## Description

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

The modular-qudit CWS construction takes in \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \), where \(\mathcal{G}\) is a graph, and where \(\mathcal{C}\) is an \((n,K,d)_{\mathbb{Z}_q}\) \(q\)-ary code over \(\mathbb{Z}_q\). From the graph, we form the modular-qudit cluster state \( |\mathcal{G} \rangle \). From the \(q\)-ary code, we form modular-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 \( | i \rangle = W_i | \mathcal{G} \rangle \).

In an alternative convention (not used here), CWS codes are defined from an underlying modular-qudit stabilizer state that is not a necessarily a cluster state.

## Notes

## Parent

- Modular-qudit USt code — Any modular-qudit CWS code can be written as a modular-qudit USt whose (\(K=1\)) stabilizer code is the modular-qudit cluster state and whose coset representatives are constructed from the \(q\)-ary classical code over \(\mathbb{Z}_q\). Prime-dimensional modular-qudit CWS codes have a unique representation as USt codes [3]. Conversely, modular-qudit USt codes are equivalent to modular-qudit CWS codes via a single-Galois-qudit Clifford circuit as follows [4,5][6; Sec. 10.4]. The set of coset representatives of any modular-qudit USt can be extended to a larger set iterating over the underlying stabilizer code such that all codewords can be obtained from a single stabilizer state. Then, one can apply a single-qubit Clifford transformation to map said modular-qudit stabilizer state into a modular-qudit cluster state.

## Child

- Codeword stabilized (CWS) code — Modular-qudit CWS codes reduce to CWS codes for \(q=2\).

## Cousins

- Modular-qudit cluster-state code — A single modular-qudit cluster state is used to construct a modular-qudit CWS code.
- Perfect quantum code — Generalized concatenatenations of modular-qudit CWS codes yield a family of codes that have larger logical dimension than stabilizer codes and that asymptotically approach the modular-qudit Hamming bound [7].
- Concatenated quantum code — Generalized concatenatenations of modular-qudit CWS codes yield a family of codes that have larger logical dimension than stabilizer codes and that asymptotically approach the modular-qudit Hamming bound [7].
- Modular-qudit stabilizer code — Modular-qudit CWS codes whose underlying classical code is a linear \(q\)-ary code over \(\mathbb{Z}_q\) are modular-qudit stabilizer codes containing a cluster-state codeword; see [8; Corr. 4-5], which defines CWS codes as admitting an underlying stabilizer state that is not a necessarily a cluster state.

## References

- [1]
- S. Y. Looi et al., “Quantum-error-correcting codes using qudit graph states”, Physical Review A 78, (2008) arXiv:0712.1979 DOI
- [2]
- D. Hu et al., “Graphical nonbinary quantum error-correcting codes”, Physical Review A 78, (2008) arXiv:0801.0831 DOI
- [3]
- S. Beigi et al., “Symmetries of Codeword Stabilized Quantum Codes”, (2013) arXiv:1303.7020 DOI
- [4]
- Y. Li, I. Dumer, and L. P. Pryadko, “Clustered Error Correction of Codeword-Stabilized Quantum Codes”, Physical Review Letters 104, (2010) arXiv:0907.2038 DOI
- [5]
- Li, Yunfan. Codeword Stabilized Quantum Codes and Their Error Correction. Diss. UC Riverside, 2010.
- [6]
- M. Grassl and M. Rötteler, “Nonadditive quantum codes”, Quantum Error Correction 261 (2013) DOI
- [7]
- M. Grassl et al., “Generalized concatenated quantum codes”, Physical Review A 79, (2009) arXiv:0901.1319 DOI
- [8]
- D. F. G. Santiago and G. S. S. Otoni, “A new approach to codeword stabilized quantum codes using the algebraic structure of modules”, (2015) arXiv:1505.00283

## Page edit log

- Victor V. Albert (2024-03-28) — most recent
- Victor V. Albert (2024-02-11)

## Cite as:

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