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arXiv:2610.10531

Efficient Estimation of Logical Sensitivities Through Fault-Counting

Winston Fu, J. Wilson Staples, Jeff D. Thompson

quant-phSubmitted Oct 7, 2026• Announced Oct 8, 2026• 0 citations
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Abstract

Quantum error-correcting circuits are affected by multiple physical noise mechanisms, whose contributions to logical failure must be understood to evaluate code performance and guide improvements in hardware. The individual error budget contributions for an error type $i$ can be characterized by its logical sensitivity $ν_i = \frac{\partial p_L}{\partial p_i}$, which measures the response of the logical error rate $p_L$ to each physical noise parameter $p_i$. Normally, $ν_i$ is measured using linear fits such as finite differences, where $p_L$ is measured at two or more values of $p_i$ to calculate partial derivatives. In this paper, we develop a differentiable estimator to obtain all components of $\nabla_{\mathbf p}p_L$ simultaneously from a single Monte Carlo data set at one noise configuration, using information about the underlying fault configurations. In surface code simulations with circuit-level noise, the estimator agrees with conventional finite differences while requiring one to two orders of magnitude fewer shots to achieve the same variance. We apply this technique to quantify error budgets, resolve sensitivities at the individual qubit level, and infer effective code distance. Finally, we incorporate the sensitivities into Newton root finding to locate and trace threshold contours in multidimensional noise models. These results provide an efficient method for extracting and applying logical sensitivity information from standard quantum error correction simulations.

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Logical error rate is the standard benchmark for quantum error correction (QEC), but it is an aggregate quantity: it says nothing about which circuit components actually drive logical failure. Recent work introduced an error attribution scheme that computes per-component sensitivities $\partial P_L/\partial p_i$ of the logical error rate jointly from the same Monte Carlo samples used to estimate $P_L$, and showed that halving noise on the 5%–7% most sensitive components reduces logical error rates by roughly 15%–25%, about twice the improvement from randomly chosen components [1],[3]. This paper develops the quantitative structure underlying that result. We formalize the sensitivity map, derive the first-order intervention gain for targeted versus random noise-reduction budgets, and show analytically that the targeted-to-random advantage is $\rho = r/(fr + 1 - f)$, where $f$ is the hotspot fraction and $r$ the sensitivity contrast; for $f = 0.06$ and $r = 2.136$ this ratio is exactly $2$, matching the reported factor. We further show that the reported 15%–25% reductions exceed the first-order prediction of about 6% by an amplification factor of approximately $3.15$, which we attribute to nonlinear, multiplicative error propagation absent from independent-error models. We derive Monte Carlo sample-complexity requirements for the sensitivity estimator, situate attribution among channel-optimized, reinforcement-learning, and robustness-optimized QEC strategies, and identify failure modes: coherent noise, estimator variance at low $P_L$, sensitivity redistribution, and non-stationary drift. ## 1. Introduction Quantum error correction protects quantum information by encoding logical qubits redundantly and decoding measured syndromes, and its performance is almost universally reported as a single number: the logical error rate $P_L$, the probability that a logical operator flips during a memory experiment or computation cycle. This aggregate benchmark is necessary but i Full text and updates: papers.qnfo.org/papers/error-attribution-as-a-resource-allocation-principle-for-quantum-error-correctio/

