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The controlled-not gate $$ CNOT = \begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 1 & 0 \end{bmatrix} $$ is important in many quantum computing protocols. So, given a quantum error-correcting code, it is of particular interest to have a fault tolerant implementations of $ CNOT $. For CSS codes this can be done with transversal $ CNOT $ CSS codes are the only stabilizer codes with transversal CNOT?.

But there are lots of stabilizer codes that are not CSS. Can anyone share a way to implement $ CNOT $ fault tolerantly on a non-CSS code? Or I suppose I would also be interested in a non-transversal but still fault-tolerant way of implementing $ CNOT $ on a CSS code.

Ian Gershon Teixeira
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A non-transversal and fault-tolerant CNOT gate on a CSS code can be achieved through lattice surgery. See Section 4 of this paper.

Navneeth Ramakrishnan
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