Measurement-Based Entanglement Distillation and Constant-Rate Quantum Repeaters over Arbitrary Distances

Measurement-based quantum repeaters employ entanglement distillation and swapping across links using locally prepared resource states of minimal size and local Bell measurements. In this Letter, we introduce a systematic protocol for measurement-based entanglement distillation and its application to repeaters that can leverage any stabilizer code. Given a code, we explicitly define the corresponding resource state and derive an error-recovery operation based on all Bell measurement outcomes. Our approach offers deeper insights into the impact of resource state noise on repeater performance while also providing strategies for efficient preparation and fault-tolerant preservation of resource states. As an application, we propose a measurement-based repeater protocol based on quantum low-density parity-check (QLDPC) codes, enabling constant-yield Bell state distribution over arbitrary distances. Numerical simulations identify a fault-tolerant threshold on the total physical error per repeater segment - including errors on resource states, remotely generated Bell states, and Bell measurements - and confirm that increasing the QLDPC code size further suppresses the logical error while maintaining a fixed encoding rate. This Letter establishes a scalable backbone for future global-scale fault-tolerant quantum networks.