Backlog Metastability in Windowed Quantum Error Correction Decoding

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.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.

Sep 14, 20260 citations

The Syndrome Buffer Sizing Problem in Quantum Error Correction

Sumeet Shirgure, Siyuan Niu

A fault-tolerant quantum computer must continuously decode a stream of syndrome data, yet the computational work required by each decoding window is random. Although average throughput determines whether the syndrome backlog is stable (doesn’t grow without bound), it does not determine the buffer capacity needed to meet a target overflow probability. Here, we model syndrome buffering as a queue and show how its required depth depends on the tail of the decoding-work distribution. For heavy-tailed work, established queueing results imply that sliding-window decoding can require buffer depths that grow superpolynomially in the logarithm of the inverse overflow probability. Parallel-window decoding suppresses the overflow tail through concurrent service, with the suppression exponent determined by the number of spare decoders; under out-of-order release, the number of spare decoders required for a fixed buffer and target overflow probability grows logarithmically with the inverse target probability. We find evidence of heavy-tailed decoding work in Relay-BP on the gross-code and minimum-weight perfect matching on the surface code. These results identify syndrome-buffer capacity and memory-release policy as important design considerations for real-time quantum error correction.A fault-tolerant quantum computer must continuously decode a stream of syndrome data, yet the computational work required by each decoding window is random. Although average throughput determines whether the syndrome backlog is stable (doesn’t grow without bound), it does not determine the buffer capacity needed to meet a target overflow probability. Here, we model syndrome buffering as a queue and show how its required depth depends on the tail of the decoding-work distribution. For heavy-tailed work, established queueing results imply that sliding-window decoding can require buffer depths that grow superpolynomially in the logarithm of the inverse overflow probability. Parallel-window decoding suppresses the overflow tail through concurrent service, with the suppression exponent determined by the number of spare decoders; under out-of-order release, the number of spare decoders required for a fixed buffer and target overflow probability grows logarithmically with the inverse target probability. We find evidence of heavy-tailed decoding work in Relay-BP on the gross-code and minimum-weight perfect matching on the surface code. These results identify syndrome-buffer capacity and memory-release policy as important design considerations for real-time quantum error correction.

Oct 15, 20260 citations

Kinetics of sliding-window quantum error correction

Adithya Sriram, Charles Stahl, Aleksander Kubica +1 more

Practical implementations of quantum error correction (QEC) require rapid measurement and continuous processing of the syndrome information in order to prevent a backlog of unprocessed data. While ``static'' QEC is theoretically well understood via mappings to equilibrium statistical mechanics models, such an understanding of ``real-time'' QEC is currently lacking. Here, we study the kinetics of sliding window decoding (SWD), an implementation of real-time decoding that acts on temporally local windows of noisy syndrome information and commits to corrections irreversibly at a nonzero rate. We propose an effective description of SWD in terms of a stochastic kinetic process, where $\mathbb{Z}_2$ -charged point particles undergo parity-conserving reaction and diffusion. This model describes dynamics at length and time scales large compared to the window size $W$ , whereas physics at scales smaller than $W$ leads to nontrivial $W$ -dependent scaling of the effective parameters. We identify the rate of decoding $1/W$ as a relevant perturbation to the decodable phase. We also show broad applicability of our results by changing many microscopic details of SWD without affecting the effective description.Practical implementations of quantum error correction (QEC) require rapid measurement and continuous processing of the syndrome information in order to prevent a backlog of unprocessed data. While ``static'' QEC is theoretically well understood via mappings to equilibrium statistical mechanics models, such an understanding of ``real-time'' QEC is currently lacking. Here, we study the kinetics of sliding window decoding (SWD), an implementation of real-time decoding that acts on temporally local windows of noisy syndrome information and commits to corrections irreversibly at a nonzero rate. We propose an effective description of SWD in terms of a stochastic kinetic process, where $\mathbb{Z}_2$-charged point particles undergo parity-conserving reaction and diffusion. This model describes dynamics at length and time scales large compared to the window size $W$, whereas physics at scales smaller than $W$ leads to nontrivial $W$-dependent scaling of the effective parameters. We identify the rate of decoding $1/W$ as a relevant perturbation to the decodable phase. We also show broad applicability of our results by changing many microscopic details of SWD without affecting the effective description.

quant-pharXiv:2608.10081Aug 12, 20260 citations

Decoding under Backlog Constraints as a Sequential Control Problem: A Deep Reinforcement Learning Approach for Quantum Error Correction

