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

Code distance \(\mathcal{Q} = ( \mathcal{G},\mathcal{C}) \) is upper bounded by the distance of the classical code \(\mathcal{C} \). The diagonal distance is upper bounded by \(\delta + 1\), where \(\delta\) is the minimum degree of \(\mathcal{G}\). Computing the distance is generally NP-complete, and is NP-hard for non-degenerate codes [2].

Encoding

If \( \mathcal{C} \) has an efficient classical encoder, then so does the CWS code \( \mathcal{Q} = (\mathcal{G},\mathcal{C}) \).

Decoding

There is no known efficient algorithm to decode non-additive (non-stabilizer) CWS codes.

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
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Zoo code information

Internal code ID: cws

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Zoo Code ID: cws

Cite as:
“Codeword stabilized (CWS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/cws
BibTeX:
@incollection{eczoo_cws, title={Codeword stabilized (CWS) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/cws} }
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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.