## Description

This family of codes strictly generalizes stabilizer codes. They are usually denoted by \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \) where \(\mathcal{G}\) is a graph and \(\mathcal{C}\) is a \( (n,K,d) \) binary classical code. From the graph we form the unique graph state (stabilizer state) \( |G \rangle \). From the classical code we form 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 bit of the \(i\)-th classical codeword. The CWS codewords are then \( | i \rangle = W_i | G \rangle \).

There is an alternative description to the one above that is locally Clifford-equivalent. In particular, we can describe CWS codes as \( \mathcal{Q} = (S,\mathcal{W})\) where \(S\) is a stabilizer group and \( \mathcal{W} = \{ w_\ell \}_{\ell = 1}^K \) is a family of \(K\) \(n\)-qubit Pauli strings. We then form CWS codeswords as \( | i \rangle = w_i | S \rangle \), where \( | S \rangle \) is the (unique) stabilizer state of \(S\).

The term CWS was coined in Ref. [2], and their approach is equivalent to another approach [1] based on Boolean functions (see Ref. [4]).

## Protection

## Encoding

## Decoding

## Notes

## Parents

- Quotient space quantum code (QSQC) — Restricting the underlying classical code to be symplectic self-dual reduces the \(((n,2^k L,d))\) QSQC construction to the \(((n,L,d))\) CWS construction [8; Sec. IV.E].
- Qudit CWS code — Qudit CWS codes reduce to CWS codes for \(q=2\).

## Children

- \(((5,6,2))\) qubit code
- \(((9,12,3))\) qubit code — The \(((9,12,3))\) qubit code is a cyclic CWS code [2].
- Smolin-Smith-Wehner code
- Qubit stabilizer code — If the CWS set \( \mathcal{W} \) is an Abelian group not containing \(-I\), then the CWS code is a stabilizer code.

## Cousins

- Movassagh-Ouyang Hamiltonian code — The Movassagh-Ouyang codes overlap the CWS codes but neither family is contained in the other.
- Spacetime code (STC) — CWS codes have been considered in the context of spacetime replication of quantum data [9,10], while STCs are designed to replicate classical data.
- XP stabilizer code — The orbit representatives of XP codes play a similar role to the word operators of CWS codes.

## References

- [1]
- V. Aggarwal and A. R. Calderbank, “Boolean Functions, Projection Operators, and Quantum Error Correcting Codes”, IEEE Transactions on Information Theory 54, 1700 (2008) arXiv:cs/0610159 DOI
- [2]
- A. Cross et al., “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009) arXiv:0708.1021 DOI
- [3]
- S. Yu, Q. Chen, and C. H. Oh, “Graphical Quantum Error-Correcting Codes”, (2007) arXiv:0709.1780
- [4]
- Zeng, Bei. Quantum operations and codes beyond the Stabilizer-Clifford framework. Diss. Massachusetts Institute of Technology, 2009.
- [5]
- U. Kapshikar and S. Kundu, “On the Hardness of the Minimum Distance Problem of Quantum Codes”, IEEE Transactions on Information Theory 69, 6293 (2023) arXiv:2203.04262 DOI
- [6]
- Y. Li et al., “Structured error recovery for code-word-stabilized quantum codes”, Physical Review A 81, (2010) arXiv:0912.3245 DOI
- [7]
- M. Grassl and M. Rötteler, “Nonadditive quantum codes”, Quantum Error Correction 261 (2013) DOI
- [8]
- J.-L. Xia, “Quotient Space Quantum Codes”, (2023) arXiv:2311.07265
- [9]
- P. Hayden and A. May, “Summoning information in spacetime, or where and when can a qubit be?”, Journal of Physics A: Mathematical and Theoretical 49, 175304 (2016) arXiv:1210.0913 DOI
- [10]
- P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016) arXiv:1601.02544 DOI

## Page edit log

- Victor V. Albert (2022-04-21) — most recent
- Victor V. Albert (2021-12-16)
- Eric Kubischta (2021-12-15)

## Cite as:

“Codeword stabilized (CWS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/cws