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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?

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  • $\begingroup$ 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. $\endgroup$
    – squiggles
    Commented Mar 24, 2023 at 15:50
  • $\begingroup$ @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? $\endgroup$
    – Ian Gershon Teixeira
    Commented Mar 24, 2023 at 15:58
  • $\begingroup$ 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 $\langle ZZZZI, XXXXI, ZZIII, IIIIZ \rangle$. $\endgroup$
    – squiggles
    Commented Mar 24, 2023 at 18:45

2 Answers 2

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

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    $\begingroup$ 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 $\endgroup$
    – Ian Gershon Teixeira
    Commented Mar 24, 2023 at 12:27
  • $\begingroup$ 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. $\endgroup$
    – Jan Olle
    Commented Aug 17, 2023 at 14:38
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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

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  • $\begingroup$ This looks like a planar surface code. $\endgroup$
    – Victor V Albert
    Commented Apr 5 at 19:50
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    $\begingroup$ @VictorVAlbert it is! they;re easier to construct and generalize as hypergraph product codes $\endgroup$
    – unknown
    Commented Apr 5 at 23:07
  • $\begingroup$ do u have a ref that constructs these? $\endgroup$
    – Victor V Albert
    Commented Apr 7 at 1:10
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    $\begingroup$ BTW, 2 more [[5,1,2]] CSS codes in these refs: arxiv.org/abs/2112.01446 and arxiv.org/abs/2011.00197 $\endgroup$
    – Victor V Albert
    Commented Apr 7 at 1:12
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    $\begingroup$ @VictorVAlbert example 5 in arxiv.org/abs/1202.0928 $\endgroup$
    – unknown
    Commented Apr 8 at 17:14

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