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A code is a non-stabilizer code if it is not equivalent by local unitaries to a stabilizer code.

What is an example of a non-stabilizer code with distance $ d \geq 2 $?

Are there any non-stabilizer codes that are known to have especially good properties, for example better parameters than any known stabilizer code?

If you have a code in mind that seems like it is non-stabilizer then feel free to share! (even if you don't have a proof that it's not equivalent to a stabilizer code by local unitaries, I'm mostly just interested in seeing code constructions that seem essentially different from stabilizer codes).

Ian Gershon Teixeira
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You're looking for non-additive quantum codes. There are many examples but I'll refer you to the seminal paper by Rains, Hardin, Shor & Sloane: https://arxiv.org/abs/quant-ph/9703002.

There's also a more recent framework by Grassl and Roetteler.

Sanchayan Dutta
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  • ran into this question which has your examples and a few others https://quantumcomputing.stackexchange.com/a/8282/19675 also this https://arxiv.org/pdf/quant-ph/9710031.pdf has a "strongly" non-additive [[11,1,3]] code for what that's worth – Ian Gershon Teixeira Aug 05 '22 at 13:27
  • Graphical Quantum Error-Correcting Codes https://arxiv.org/pdf/0709.1780.pdf has a ((9,12,3)) code beating the best [[9,3,3]] stabilizer code, – Ian Gershon Teixeira Aug 05 '22 at 13:35
  • https://arxiv.org/pdf/quant-ph/0210097.pdf has a nice general framework that involves taking a stabilizer code then twisting it with different characters and then then taking the direct sum of all the resulting spaces. This approach includes what is done in the RHSS non additive $ ((5,6,2)) $ code as a special case. – Ian Gershon Teixeira Aug 10 '22 at 18:57