Quantum Error Correction for Second-Generation Quantum Repeaters

We consider 2nd generation (2G) Quantum Repeaters (QR) for creating long-distance entanglement in quantum networks. Combining a distance-dependent depolarizing error model with the non-local Bell-state purification procedure required by 2G QRs leads to an error model consisting of correlated and biased errors. To correct correlated errors, non-symmetric 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 non-local logical CNOT gates across the two stations, or alternatively, non-local CZ gates combined with logical Hadamards. We provide a complete characterization of pairs of Calderbank-Steane-Shor (CSS) codes that allow CNOT- or CZ-transversality, and examine an explicit family of mirrored CSS codes allowing CZ-transversality. We verify the Hadamard gate transversality using our framework and show the importance of the mapping matrix. Also, we conclude that using different QECCs can not achieve transversal universal computation with Clifford + <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><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 non-symmetric and mirrored structure QECCs outperform the conventional approach of using symmetric CSS codes at the repeater stations.