Union stabilizer (USt) code[15] 

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

Description

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.

Protection

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

Decoding

Error-detection algorithm [810].

Notes

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

Parents

Children

  • 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.

Cousin

  • 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].

References

[1]
E. M. Rains et al., “A Nonadditive Quantum Code”, Physical Review Letters 79, 953 (1997) arXiv:quant-ph/9703002 DOI
[2]
M. Grassl and T. Beth, “A Note on Non-Additive Quantum Codes”, (1997) arXiv:quant-ph/9703016
[3]
V. P. Roychowdhury and F. Vatan, “On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes”, (1997) arXiv:quant-ph/9710031
[4]
V. Arvind, P. P. Kurur, and K. R. Parthasarathy, “Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group”, (2002) arXiv:quant-ph/0210097
[5]
M. Grassl and M. Rotteler, “Non-additive quantum codes from Goethals and Preparata codes”, 2008 IEEE Information Theory Workshop (2008) arXiv:0801.2144 DOI
[6]
M. Grassl and M. Rötteler, “Nonadditive quantum codes”, Quantum Error Correction 261 (2013) DOI
[7]
J.-L. Xia, “Quotient Space Quantum Codes”, (2024) arXiv:2311.07265
[8]
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
[9]
Y. Li et al., “Clustered bounded-distance decoding of codeword-stabilized quantum codes”, 2010 IEEE International Symposium on Information Theory (2010) DOI
[10]
Li, Yunfan. Codeword Stabilized Quantum Codes and Their Error Correction. Diss. UC Riverside, 2010.
[11]
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
BibTeX:
@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.