Hiroshi Tanaka, Anwar Ali, Muhammad Imran

The practical realization of fault-tolerant quantum computing relies fundamentally on the rapid and accurate decoding of quantum error-correcting codes. However, existing analyses of decoding algorithms often treat the problem as a sequence of independent, state-less optimization tasks, typically optimizing for a scalar metric such as logical error rate or average latency. In reality, the decoding process in a continuously operating quantum computer is tightly constrained by the physical rate of syndrome extraction and the finite capacity of the classical processing buffer, creating a non-trivial backlog dynamic. In this work, we present a rigorous mathematical framework that formulates the quantum decoding problem under strict backlog constraints as a discrete-time Markov Decision Process (MDP). By augmenting the state space to include both the real-time syndrome weight and the instantaneous buffer backlog, we shift the paradigm from one-dimensional thresholding to a multi-dimensional sequential control problem. We demonstrate mathematically and empirically that when the queue dynamics are explicitly modeled, the optimal policy diverges significantly from simple syndrome-weight thresholds. Furthermore, we employ Deep Q-Networks (DQN) to discover non-linear policies that inherently balance the trade-off between logical error suppression and queue stability. Our control-theoretic analysis reveals that incorporating state-space representations of the backlog allows RL agents to robustly prevent buffer overflow while maintaining near-optimal logical fidelity. Through extensive simulation on surface codes under circuit-level noise, our DQN agent strictly outperforms naive two-dimensional grid-search baselines, achieving a zero overflow rate in critical regimes where traditional thresholding fails catastrophically.The practical realization of fault-tolerant quantum computing relies fundamentally on the rapid and accurate decoding of quantum error-correcting codes. However, existing analyses of decoding algorithms often treat the problem as a sequence of independent, state-less optimization tasks, typically optimizing for a scalar metric such as logical error rate or average latency. In reality, the decoding process in a continuously operating quantum computer is tightly constrained by the physical rate of syndrome extraction and the finite capacity of the classical processing buffer, creating a non-trivial backlog dynamic. In this work, we present a rigorous mathematical framework that formulates the quantum decoding problem under strict backlog constraints as a discrete-time Markov Decision Process (MDP). By augmenting the state space to include both the real-time syndrome weight and the instantaneous buffer backlog, we shift the paradigm from one-dimensional thresholding to a multi-dimensional sequential control problem. We demonstrate mathematically and empirically that when the queue dynamics are explicitly modeled, the optimal policy diverges significantly from simple syndrome-weight thresholds. Furthermore, we employ Deep Q-Networks (DQN) to discover non-linear policies that inherently balance the trade-off between logical error suppression and queue stability. Our control-theoretic analysis reveals that incorporating state-space representations of the backlog allows RL agents to robustly prevent buffer overflow while maintaining near-optimal logical fidelity. Through extensive simulation on surface codes under circuit-level noise, our DQN agent strictly outperforms naive two-dimensional grid-search baselines, achieving a zero overflow rate in critical regimes where traditional thresholding fails catastrophically.

Jul 21, 20260 citations

Decoding under Backlog Constraints as a Sequential Control Problem: A Deep Reinforcement Learning Approach for Quantum Error Correction

Ali Anwar, Tanaka Hiroshi, Imran Muhammad

Jul 21, 20260 citations

Demonstrating real-time and low-latency quantum error correction with superconducting qubits

Laura Caune, Luka Skorić, Nick S. Blunt +22 more

Quantum error correction will be essential for quantum computers to realise their full potential. As quantum computers advance towards demonstrating a universal fault-tolerant logical gate set, implementing scalable and low-latency real-time decoding will be crucial to avoid an exponential slowdown and maintain a fast logical clock rate. Here, we demonstrate low-latency feedback with a scalable FPGA decoder integrated into the control system of a superconducting quantum processor. We perform an 8-qubit stability experiment with up to 25 decoding rounds and a sub-microsecond mean decoding time per round, providing strong evidence that the backlog problem will be avoided when the decoder is operated as a streaming decoder on a superconducting hardware with the strictest speed requirements. We observe logical error suppression as the number of decoding rounds is increased. We also implement and time a fast-feedback experiment demonstrating a decoding response time of 9.6 μs for a total of 9 measurement rounds.Quantum error correction will be essential for quantum computers to realise their full potential. As quantum computers advance towards demonstrating a universal fault-tolerant logical gate set, implementing scalable and low-latency real-time decoding will be crucial to avoid an exponential slowdown and maintain a fast logical clock rate. Here, we demonstrate low-latency feedback with a scalable FPGA decoder integrated into the control system of a superconducting quantum processor. We perform an 8-qubit stability experiment with up to 25 decoding rounds and a sub-microsecond mean decoding time per round, providing strong evidence that the backlog problem will be avoided when the decoder is operated as a streaming decoder on a superconducting hardware with the strictest speed requirements. We observe logical error suppression as the number of decoding rounds is increased. We also implement and time a fast-feedback experiment demonstrating a decoding response time of 9.6 μs for a total of 9 measurement rounds.

arXiv:2410.05202Nature CommunicationsJun 1, 202612 citations

\mathbb{Z}_2

-charged point particles undergo parity-conserving reaction and diffusion. This model describes dynamics at length and time scales large compared to the window size

W

, whereas physics at scales smaller than

W

leads to nontrivial

W

-dependent scaling of the effective parameters. We identify the rate of decoding

1/W

as a relevant perturbation to the decodable phase. We also show broad applicability of our results by changing many microscopic details of SWD without affecting the effective description.