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Every $ [[5,1,3]] $ code is equivalent to the perfect five qubit code with stabilizer generators $$ XZZXI \\ IXZZX \\ XIXZZ \\ ZXIXZ $$ see for example

Equivalence between Quantum Error Correcting codes and uniqueness of the $[\![5,1,3]\!]$ code

The $ [[5,1,3]] $ code is not equivalent to any CSS code. See

CSS Code in disguise

So no $ [[5,1,3]] $ CSS code exists.

My question is:

Does a $ [[5,1,2]] $ CSS code exist?

Ian Gershon Teixeira
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  • Just curious, why not take the $[[4,1,2]]$ and extend it by a single fixed qubit? You can always make codes 'worse' in this way. – squiggles Mar 24 '23 at 15:50
  • @squiggles ya I thought about that and I realized I actually don't know how to just take a CSS code and extend it to a CSS code on one more quibt! What is the stabilizer for a $ [[5,1,2]] $ CSS code extended from a $ [[4,1,2]] $ CSS code? – Ian Gershon Teixeira Mar 24 '23 at 15:58
  • e.g. for the $[[4,1,2]]$ given by $\langle ZZZZ, XXXX, ZZII\rangle$, you can make the $[[5,1,2]]$ code that is just appending a fixed '0' qubit $\langle ZZZZI, XXXXI, ZZIII, IIIIZ \rangle$. – squiggles Mar 24 '23 at 18:45

2 Answers2

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A $[[5,1,2]]$ code occurs as a member of a family of hypergraph product codes with parameters $[[2d^2-2d+1,1,d]]$ codes : $[[5,1,2]],[[13,1,3]],[[25,1,4]],\cdots$. These are all CSS codes. Here are (quantum) tanner graphs of the first three :enter image description here

unknown
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$S=\langle ZZZZZ, XXXXI, IXXXX, XXIXX\rangle $ with logicals $L_X = XXIII, L_Z= IZIZI$ should do the trick.

Jonas Anderson
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    oh nice this even gives a $ [[5,2,2]] $ CSS code $S=\langle ZZZZZ, XXXXI, IXXXX \rangle $ and in general gives an $ [[2n+1,2n-2,2]] $ CSS code – Ian Gershon Teixeira Mar 24 '23 at 12:27
  • You can also obtain a $[![5,2,2]!]$ CSS code by extending the $[![4,2,2]!]$ (CSS) code $S = \langle XXXX, ZZZZ \rangle$ by a single fixed qubit. – Jan Olle Aug 17 '23 at 14:38