# Codeword stabilized (CWS) code[1]

## 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 \(S\).

## Protection

## Encoding

## Decoding

## Parent

## Child

- 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 [3][4], 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]
- A. Cross et al., “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009). DOI; 0708.1021
- [2]
- Upendra Kapshikar and Srijita Kundu, “Diagonal distance of quantum codes and hardness of the minimum distance problem”. 2203.04262
- [3]
- 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). DOI; 1210.0913
- [4]
- P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016). DOI; 1601.02544

## Page edit log

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

## Zoo code information

## Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/cws.yml.