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It is well known that CSS codes can have lots of transversal entangling gates. For example, $ CNOT $ is exactly transversal on 2 blocks of any $ [[n,1,d]] $ CSS code. And https://arxiv.org/abs/1304.3709 shows that $ CCZ $ is exactly transversal on 3 blocks of any triorthogonal CSS code. Also $$ T \otimes T^\dagger \otimes T^\dagger \otimes T \otimes T^\dagger \otimes T \otimes T \otimes T^\dagger $$ implements logical $ CCZ $ on the $ [[8,3,2]] $ colour code, see https://earltcampbell.com/2016/09/26/the-smallest-interesting-colour-code/, and $ S^{\otimes 4} $ on the $ [[4,2,2]] $ code implements logical $ CZ $ followed by logical $ ZZ $, see https://arxiv.org/abs/1610.03507.

However I have never seen an example of a transversal gate implementing a logical entangling gate on a non-CSS stabilizer code.

Are transversal entangling gates possible for stabilizer codes other than CSS?

Ian Gershon Teixeira
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2 Answers2

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Here's a shot, and please let me know if there's an error. I welcome edits adding a picture.

I'm pretty sure that 2d color codes, by virtue of their weak self-duality, enjoy a strongly transversal $CZ$ gate between two code blocks. I'm imagining a planar 2d color code on a hexagonal lattice in particular.

Next, within a code block, let's consider the code created by taking $CZ$ on every edge of the color code. The $Z$ stabilizer generators stay the same (six $Z$ around each hexagon). After conjugation by the $CZ$'s, the $X$ stabilizer generators pick up $Z$'s on the six vertices adjacent to the hexagon. These extra $Z$'s cannot be removed by permuting qubits and/or by the action of single qubit gates, so, by Ian's definition noted in this question, the resulting code is not a CSS code.

However, since $CZ$ gates are diagonal, they also commute with the $CZ$ gates between code blocks that are used to perform transversal $CZ$, so this code maintains the same transversal $CZ$ between code blocks that the 2d color code enjoys.

user196574
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Entangling gates seem to be very common for non-CSS codes I think.

For example, using this notebook you can see some examples of swap-transversal gates for the best known distance codes using code automorphisms.

If you try the $[[12,4,4]]$ code (just setting n=12, k=4 is enough), you will get that transversal HS (with a Pauli correction) has this logical action:

circuit That circuit doesn't seem that useful though. So looking at a more interesting example, Gottesman's $[[8,3,3]]$ code (EC zoo link) is non-CSS, and it has addressable logical CZ gates between each of its qubits through qubit permutations (the example is also in the above notebook).

user35159
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