A generalization of stabilizer codes are the codeword stabilized codes (CWS), see https://errorcorrectionzoo.org/c/cws.

These encompass stabilizer codes but more broadly also contain some non-additive codes.

What is a simple example of a non-additive quantum code that is not a CWS code?


1 Answer 1


All $((n,K,d))$ CWS codewords can be written as $|i \rangle = Z_{\gamma_i} | G \rangle$ where $| G \rangle$ is a graph state, $1 \leq i \leq K$, and $\gamma_i \subset \{1,2, \cdots, n\}$. To be sure, $Z_{\gamma_i}$ is a operator that acts with Pauli $Z$ on each qubit in $\gamma_i$ (and identity elsewhere).

On the other hand, graph states are defined up to normalization by $$ | G \rangle = \sum_{\mu = 00\cdots 0}^{11\cdots 1} (-1)^{\frac{1}{2} \mu \Gamma \mu} | \mu \rangle, $$ where $\Gamma$ is the adjacency matrix for the graph $G$.

It follows that CWS codes can always be written in a form for which all codewords have coefficients of $\pm 1$ (up to normalization).

Therefore, any code that does not admit such a basis will be non-CWS.


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