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quant-pharXiv:2602.19722•Feb 24, 2026•0 citations
Quantum error-correcting circuits are affected by multiple physical noise mechanisms, whose contributions to logical failure must be understood to evaluate code performance and guide improvements in hardware. The individual error budget contributions for an error type iii can be characterized by its logical sensitivity νi=∂pL∂piν_i = \frac{\partial p_L}{\partial p_i}νi​=∂pi​∂pL​​, which measures the response of the logical error rate pLp_LpL​ to each physical noise parameter pip_ipi​. Normally, νiν_iνi​ is measured using linear fits such as finite differences, where pLp_LpL​ is measured at two or more values of pip_ipi​ to calculate partial derivatives. In this paper, we develop a differentiable estimator to obtain all components of ∇ppL\nabla_{\mathbf p}p_L∇p​pL​ simultaneously from a single Monte Carlo data set at one noise configuration, using information about the underlying fault configurations. In surface code simulations with circuit-level noise, the estimator agrees with conventional finite differences while requiring one to two orders of magnitude fewer shots to achieve the same variance. We apply this technique to quantify error budgets, resolve sensitivities at the individual qubit level, and infer effective code distance. Finally, we incorporate the sensitivities into Newton root finding to locate and trace threshold contours in multidimensional noise models. These results provide an efficient method for extracting and applying logical sensitivity information from standard quantum error correction simulations.
Logical error rate is a standard benchmark for quantum error correction, providing an aggregate measure of the overall effect of physical noise. In actual QEC architectures, errors at different circuit locations can have markedly different effects on logical failure even when their probabilities are equal. Here we introduce an error attribution scheme to quantitatively characterize the influence of individual circuit components on the logical error rate and use this refined information to guide targeted intervention and design for improved QEC performance. Our attribution estimator computes all component sensitivities jointly from the same Monte Carlo samples used to estimate the logical error rate. The resulting sensitivity maps identify hotspots where targeted noise reduction can most effectively improve logical performance. We test the scheme in circuit simulations of rotated surface code memories and high-rate lifted product codes: halving the noise strength on about 5\%-7\% of physical components selected by sensitivity reduces logical error rates by approximately 15\%-25\%, roughly twice the mean reduction from the same intervention on an equal number of randomly selected components. By connecting performance benchmarking to targeted optimization, our scheme provides a practical strategy for improving quantum error correction with limited resources.
Characterizing errors in quantum circuits is essential for device calibration, yet detecting rare error events requires a large number of samples. This challenge is particularly severe in calibrating fault-tolerant, error-corrected circuits, where logical error probabilities are suppressed to higher order relative to physical noise and are therefore difficult to calibrate through direct logical measurements. Recently, Wagner et al. [PRL 130, 200601 (2023)] showed that, for phenomenological Pauli noise models, the logical channel can instead be inferred from syndrome measurement data generated during error correction. Here, we extend this framework to realistic circuit-level noise models. From a unified code-theoretic perspective and spacetime code formalism, we derive necessary and sufficient conditions for learning the logical channel from syndrome data alone and explicitly characterize the learnable degrees of freedom of circuit-level Pauli faults. Using Fourier analysis and compressed sensing, we develop efficient estimators with provable guarantees on sample complexity and computational cost. We further present an end-to-end protocol and demonstrate its performance on several syndrome-extraction circuits, achieving orders-of-magnitude sample-complexity savings over direct logical benchmarking. Our results establish syndrome-based learning as a practical approach to characterizing the logical channel in fault-tolerant quantum devices.
Quantum error correction codes are essential for fault-tolerant quantum computing. They are a method of detecting and correcting errors in a quantum computer by encoding logical qubits across multiple physical qubits. Several prominent error correction codes have been developed and larger and more complex codes are being developed continuously. However, evaluating the fault-tolerance of these larger codes is becoming exponentially more complicated. While conventional methods of proving and evaluating the fault-tolerance of these circuits have worked thus far, they are soon to be obsolete. The aim of this thesis is to develop a framework that automatically evaluates the fault-tolerance of a quantum error correction circuit across different hardware architectures. The framework works as follows: it takes a noiseless quantum error correction code; injects hardware-realistic noise into it; and evaluates the fault-tolerance of the resulting noisy circuit by estimating its logical error rate through Monte Carlo sampling. For the injection of hardware-realistic noise, this thesis considers noise stemming from five different quantum computing hardware architectures: superconducting qubits, neutral atoms, trapped ions, photonic qubits, and spin qubits. The noise model was constructed by extracting single-qubit gate, two-qubit gate and measurement error fidelities from current experimental literature. The noise model and circuit generation is implemented in Python, circuit representation and noise injection is executed in Stim and decoding and estimating the logical error rate is achieved through PyMatching. The framework is evaluated on the repetition code, the surface code, and the color code; the logical error rate is estimated across all five hardware architectures under the noise model; and the results are discussed in the context of hardware compatibility for the given codes.
The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g. 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g. billions) of logical gates on many (e.g. thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.
Logical error rate is the standard benchmark for quantum error correction (QEC), but it is an aggregate quantity: it says nothing about which circuit components actually drive logical failure. Recent work introduced an error attribution scheme that computes per-component sensitivities ∂PL/∂pi\partial P_L/\partial p_i∂PL​/∂pi​ of the logical error rate jointly from the same Monte Carlo samples used to estimate PLP_LPL​, and showed that halving noise on the 5%–7% most sensitive components reduces logical error rates by roughly 15%–25%, about twice the improvement from randomly chosen components [1],[3]. This paper develops the quantitative structure underlying that result. We formalize the sensitivity map, derive the first-order intervention gain for targeted versus random noise-reduction budgets, and show analytically that the targeted-to-random advantage is ρ=r/(fr+1−f)\rho = r/(fr + 1 - f)ρ=r/(fr+1−f), where fff is the hotspot fraction and rrr the sensitivity contrast; for f=0.06f = 0.06f=0.06 and r=2.136r = 2.136r=2.136 this ratio is exactly 222, matching the reported factor. We further show that the reported 15%–25% reductions exceed the first-order prediction of about 6% by an amplification factor of approximately 3.153.153.15, which we attribute to nonlinear, multiplicative error propagation absent from independent-error models. We derive Monte Carlo sample-complexity requirements for the sensitivity estimator, situate attribution among channel-optimized, reinforcement-learning, and robustness-optimized QEC strategies, and identify failure modes: coherent noise, estimator variance at low PLP_LPL​, sensitivity redistribution, and non-stationary drift. ## 1. Introduction Quantum error correction protects quantum information by encoding logical qubits redundantly and decoding measured syndromes, and its performance is almost universally reported as a single number: the logical error rate PLP_LPL​, the probability that a logical operator flips during a memory experiment or computation cycle. This aggregate benchmark is necessary but i Full text and updates: papers.qnfo.org/papers/error-attribution-as-a-resource-allocation-principle-for-quantum-error-correctio/
Accurate noise estimation is essential for fault-tolerant quantum computing, since decoding per formance depends critically on the fidelity of circuit-level noise parameters. We introduce a differ entiable Maximum Likelihood Estimation (dMLE) framework that learns these parameters directly from experimental syndrome data using likelihood gradients, with no decoder in the training loop. Mapping syndrome likelihoods to statistical-mechanical models enables exact and efficient evalua tion via planar solvers for repetition codes and tensor networks (TNs) for surface codes, extending the exact treatment to code distance d = 5 with a small number of rounds, and, via spatial subsam pling combined with time-translation symmetry, to larger codes and larger round counts. Across four experimental superconducting datasets, dMLE reduces logical error rates by up to 30.6% for repetition codes. On the Sycamore d = 3 surface code, a TN decoder using the dMLE prior attains a logical error rate comparable to that reported for state-of-the-art AlphaQubit, using standard syndromes alone without auxiliary leakage data. On the larger d = 5 surface code the dMLE prior lowers the logical error rate of the TN and Tesseract decoders by 7.18% and 5.23% relative to a prior optimized by reinforcement learning; at d = 7 it lowers that of Tesseract by 4.56% and 4.83% in the X and Z bases.