  Fast classical processing is essential for most quantum fault-tolerance
architectures. We introduce a sliding-window decoding scheme that provides fast
classical processing for the surface code through parallelism. Our scheme
divides the syndromes in spacetime into overlapping windows along the time
direction, which can be decoded in parallel with any inner decoder. With this
parallelism, our scheme can solve the decoding throughput problem as the code
scales up, even if the inner decoder is slow. When using min-weight perfect
matching and union-find as the inner decoders, we observe circuit-level
thresholds of $0.68\%$ and $0.55\%$, respectively, which are almost identical
to $0.70\%$ and $0.55\%$ for the batch decoding.
