Union stabilizer (USt) code[15] 

Also known as Non-stabilizer code, Quotient space quantum code (QSQC).


A qubit code whose codespace consists of a direct sum of a 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 stabilizer code \([[n,k]]\) with codespace \(\mathsf{C}\) and stabilizer group \(\mathsf{S}\), one can construct the USt with codespace [6; Def. 10.1] \begin{align} \mathsf{C}_{\text{USt}}=\bigoplus_{t\in T}t\mathsf{C}~. \tag*{(1)}\end{align} The parameters of the USt are \(((n,2^k |T|))\), where \(|T|\) is the number of chosen coset representatives. A 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 codes constructed in Ref. [4] include the \(((33, 155, 3))\) and \(((15, 8, 3))\) codes.


Distance bounds are calculated in Refs. [6,7] using various formulations.


Error-detection algorithm [810].


See Ref. [6] for an overview of union stabilizer codes.



  • Codeword stabilized (CWS) code — Any 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 binary classical code. Conversely, USt codes are equivalent to CWS codes via a single-qubit Clifford circuit as follows [8,10][6; Sec. 10.4]. The set of coset representatives of any 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.
  • \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code
  • Qubit stabilizer code — A stabilizer code with stabilizer group \(\mathsf{S}\) can be thought of as a USt with only the identity coset representative. Conversely, if the set of coset representatives of a USt form a linear binary code, then they can be absorbed into a stabilizer group that defines the USt.


  • Qubit CSS code — An \([[n,2k-n,d]]\) CSS code can be converted to a \([[n,k+k^{\prime}−n,\min(d,\left\lceil 3d^{\prime}/2\right\rceil )]]\) code for particular \(k^{\prime}\) and \(d^{\prime}\) via the Steane enlargement construction [11]. This code can be treated as a union stabilizer code [5].


E. M. Rains et al., “A Nonadditive Quantum Code”, Physical Review Letters 79, 953 (1997) arXiv:quant-ph/9703002 DOI
M. Grassl and T. Beth, “A Note on Non-Additive Quantum Codes”, (1997) arXiv:quant-ph/9703016
V. P. Roychowdhury and F. Vatan, “On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes”, (1997) arXiv:quant-ph/9710031
V. Arvind, P. P. Kurur, and K. R. Parthasarathy, “Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group”, (2002) arXiv:quant-ph/0210097
M. Grassl and M. Rotteler, “Non-additive quantum codes from Goethals and Preparata codes”, 2008 IEEE Information Theory Workshop (2008) arXiv:0801.2144 DOI
M. Grassl and M. Rötteler, “Nonadditive quantum codes”, Quantum Error Correction 261 (2013) DOI
J.-L. Xia, “Quotient Space Quantum Codes”, (2024) arXiv:2311.07265
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
Y. Li et al., “Clustered bounded-distance decoding of codeword-stabilized quantum codes”, 2010 IEEE International Symposium on Information Theory (2010) DOI
Li, Yunfan. Codeword Stabilized Quantum Codes and Their Error Correction. Diss. UC Riverside, 2010.
A. M. Steane, “Enlargement of Calderbank Shor Steane quantum codes”, (1998) arXiv:quant-ph/9802061
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Zoo Code ID: non_stabilizer

Cite as:
“Union stabilizer (USt) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/non_stabilizer
@incollection{eczoo_non_stabilizer, title={Union stabilizer (USt) code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/non_stabilizer} }
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“Union stabilizer (USt) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/non_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/nonstabilizer/union_stabilizer/non_stabilizer.yml.