## 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

## Decoding

## Notes

## Parents

- Qubit code
- Modular-qudit USt code — Modular-qudit union stabilizer codes reduce to union stabilizer codes for \(q=2\).
- Galois-qudit USt code — Galois-qudit union stabilizer codes reduce to union stabilizer codes for \(q=2\).

## 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.

## Cousins

- 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 Steane enlargement. This code can be treated as a union stabilizer code [5].
- Hybrid stabilizer code — The algebraic structure of a hybrid stabilizer code is the same as that of a USt code whose cosets are indexed by a linear binary code [11].

## 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]
- I. Kremsky, M.-H. Hsieh, and T. A. Brun, “Classical enhancement of quantum-error-correcting codes”, Physical Review A 78, (2008) arXiv:0802.2414 DOI

## Page edit log

- Victor V. Albert (2024-03-29) — most recent

## Cite as:

“Union stabilizer (USt) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/non_stabilizer