Quantum low-density parity-check codes are a promising candidate for fault-tolerant quantum computing with considerably reduced overhead compared to the surface code. However, the lack of a practical decoding algorithm remains a barrier to their implementation. In this work, we introduce localized statistics decoding, a reliability-guided inversion decoder that is highly parallelizable and applicable to arbitrary quantum low-density parity-check codes. Our approach employs a parallel matrix factorization strategy, which we call on-the-fly elimination, to identify, validate, and solve local decoding regions on the decoding graph. Through numerical simulations, we show that localized statistics decoding matches the performance of state-of-the-art decoders while reducing the runtime complexity for operation in the sub-threshold regime. Importantly, our decoder is more amenable to implementation on specialized hardware, positioning it as a promising candidate for decoding real-time syndromes from experiments.

翻译：量子低密度奇偶校验码是一种有前景的容错量子计算候选方案，与表面码相比，其开销显著降低。然而，缺乏实用的解码算法仍是其实现的主要障碍。在本工作中，我们提出了局部统计解码，这是一种高度可并行化且适用于任意量子低密度奇偶校验码的可靠性引导反转解码器。我们的方法采用一种并行矩阵分解策略——我们称之为即时消元法——来识别、验证并求解解码图上的局部解码区域。通过数值模拟，我们证明局部统计解码在性能上与最先进的解码器相当，同时降低了在亚阈值区域内运行的运行时复杂度。重要的是，我们的解码器更易于在专用硬件上实现，这使其成为解码实验中实时校验子信号的一个有前景的候选方案。