Codeword stabilized (CWS) code[1] 


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 \(S\).


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].


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


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


See Ref. [3] for an introduction to CWS codes.




  • 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 [4,5], 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.


A. Cross et al., “Codeword Stabilized Quantum Codes”, IEEE Transactions on Information Theory 55, 433 (2009) arXiv:0708.1021 DOI
U. Kapshikar and S. Kundu, “Diagonal distance of quantum codes and hardness of the minimum distance problem”, (2022) arXiv:2203.04262
M. Grassl and M. Rötteler, “Nonadditive quantum codes”, Quantum Error Correction 261 (2013) DOI
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
P. Hayden et al., “Spacetime replication of continuous variable quantum information”, New Journal of Physics 18, 083043 (2016) arXiv:1601.02544 DOI
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Zoo Code ID: cws

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“Codeword stabilized (CWS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
  title={Codeword stabilized (CWS) code},
  booktitle={The Error Correction Zoo},
  editor={Albert, Victor V. and Faist, Philippe},
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“Codeword stabilized (CWS) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.