A new study addresses the “repairability” of approximate numerical solutions used in recursive state estimation, a fundamental process in control and filtering systems. Researchers have characterized when a correction within a prescribed subspace and norm budget can meet a local admissibility tolerance. This approach is crucial because, in these systems, high accuracy in individual steps does not necessarily guarantee better overall performance. The work focuses on a fixed linear Kalman model, analyzing how introduced defects affect the covariance response over a finite horizon.

The study reveals that by centering each defect on the exact gain for the implemented covariance, the current solve error is separated from inherited gain drift. Expanding the exact residual-drift identity shows opposing quartic contributions beyond the quadratic response: innovation-covariance inflation enters positively, while local-gain reoptimization enters subtractively. Under matched initialization, an absolute sixth-order remainder bound, uniform over bounded defect sequences at a fixed horizon, provides sufficient conditions for quadratic under- or overprediction.

The research also explores the use of machine learning to propose bounded corrections. A learner-independent residual certificate and verified fallback mechanism govern the execution of classical and quantum candidates without changing the reference estimator. In a power-grid tolerance study, learned correction lowered the minimum conjugate-gradient iteration count for deployment without fallback relative to uncorrected solves under the same residual certificate. Furthermore, gains reconstructed from a variational quantum linear solver and from an annealing-based binary encoding, with small-scale terminal measurements on superconducting hardware and sampling on a quantum annealer, were executed through the same interface. This framework, by linking local repairability to nonlinear error propagation, evaluates approximate solvers and learned corrections through independent certification and finite-horizon response, providing a practical basis for studying hybrid quantum-classical computation.