Codeword stabilized (CWS) code[13] 

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

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 [5].

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.Structured error recovery [6], which reduces to syndrome-based recovery for additive (i.e., stabilizer) CWS codes.

Notes

See Ref. [7] for an introduction to CWS codes.

Parents

Children

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
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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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“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/edit/main/codes/quantum/qubits/nonstabilizer/cws.yml.