A fault-tolerant quantum computer must decode and correct errors faster than they appear. The faster errors can be corrected, the more time the computer can do useful work. The Union-Find (UF) decoder is promising with an average time complexity slightly higher than $O(d^3)$. We report a distributed version of the UF decoder that exploits parallel computing resources for further speedup. Using an FPGA-based implementation, we empirically show that this distributed UF decoder has a sublinear average time complexity with regard to $d$, given $O(d^3)$ parallel computing resources. The decoding time per measurement round decreases as $d$ increases, a first time for a quantum error decoder. The implementation employs a scalable architecture called Helios that organizes parallel computing resources into a hybrid tree-grid structure. We are able to implement $d$ up to 21 with a Xilinx VCU129 FPGA, for which an average decoding time is 11.5 ns per measurement round under phenomenological noise of 0.1\%, significantly faster than any existing decoder implementation. Since the decoding time per measurement round of Helios decreases with $d$, Helios can decode a surface code of arbitrarily large $d$ without a growing backlog.

翻译：容错量子计算机必须在错误出现之前对其进行解码和纠正。纠错速度越快，计算机就能有更多时间执行有用计算。Union-Find（UF）解码器具有平均时间复杂度略高于$O(d^3)$的潜力。我们提出了一种利用并行计算资源进一步加速的分布式UF解码器版本。通过基于FPGA的实现，我们实证表明，在拥有$O(d^3)$并行计算资源的情况下，该分布式UF解码器的平均时间复杂度关于$d$呈次线性关系。每轮测量解码时间随$d$增加而减少，这在量子纠错解码器中尚属首次。该实现采用名为Helios的可扩展架构，将并行计算资源组织成混合树-网格结构。我们能够在Xilinx VCU129 FPGA上实现最大$d=21$的解码，在0.1%现象学噪声下每轮测量平均解码时间为11.5纳秒，显著快于现有任何解码器实现。由于Helios每轮测量解码时间随$d$增加而减少，它能够解码任意大$d$的表面码而不会产生积压增长。