Surface Code Threshold in the Presence of Correlated Errors

We study the fidelity of the surface code in the presence of correlated errors induced by the coupling of physical qubits to a bosonic environment. By mapping the time evolution of the system after one quantum error correction cycle onto a statistical spin model, we show that the existence of an error threshold is related to the appearance of an order-disorder phase transition in the statistical model in the thermodynamic limit. This allows us to relate the error threshold to bath parameters and to the spatial range of the correlated errors.

Figure 1

(a) Star ($A$) and plaquette ($B$) operators. (b) Examples of logical $\overline{X}$ and $\overline{Z}$ operators. The crosses and shades indicate the qubits involved in the respective operator.

Figure 2

Example of states that contribute to the $S_{+}$ (a) and $S_{-}$ (b) sums [see Eq. (15)]. The light (dark) circles indicate qubits in $+$ ($-$) ${\sigma }^x$ eigenstates. The dark line indicates the path of the logical error ${\overline{Z}}_{\gamma }$, which is active in (b) and inactive in (a).