Entanglement Purification with Quantum LDPC Codes and Iterative Decoding*

Recent constructions of quantum LDPC (QLDPC) codes provide optimal scaling of the code parameters, enabling fault-tolerant quantum systems with minimal resource overhead. However, the hardware path to long-range-interaction-demanding QLDPC codes is likely a challenging one. Given the practical difficulty in building a monolithic architecture for quantum systems, such as computers, based on optimal QLDPC codes, it is worth considering a distributed implementation of such codes over a network of interconnected medium-sized quantum processors. In such a setting, all syndrome measurements and logical operations must be performed through the use of high-fidelity shared entangled states between the processing nodes. Since probabilistic many-to-1 distillation schemes for purifying entanglement are inefficient, we investigate quantum error correction based entanglement purification in this work. Specifically, we employ QLDPC codes to distill GHZ states, as the resulting high-fidelity logical GHZ states can interact directly with the code used to perform distributed quantum computing (DQC), e.g. for fault-tolerant Steane syndrome extraction. We use the min-sum decoder for distilling 3-qubit GHZ states using a rate 0.118 family of lifted product QLDPC codes and obtain a threshold of 10.7% under depolarizing noise. Our results can be extended to larger size GHZ states as well.