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Error Propagation Paths in Quantum Error Correction: A Five-Stage Model for Quantifying Noise Suppression from Physical Source to Logical Qubit

Gunjan Jain

Published Apr 24, 2026• 0 citations
Publisher

Abstract

We introduce a five-stage error propagation path model that traces quantum errors from noise origin through physical qubit, syndrome extraction, QEC decoding, to logical qubit. Each stage carries a quantified error rate. The suppression factor S = log10(p_physical / p_logical) captures QEC effectiveness in a single scalar. We catalog propagation paths across 12 noise types and 6 QEC codes, revealing suppression factors from S > 9 (excellent) to S < 1 (weak/reachable). The model enables bottleneck identification: for a typical superconducting processor, measurement error is the bottleneck (S = 4.0) while all other noise types achieve S > 9. Improving measurement readout by 2x yields 30x improvement in bottleneck logical error rate — more cost-effective than increasing code distance.

DOI
10.5281/zenodo.19719190

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We introduce a five-stage error propagation path model that traces quantum errors from noise origin through physical qubit, syndrome extraction, QEC decoding, to logical qubit. Each stage carries a quantified error rate. The suppression factor S = log10(p_physical / p_logical) captures QEC effectiveness in a single scalar. We catalog propagation paths across 12 noise types and 6 QEC codes, revealing suppression factors from S > 9 (excellent) to S < 1 (weak/reachable). The model enables bottleneck identification: for a typical superconducting processor, measurement error is the bottleneck (S = 4.0) while all other noise types achieve S > 9. Improving measurement readout by 2x yields 30x improvement in bottleneck logical error rate — more cost-effective than increasing code distance.
Abstract Practical quantum computing will require error rates well below those achievable with physical qubits. Quantum error correction 1,2 offers a path to algorithmically relevant error rates by encoding logical qubits within many physical qubits, for which increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low for logical performance to improve with increasing code size. Here we report the measurement of logical qubit performance scaling across several code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find that our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, in terms of both logical error probability over 25 cycles and logical error per cycle ((2.914 ± 0.016)% compared to (3.028 ± 0.023)%). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a 1.7 × 10 −6 logical error per cycle floor set by a single high-energy event (1.6 × 10 −7 excluding this event). We accurately model our experiment, extracting error budgets that highlight the biggest challenges for future systems. These results mark an experimental demonstration in which quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation.
We introduce noise reachability analysis — a method for determining which physical qubit errors propagate past quantum error correction codes to become logical errors. The engine classifies noise into three source categories (channel, gate, environment) with 12 sub-types, evaluates each against six QEC code families at multiple code distances (720 evaluations per scan), and determines which errors are "reachable" (propagate past the best available correction). We demonstrate automatic noise bias detection, correlated burst noise handling, and noise-aware QEC code selection. The engine recommends optimal codes with 24 source-specific compensating controls beyond code distance increase. Implemented within QCrypton and validated across multiple noise scenarios, the analysis completes in under 2ms.
Quantum computers could solve problems beyond the reach of classical devices, but this potential depends on quantum error correction (QEC) to protect fragile quantum states from noise. A central challenge in QEC is decoding: inferring likely physical errors from syndrome patterns generated by repeated stabilizer measurements. Existing decoders, including graph-based and neural approaches, typically return a single correction hypothesis and therefore discard the richer posterior structure of the error distribution conditioned on the observed syndrome. Here we recast QEC decoding as posterior inference using discrete denoising diffusion, exploiting the analogy between stochastic error accumulation and the forward diffusion process. We introduce DiffQEC, a generative decoder that combines a syndrome processor for multi-round spatial-temporal syndrome histories with syndrome feature modulation to condition denoising on the observed syndrome throughout inference. On experimental data from Google's superconducting quantum processor, DiffQEC reduces logical error rates by up to 10.2% relative to minimum-weight perfect matching and by about 5% relative to tensor-network decoding. These improvements persist for larger code distances up to 17 under depolarizing noise and for logical circuits of increasing depth. Beyond accuracy, the learned posterior provides confidence estimates for post-selection and reveals physically meaningful error structure, establishing posterior generative decoding as a practical framework for QEC.
We present a general framework for applying linear quantum error mitigation (QEM) techniques directly to physical qubits within a logical qubit to suppress logical errors. By exploiting the linearity of quantum error correction (QEC), we demonstrate that any linear QEM method$\unicode{x2014}$including probabilistic error cancellation (PEC), zero-noise extrapolation (ZNE), and symmetry verification$\unicode{x2014}$can be integrated into the physical layer without requiring modifications to the subsequent QEC decoder. Applying this framework to memory experiments using PEC, we analytically prove and numerically verify that the leading-order contribution to the logical error can be removed, increasing the effective code distance by 2. Our simulations on repetition and rotated surface codes show that a distance-3 code with physical-level PEC achieves logical error rates lower than or similar to a distance-5 unmitigated code while using 40% and 64% fewer qubits, respectively. These results establish physical-level QEM as a widely compatible and resource-efficient strategy for enhancing logical performance in early fault-tolerant architectures.
The ability of quantum error correction (QEC) to protect logical qubits against physical noise is traditionally quantified by a noise threshold: a critical noise level below which the logical error rate decreases exponentially with code distance. This paper proposes that QEC itself can undergo a phase transition analogous to a physical phase change. We develop a theoretical framework that treats the logical error rate as an order parameter and maps the behavior of large stabilizer codes onto a statistical mechanics model. By analyzing the scaling of logical error probabilities with code distance and noise strength, we identify a continuous phase transition at the threshold, characterized by critical exponents that depend on the code family and noise model. The transition separates a "fault‑tolerant" phase where logical errors are exponentially suppressed from a "fault‑unreliable" phase where logical errors proliferate. We discuss implications for the design of large‑scale quantum computers, potential experimental signatures in current quantum hardware, and the prospect of discovering new quantum information physics through this phase‑transition perspective.
Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates in the 10−1510^{-15}10−15 regime, but state-of-the-art quantum platforms typically have physical error rates near 10−310^{-3}10−3. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be detected and corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local and that performance is maintained over many rounds of error correction, two major outstanding experimental challenges. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than 100×100\times100× when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.