One Simulation Estimated Every Logical Error Sensitivity

Quantum error budgets usually vary each physical noise parameter and rerun simulations to estimate how logical failure responds. This work derives a differentiable estimator that extracts every component of that gradient from one Monte Carlo data set by retaining information about the underlying fault configurations. In circuit-level surface-code simulations, the estimator agreed with conventional finite differences while requiring ten to one hundred times fewer shots for comparable variance. The authors use it to resolve sensitivities down to individual qubits, infer effective code distance and trace threshold contours in multidimensional noise models with Newton updates. The evidence is simulation, but the method could make detailed error-budget diagnosis much cheaper.