Quantum Error Correction for Second-Generation Quantum Repeaters

In this article, we consider second-generation (2G) quantum repeaters (QRs) for creating long-distance entanglement in quantum networks. Combining a distance-dependent depolarizing error model with the nonlocal Bell state purification procedure required by 2G QRs leads to an error model consisting of correlated and biased errors. To correct correlated errors, nonsymmetric Calderbank&#x2013;Steane&#x2013;Shor (CSS) codes with joint decoding between stations can be used. The dominating errors are biased, such that different repeater stations suffer from different types of errors. To mitigate this, different quantum codes can be used at the stations, optimized for the specific error model of the station. To comply with the 2G QR procedure, the codes used in neighboring stations must allow for the transversal implementation of nonlocal logical <sc>cnot</sc> gates across the two stations or, alternatively, nonlocal <sc>cz</sc> gates combined with logical Hadamard gates. We provide a complete characterization of pairs of CSS codes that allow <sc>cnot</sc> or <sc>cz</sc> transversality, and examine an explicit family of mirrored CSS codes allowing <sc>cz</sc> transversality. We verify Hadamard gate transversality using our framework and show the importance of the logical qubit mapping matrix. Also, we conclude that using different QECCs does not lead to universal computation with the Clifford + <inline-formula><tex-math notation="LaTeX">$T$</tex-math></inline-formula> gate set. Finally, we study the entanglement generation rate (EGR) in 2G QRs with limited quantum memory, minimizing the number of intermediate stations for a given fidelity and EGR. By simulation, we observe that nonsymmetric and mirrored structure QECCs outperform the conventional approach of using symmetric CSS codes at the repeater stations.