Timeline for Do non-stabilizer codes have integer weight enumerator?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 24, 2023 at 18:28 | comment | added | Ian Gershon Teixeira | Also looking back at the $ ((5,6,2)) $ code paper they give the unnormalized weight enumerator in equation (6) as $ 36u^5 + 60uv^4 + 96v^5$. If we normalize (dividing by $ K^2=6^2=36$) then we get the standard weight enumerator $ (1,0,0,0,5/3,8/3) $ confirming your claim above that $ A_4=5/3 $. | |
| Apr 22, 2023 at 23:30 | history | edited | Adam Zalcman | CC BY-SA 4.0 |
added 66 characters in body
|
| Apr 22, 2023 at 23:27 | comment | added | Adam Zalcman | Thank you for catching this! I have corrected the post. I chose to keep the calculation for the $[\![4,2]\!]$ code to make the correction clear (see diff). I think the calculation was correct all the way to $(9)$. Then we add together a term $q$ from the first sum over $Q_4$ in $(9)$ to the corresponding term $(-1)^{\mathrm{wt}_Z(q)}q$ from the second sum and should get zero (if the signs disagree) or $2q$ (if the signs agree). The error was that I missed the factor of two, i.e. I carried on as if $q+q=q$. Oops... Apologies for the error and thank you again for catching it. | |
| Apr 22, 2023 at 23:20 | history | edited | Adam Zalcman | CC BY-SA 4.0 |
added 66 characters in body
|
| Apr 22, 2023 at 17:35 | comment | added | Ian Gershon Teixeira | A counterexample of the sort asked for in the original question is the $ ((11,K=2,3)) $ non-stabilizer code given in arxiv.org/abs/quant-ph/9710031 which has weight enumerator $ (1,0,0,0,\frac{110}{3},0,88,0,605,0,\frac{880}{3},0) $ | |
| Apr 22, 2023 at 17:31 | comment | added | Ian Gershon Teixeira | The code projector you have in line (5) gives an $ ((n=4,K=4,2)) $ code and all such codes are equivalent to the $ [[4,2,2]] $ stabilizer code by Theorem 8 of arxiv.org/abs/quant-ph/9704043 and so in particular the code you are describing has weight enumerator $ (1,0,0,0,3) $. There is an error somewhere, probably line (8), and the expression in line (10) is not equal to the code projector given in line (5). | |
| Mar 5, 2023 at 23:33 | history | edited | Adam Zalcman | CC BY-SA 4.0 |
deleted 3 characters in body
|
| Mar 5, 2023 at 18:33 | comment | added | Adam Zalcman | I've encountered them in Chapter 10 in the red book "Quantum Error Correction" edited by by Daniel A. Lidar and Todd A. Brun, but yeah the papers you link to look relevant too. Also, AFAIR "codeword stabilized codes" are the same as "union stabilizer codes" (the construction is different, but IIRC they work out to be the same thing). | |
| Mar 5, 2023 at 18:28 | comment | added | Ian Gershon Teixeira | That is very interesting! Seems like exactly something I would be interested in. Is the original union stabilizer codes paper you have in mind arxiv.org/abs/quant-ph/9703016 ? Seems like arxiv.org/pdf/0708.1021.pdf on codeword stabilized quantum codes is a similar idea. I'll definitely look into this more. | |
| Mar 5, 2023 at 17:39 | comment | added | Adam Zalcman | I think most (all?) known non-stabilizer codes are examples of union stabilizer codes which can be constructed as the direct sum of stabilizer codes. In those codes you can always find codewords that are stabilizer states. However, their stabilizer groups will in general be different (and in particular cannot be extended to a stabilizer group of the whole code). | |
| Mar 5, 2023 at 14:13 | comment | added | Ian Gershon Teixeira | Indeed the codewords for both your codes are even $ \pm 1 $ signed superpositions over the computational basis kets corresponding to an affine binary vector space (which would explain why the support is a power of $ 2 $). Again I'm not claiming the codewords are stabilizer states, only that they have a very similar form. It's definitely an interesting pattern that every non stabilizer code I've ever seen has a basis of codewords that have this stabilizer state-esque form. | |
| Mar 5, 2023 at 13:53 | comment | added | Ian Gershon Teixeira | This is exactly what I was looking for, thanks so much. Interesting to note that even though these codes are fairly exotic the codewords still have a form very similar to stabilizer states. In particular there is basis of codewords for both codes above in which the basis codewords have support with size a power of $ 2 $ and are just a uniform $ \pm 1 $ signed superposition over their support, times the a global scalar $ 1/\sqrt{|Support|}$. This observation is essentially my question quantumcomputing.stackexchange.com/q/30218/19675 which you might be interested in. | |
| Mar 5, 2023 at 13:10 | history | bounty ended | Ian Gershon Teixeira | ||
| Mar 5, 2023 at 13:10 | vote | accept | Ian Gershon Teixeira | ||
| Mar 5, 2023 at 9:58 | history | answered | Adam Zalcman | CC BY-SA 4.0 |