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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?

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A simple example of a non-additive code that is not a CWS code is the $ ((11,2,3)) $ code given in On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes .

Theorem 6 of Codeword stabilized quantum codes: algorithm and structure says "All ((n, 2, d)) CWS codes are additive."

Thus, any non-additive $ ((n,2,d)) $ code is not CWS. An interesting example, pointed out in the same paper, is the $ ((11,2,3)) $ code from On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes . This code is non-additive, which one can check for example by observing that the weight enumerators are not integers. Thus the code is not CWS.

This $ ((11,2,3)) $ code example is especially interesting because the codewords are a uniform superpositions of computational basis states, so the test you give in your answer would fail to detect that this $ ((11,2,3)) $ code is not CWS.

In general the test you describe is agnostic about any code whose codewords are (signed) uniform superpositions of computational basis states. Given this $ ((11,2,3)) $ example it seems likely that there may be many codes whose codewords are signed uniform superpositions of computational basis states, but which are not CWS codes.

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  • $\begingroup$ If the answer is modified to "A CWS code is one that admits a signed uniform superposition and the basis is a power of 2", would that be correct? $\endgroup$ Commented May 20 at 18:49
  • $\begingroup$ @EricKubischta I think even then there might be weird exceptions because you could have codewords that have power of 2 support but are "fake stabilizer states" like this question quantumcomputing.stackexchange.com/questions/27666/… Probably you need to explicitly require a basis of stabilizer states and then I bet the code must be CWS. That equivalence sounds about right but I'm not totally sure so I asked a new question quantumcomputing.stackexchange.com/questions/38410/… $\endgroup$ Commented May 20 at 22:02
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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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