Modeling coherent errors in quantum error correction

Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Paulimodel. Herewe examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ò)under the repetition code.We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n, in the small error limit.Wefind that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than ( ) d 1 n error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failuremore quickly than the Paulimodel would predict, and thismay need to be combated with coherent suppressionmethods at the physical level or larger codes.