CWS code for Galois qudits, defined using a Galois-qudit cluster state and a set of Galois-qudit \(Z\)-type Pauli strings defined by a \(q\)-ary classical code.

This entry has not yet been developed in the literature, so the formulation below is a conjecture. The Galois-qudit CWS construction takes in \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \), where \(\mathcal{G}\) is a graph, and where \(\mathcal{C}\) is an \((n,K,d)_q\) \(q\)-ary code. From the graph, we form the Galois-qudit cluster state \( |\mathcal{G} \rangle \). From the \(q\)-ary code, we form Galois-qudit Pauli \(Z\)-type operators \( W_i = Z_{c_{i,1}} \otimes \cdots \otimes Z_{c_{i,n}} \), where \(c_{i,j} \) is the \(j\)-th coordinate of the \(i\)-th classical codeword. The codewords are then \( | i \rangle = W_i | \mathcal{G} \rangle \).

Parent

  • Galois-qudit USt code — Any Galois-qudit CWS code can be written as a USt whose (\(K=1\)) stabilizer code is the cluster state and whose coset representatives are constructed from the \(q\)-ary classical code.

Child

Cousins

  • Group-based cluster-state code — A group-based cluster-state codeword for \(G=GF(q)\) is used to construct a modular-qudit CWS code.
  • Galois-qudit stabilizer code — Galois-qudit CWS codes whose underlying classical code is a linear \(q\)-ary code are Galois-qudit stabilizer codes containing a cluster-state codeword.
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Zoo Code ID: galois_cws

Cite as:
“Galois-qudit CWS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_cws
BibTeX:
@incollection{eczoo_galois_cws, title={Galois-qudit CWS code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/galois_cws} }
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Permanent link:
https://errorcorrectionzoo.org/c/galois_cws

Cite as:

“Galois-qudit CWS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_cws

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/nonstabilizer/galois_cws.yml.