Real-time decoding of the gross code memory with FPGAs

We introduce a prototype FPGA decoder implementing the recently discovered Relay-BP algorithm and targeting memory experiments on the $[[144,12,12]]$ bivariate bicycle quantum low-density parity check code. The decoder is both fast and accurate, achieving a belief propagation iteration time of 24ns. It matches the logical error performance of a floating-point implementation despite using reduced precision arithmetic. This speed is sufficient for an average per cycle decoding time under $1\mu \mathrm{s}$ assuming circuit model error probabilities are less than $3\times {10}^{-3}$. This prototype decoder offers useful insights on the path toward decoding solutions for scalable fault-tolerant quantum computers.