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.

翻译：近期构建的量子低密度奇偶校验码在码长方面实现了逻辑量子比特数与最小距离的最优缩放，从而为资源开销最小的容错量子系统开辟了道路。然而，从基于最近邻连接的拓扑码过渡到需要长程相互作用的量子LDPC码在硬件实现上颇具挑战。鉴于基于最优量子LDPC码构建量子计算机单片架构的实际困难，值得考虑在互连量子处理器网络中分布式实现此类码的方案。在此类场景下，所有综合征测量与逻辑运算必须通过处理节点间高保真共享纠缠态完成。由于概率性多对一蒸馏方案在纯化纠缠方面效率低下，本文研究基于量子纠错的纠缠纯化方法。具体而言，我们采用量子LDPC码蒸馏GHZ态，所得高保真逻辑GHZ态可直接与分布式量子计算所用的编码交互，例如用于容错Steane综合征提取。该协议的应用范围超越分布式量子计算，因为纠缠纯化是任何量子网络的核心任务。我们使用基于最小和算法的迭代译码器，采用码率0.118的乘积提升量子LDPC码族蒸馏3量子比特GHZ态，在独立同分布单量子比特退极化噪声下获得了约0.7974的输入阈值。对于任何GHZ纯化协议，这是0.118产率对应的最佳阈值。我们的结果同样适用于更大规模的GHZ态，其中通过将关于3量子比特GHZ态测量性质的技术结论加以扩展，构建了可扩展的GHZ纯化协议。