Scalable Quantum Error Correction for Surface Codes Using FPGA

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.