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Two codes are said to be equivalent if their code spaces are related by a non-entangling gate, i.e., a gate from $U(2)^{\otimes n} \rtimes S_n$, the local unitaries together with permutations.

The paper Projective Plane and Planar Quantum Codes lists three non-equivalent $ [[9,1,3]] $ CSS codes coming from different cellulations of the projective plane (Figures 2, 3 and 4). The code from Figure 4 is the Shor code.

In the answer here https://quantumcomputing.stackexchange.com/a/27699/19675 Adam Zalcman lists yet another $ [[9,1,3]] $ surface code with explicit stabilizer generators given in the comments.

All four of these codes are non-equivalent. The codes from figure 2,3 are both odd codes but the code in figure 3 has fewer $ Z $ type stabilizer generators (because there are fewer vertices in the cellulation, in general the number if $ Z $ type stabilizer generators is the number of vertices minus $ 1 $). Similarly, the Shor code and the code described by Adam Zalcman are both even but the Shor code given in figure 4 has fewer $ Z $ type generators.

I'm curious how many different $ [[9,1,3]] $ surface codes there are. Is it just these four? Or are there other $ [[9,1,3]] $ surface codes not equivalent to any of these four? Is there some way to count the number of inequivalent $ [[9,1,3]] $ surface codes? Is it possible that all $ [[9,1,3]] $ CSS codes are equivalent to some surface code?

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Not sure if this is too trivial, but take the $[[7,1,3]]$ code and add two dummy qubits.

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  • $\begingroup$ Can you explain for which cellulation of which surface this is a surface code? $\endgroup$ Jun 14 at 15:18
  • $\begingroup$ Hmm I think the original question asked “ Is every [[9,1,3]] CSS code a surface code?” to which the above is a (trivial) counter example. $\endgroup$
    – squiggles
    Jun 15 at 7:08
  • $\begingroup$ I see. Could you explain more about why adding two dummy qubits to the $ [[7,1,3]] $ code cannot be a surface code? That claim seems plausible but I'm just not very familiar with techniques for showing that a given CSS code cannot be any surface code. Even if you don't have a totally formal proof, I would just be curious to hear any partial proof/approaches you might suggest to showing that the [[7,1,3]] code plus 2 dummy qubits is not a surface code. If you think it's better I can make a separate question "Proof that Steane 7 qubit code is not equivalent to any surface code" $\endgroup$ Jun 15 at 15:41

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