# Example non-stabilizer code?

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).

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.

• ran into this question which has your examples and a few others quantumcomputing.stackexchange.com/a/8282/19675 also this arxiv.org/pdf/quant-ph/9710031.pdf has a "strongly" non-additive [[11,1,3]] code for what that's worth Aug 5, 2022 at 13:27
• Graphical Quantum Error-Correcting Codes arxiv.org/pdf/0709.1780.pdf has a ((9,12,3)) code beating the best [[9,3,3]] stabilizer code, Aug 5, 2022 at 13:35
• 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. Aug 10, 2022 at 18:57

A lot of non stabilizer (usually called non-additive) codes can be found in my recent work with Eric Kubischta:

A Family of Quantum Codes with Exotic Transversal Gates

The Not-So-Secret Fourth Parameter of Quantum Codes

These first two papers are all permutationally invariant codes (it was proven in Investigations on Automorphism Groups of Quantum Stabilizer Codes that all permutationally invariant codes besides the $$[[2n,2n-2,2]]$$ family with stabilizer generators $$X^{\otimes n}$$ and $$Z^{\otimes n}$$ are all non-additive)

The next two papers cover more general codes

Free Quantum Codes from Twisted Unitary t-groups

Quantum Codes and Irreducible Products of Characters