Error Attribution as a Resource-Allocation Principle for Quantum Error Correction: Sensitivity-Guided Noise Reduction and Its Analytic Limits

Abstract

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/