Loophole-Robust Certification of Quantum Advantage
Claims of quantum advantage should remain robust even when classical strategies have access to side information correlated with the benchmark under evaluation, just as Bell certification must account for measurement dependence. We formalize such correlations as benchmark dependence, a task-level generalization of measurement dependence. For every bounded-reward task, we show that the optimal benchmark-dependent classical score obeys $S_\eta\leq\min\{1,S_{\mathrm{cl}}+\eta\}$, and construct a family of tasks that saturates this bound, showing that the linear dependence on $\eta$ is tight without further assumptions. For repeated product tasks with roundwise dependence, we obtain the stronger multiplicative bound $S_\eta^{(n)}\leq(\omega_{\mathrm c}+\eta)^n$, and extend the framework to finite-sample data, mutual-information constraints, multipartite tasks, and correlations distributed along a causal path. Applying these results to aggregated IBM hardware data, we obtain positive raw-count cycle-product certificates of 0.0812 for CHSH and 0.2178 for Mermin--GHZ, while the nine-context magic-square construction remains uncertified; readout-mitigated values are reported separately as sensitivity estimates. We also analyze a non-Bell quantum-kernel benchmark, where a label-construction variable has measured conditional dependence $\widehat{\eta}_{\lambda}^{(Y)}=0.5$, above the threshold $\eta_{\mathrm{req}}=0.375$, required to close the reported score gap, and yields perfect classical classification. The framework therefore converts a quantum--classical score separation into a quantitative lower bound on the benchmark-correlated classical information required to explain the score separation.