Determine residual Pauli corrections in logical Clifford implementations

Determine whether residual Pauli operators occur when implementing two-qubit logical Clifford operations using the C$_4$-Helix code’s logical global phase and logical CZ gates, whose symplectic actions include additional logical $Z$ operators.

Background

The $[[20,2,6]]$ C$_4$-Helix code implements its complete two-qubit Clifford group using automorphism gates together with a logical global phase gate, $\overline{S}_0\overline{S}_1$. The paper notes that the physical constructions used for the global phase and logical CZ gates implement the intended symplectic transformations only up to additional logical Pauli factors: the global phase gate realizes $$\overline{S}_0\overline{S}_1\overline{Z}_0\overline{Z}_1$$, while the logical CZ realizes $\overline{CZ}_{0,1}\overline{Z}_1$.

Although these extra Pauli factors do not alter the symplectic action, their presence or absence must be tracked to ensure that compiled two-qubit logical Clifford operations have the correct full action, including Pauli-frame effects. The paper explicitly leaves unresolved whether residual Paulis remain in particular implementations.