Clifford gates are transversal What exactly does this transversal mean? What is the difference between non-Clifford gates and Clifford gates?

Question

Clifford gates are transversal What exactly does this transversal mean? What is the difference between non-Clifford gates and Clifford gates? Why is it simple for Clifford gates to implement transversal for quantum fault tolerance? Please explain the definitions, differences, and functions of the two in detail. I don't understand them very well, thank you.

Answer 1

Transversal and Clifford are not as closely linked as your question would seem to imply.

Transversal gates are those for which an error-correcting code can achieve the transformation on a logical qubit by applying that gate on each of the physical qubits. For example, in a 7-qubit code, if you can achieve logical Hadamard by applying Hadamard on each of the 7 physical qubits, that gate is transversal. These gates are the simplest to implement because they are automatically fault-tolerant: if an error occurs on one physical qubit, there is no action that can propagate it onto a different physical qubit in the same code because no two qubits in the block every interact. They are also particularly short gate sequences (just one time step), which has the knock-on effect of making the corresponding fault-tolerant threshold particularly high.

Now it happens that for some of the most common quantum error correcting codes, the gates that you can apply transversally coincide with the Clifford gates. Indeed, iirc, this is true for all self-dual CSS codes. But this is just a particular choice of error correcting code. There are other choices where it is not the Clifford gates that can be implemented transversally. In that sense, there's nothing particularly special about Clifford.