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I noticed that stabilizer codes and additive quantum codes are equivalent, but why? I am confused at the word "additive" since the operation of stabilizer genertors is multiplication.

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A stabilizer code is also called an additive code, because it is closed under the sum of its elements.

The namesake is described on page 33 of "Stabilizer Codes and Quantum Error Correction" (link).

Additionally, additive quantum codes are the quantum version of additive codes found in coding theory.

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  • $\begingroup$ What do you mean by "it is closed under the sum of its elements"? $\endgroup$ Commented Aug 5, 2022 at 17:25
  • $\begingroup$ This answer misses the point. While it's true that many QCQI theorists conflate the notion of stabilizer codes and additive codes, they're definitely not the same. I have no intention to advertise my own answer, but for future readers, it might help to read my explanation here. $\endgroup$ Commented Aug 7, 2022 at 23:12

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