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Backlog Metastability in Windowed Quantum Error Correction Decoding

Jean-Jacques Dubray

Published Sep 14, 2026• 0 citations
Publisher

Abstract

Real-time quantum error correction requires a classical decoder to consume syndrome measurements at the rate the quantum hardware produces them, roughly one round per microsecond on supercon-ducting devices. The standard stability requirement is a utilization condition: the mean decode time per round must be less than the round period, ρ < 1. We show that this condition is inadequate for windowed decoders whose per-window cost grows superlinearly in the number of detection events, which is the empirical shape of matching-based decoders under burst-like syndrome content. For this class, stability is not a threshold but a basin. The deterministic backlog dynamics have a stable operating point and an unstable boundary B∗. Noise erodes that boundary downward, and sufficiently superlinear decoders escape spontaneously. In discrete-event simulations paired under common random numbers, a decoder with cost exponent γ = 1.5 and mean utilization ρ0 = 0.6 (forty percent nominal headroom) fails without provocation, with a mean time to failure of 1,344 ± 289 rounds, about 1.3 ms of wall clock at a 1 μs cycle. These Poisson times are upper bounds: modulating the arrival rate at fixed mean load shortens them monotonically with the measured dispersion index of the arrivals, fourfold by D ≈ 3, and a dispersion excess of twelve percent already uncensors the ρ0 = 0.5 ensemble. We measure the cost exponent of a production decoder (PyMatching 2, sparse blossom) and find it anisotropic: near-linear (γ ≈ 1.1–1.2) when a window grows in length at fixed syndrome density, but approximately quadratic (γ ≈ 2) when the same window grows denser. Bursts travel along the dangerous axis, where the basin is 2–4 rounds deep for ρ0 between 0.5 and 0.8. Spatially clustered event content, the syndrome footprint of leakage and cosmic-ray bursts, is worse still: at matched event count, clustered windows cost 7.6× length-grown ones and 4.8× uniformly densified ones, so the quadratic density exponent is a conservative proxy for burst content. The anisotropy is a property of the matching class rather than of one implementation, since a second exact-MWPM decoder reproduces the density exponent, while a BP+OSD decoder is nearly event-independent: no basin, but orders of magnitude more cost per window. The basin is the price of event-adaptive decoding. Greedy catch-up (“decode everything queued”) is precisely the divergent policy. A bounded decoding window restores stability, and the measured stability edge lands on a closed-form line. The rule survives finite memory: under greedy service a queue cap acts as an implicit window, and a cap on the unstable side of the same line pins at the cap and sheds rounds indefinitely, so adding buffer memory makes the greedy policy strictly worse. The simulation model is cross-validated against Stim at the level of individual fault mechanisms.

DOI
10.20944/preprints202608.1425.v3

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Real-time quantum error correction requires a classical decoder to consume syndrome measurements at the rate the quantum hardware produces them, roughly one round per microsecond on supercon-ducting devices. The standard stability requirement is a utilization condition: the mean decode time per round must be less than the round period, ρ < 1. We show that this condition is inadequate for windowed decoders whose per-window cost grows superlinearly in the number of detection events, which is the empirical shape of matching-based decoders under burst-like syndrome content. For this class, stability is not a threshold but a basin. The deterministic backlog dynamics have a stable operating point and an unstable boundary B∗. Noise erodes that boundary downward, and sufficiently superlinear decoders escape spontaneously. In discrete-event simulations paired under common random numbers, a decoder with cost exponent γ = 1.5 and mean utilization ρ0 = 0.6 (forty percent nominal headroom) fails without provocation, with a mean time to failure of 1,344 ± 289 rounds, about 1.3 ms of wall clock at a 1 μs cycle. These Poisson times are upper bounds: modulating the arrival rate at fixed mean load shortens them monotonically with the measured dispersion index of the arrivals, fourfold by D ≈ 3, and a dispersion excess of twelve percent already uncensors the ρ0 = 0.5 ensemble. We measure the cost exponent of a production decoder (PyMatching 2, sparse blossom) and find it anisotropic: near-linear (γ ≈ 1.1–1.2) when a window grows in length at fixed syndrome density, but approximately quadratic (γ ≈ 2) when the same window grows denser. Bursts travel along the dangerous axis, where the basin is 2–4 rounds deep for ρ0 between 0.5 and 0.8. Spatially clustered event content, the syndrome footprint of leakage and cosmic-ray bursts, is worse still: at matched event count, clustered windows cost 7.6× length-grown ones and 4.8× uniformly densified ones, so the quadratic density exponent is a conservative proxy for burst content. The anisotropy is a property of the matching class rather than of one implementation, since a second exact-MWPM decoder reproduces the density exponent, while a BP+OSD decoder is nearly event-independent: no basin, but orders of magnitude more cost per window. The basin is the price of event-adaptive decoding. Greedy catch-up (“decode everything queued”) is precisely the divergent policy. A bounded decoding window restores stability, and the measured stability edge lands on a closed-form line. The rule survives finite memory: under greedy service a queue cap acts as an implicit window, and a cap on the unstable side of the same line pins at the cap and sheds rounds indefinitely, so adding buffer memory makes the greedy policy strictly worse. The simulation model is cross-validated against Stim at the level of individual fault mechanisms.
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