Surface-code decoding skips the zeros: a sparse Mamba decoder cuts logical error rate by up to 49% under SI1000 noise

Surface-code decoding no longer feeds the whole syndrome array into a neural network; it takes only the detection events that actually fired. Earlier dense neural decoders processed the full syndrome of size $d^2 \times R$ at every step, independent of how many errors occurred. Under SI1000 noise at $p=10^{-3}$ and R=120, measured detection-event densities were 2.83%–3.22% for d=3–7, so the overwhelming majority of entries are zero.

Each fired detection event becomes a 13-dimensional feature, and the ordered event sequence is passed once through a Mamba backbone. A Mamba selective state-space model is a backbone that sweeps linearly along a sequence; here it does not advance a hidden state round by round. On the public Google Sycamore experimental data, the ensemble reached mean logical error per round of $2.940\times10^{-2}$ at d=3 and $3.001\times10^{-2}$ at d=5, matching or slightly surpassing the dense Mamba decoder's reported $2.98\times10^{-2}$ and $3.03\times10^{-2}$.

The speed advantage grows with code distance because the forward pass scales only with the number of active defects k. The near-maximum-likelihood Tesseract decoder and Belief Matching scale with detector-graph size and take millisecond-scale time under SI1000 noise. On a single H200 NVL, SMD's speedup over Tesseract rises from 95× at d=3 to 467× at d=7, and over Belief Matching from 232× to 463×.

Evaluation covers the rotated surface code under depolarizing, uniform circuit-level and SI1000 synthetic noise plus the public Google Sycamore experimental data, with SI1000 experiments limited to d≤7 and R=120. All results were obtained on commodity NVIDIA GPUs, with SI1000 speed measured on a single H200 NVL and uniform circuit-level speed on an RTX 4090. The authors identify combining O(k) scaling with hardware-specific streaming decoding as the direction toward sub-microsecond sparse neural decoding.

Under SI1000 noise, the single model at d=7 is essentially level with MWPM (ratio 0.99), and the ensemble's 16% improvement falls within a relative 95% confidence interval of about ±24%. At d=5 SMD loses to MWPM at short round counts (r≤25), and at d=7 the ratio rises to 1.40 at r=140, beyond the training range. Single-shot inference at batch size 1 takes 4–12 ms, dominated by GPU kernel-launch overhead, which is incompatible with the roughly 1 µs per-cycle budget of superconducting qubits.