Decision Kernels for Quantum Error Mitigation: Why Accuracy Gains Need Not Improve Downstream Decisions
Quantum error mitigation (QEM) is usually benchmarked by expectation-value accuracy, but many near-term workflows use those values only to make downstream choices such as argmin selection, ranking, top-k filtering, optimizer-step acceptance, or phase labeling. This creates a structural mismatch: accuracy is measured in the ambient landscape space, whereas shift-invariant decisions depend only on gaps. We develop a quotient-space theory of finite-shot QEM for downstream decisions. The minimal decision-complete object is the residual gap law; in Gaussian finite-shot regimes it is summarized by effective margins and a decision kernel. The QEM-specific point is that this kernel is not free: it is the pullback of shared physical device noise through the mitigation map. We prove quotient factorization, gap-law minimality, a marginal no-go theorem, a QEM pullback theorem, Gaussian decision-risk formulas, and a fixed-allocation shot-level converse. Finite-shot Qiskit Aer simulations demonstrate the predicted divergence in the evaluated regimes. Clifford-data regression can be decision-flat while improving mean-squared error, and probabilistic error cancellation can improve accuracy while worsening decision risk through sampling overhead. Decision-aware selection modestly reduces static held-out failure relative to accuracy-based selection, often by retaining Raw, but the dynamic success target is not reached. Pre-registered stress tests under a calibrated device-noise model and on a hardware micro-cell probe robustness beyond these regimes. The operational implication in the evaluated regimes is to select QEM methods through residual gap geometry, not from expectation-value accuracy alone.