Fig. 3

Circuit structure for a error detection and b logical error correction for Pauli Z rotation gate Rz with remnant error of \({{\mathcal{E}}}_{{\rm{rem}}}\). We may further combine with the PEC method, for which the Pauli operation \({\mathcal{P}}\) with dotted square is executed virtually via updating the Pauli frame. Note that the feedback operation in (b) is also done via the Pauli frame.