Modular-qudit USt code[1,2] 


A modular-qubit code whose codespace consists of a direct sum of a modular-qubit stabilizer codespace and one or more of that stabilizer code's error spaces.

Given a subset \(T\) of coset representatives of \(\mathsf{N}(\mathsf{S})/\mathsf{S}\) of a modular-qudit stabilizer code \(((n,K))\) with codespace \(\mathsf{C}\) and stabilizer group \(\mathsf{S}\), one can construct the modular-qudit USt with codespace \begin{align} \mathsf{C}_{\text{USt}}=\bigoplus_{t\in T}t\mathsf{C}~. \tag*{(1)}\end{align} The parameters of the USt are \(((n,K|T|,d))\), where \(|T|\) is the number of chosen coset representatives. A modular-qudit USt is CSS-like when the underlying stabilizer code is CSS, so the coset representatives from the two classical codes underlying the CSS code.



  • Union stabilizer (USt) code — Modular-qudit union stabilizer codes reduce to union stabilizer codes for \(q=2\).
  • Modular-qudit CWS code — Any modular-qudit CWS code can be written as a modular-qudit USt whose (\(K=1\)) stabilizer code is the 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 stabilizer state into a cluster state.
  • Modular-qudit stabilizer code — A modular-qudit stabilizer code with stabilizer group \(\mathsf{S}\) can be thought of as a modular-qudit USt with only the identity coset representative. Conversely, if \(K = q^k\), and if the set of coset representatives of a modular-qudit USt form a \(q\)-ary linear code over \(\mathbb{Z}_q\), then they can be absorbed into a modular-qudit stabilizer group that defines the USt.


S. Y. Looi et al., “Quantum-error-correcting codes using qudit graph states”, Physical Review A 78, (2008) arXiv:0712.1979 DOI
D. Hu et al., “Graphical nonbinary quantum error-correcting codes”, Physical Review A 78, (2008) arXiv:0801.0831 DOI
S. Beigi et al., “Symmetries of Codeword Stabilized Quantum Codes”, (2013) arXiv:1303.7020 DOI
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
Li, Yunfan. Codeword Stabilized Quantum Codes and Their Error Correction. Diss. UC Riverside, 2010.
M. Grassl and M. Rötteler, “Nonadditive quantum codes”, Quantum Error Correction 261 (2013) DOI
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Zoo Code ID: qudit_non_stabilizer

Cite as:
“Modular-qudit USt code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_qudit_non_stabilizer, title={Modular-qudit USt code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Modular-qudit USt code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.