  Recent constructions of quantum low-density parity-check (QLDPC) codes
provide optimal scaling of the number of logical qubits and the minimum
distance in terms of the code length, thereby opening the door to
fault-tolerant quantum systems with minimal resource overhead. However, the
hardware path from nearest-neighbor-connection-based topological codes to
long-range-interaction-demanding QLDPC codes is a challenging one. Given the
practical difficulty in building a monolithic architecture for quantum
computers based on optimal QLDPC codes, it is worth considering a distributed
implementation of such codes over a network of interconnected quantum
processors. In such a setting, all syndrome measurements and logical operations
must be performed using 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. This
protocol is applicable beyond DQC since entanglement purification is a
quintessential task of any quantum network. We use the min-sum algorithm (MSA)
based iterative decoder for distilling $3$-qubit GHZ states using a rate
$0.118$ family of lifted product QLDPC codes and obtain an input threshold of
$\approx 0.7974$ under i.i.d. single-qubit depolarizing noise. This represents
the best threshold for a yield of $0.118$ for any GHZ purification protocol.
Our results apply to larger size GHZ states as well, where we extend our
technical result about a measurement property of $3$-qubit GHZ states to
construct a scalable GHZ purification protocol.
