Terminal-Aware Measurement Allocation for Quantum Deep Equilibrium Inference

Quantum deep equilibrium models (QDEQs) construct implicit representations through repeated evaluations of a shared parameterized quantum circuit, but finite-shot measurements perturb the numerical information used by equilibrium solvers. We first characterize the measurement precision required to resolve Broyden secant differences at fixed query points, showing that sampling fluctuations can obscure shrinking residual differences near equilibrium. We then study damped Picard inference, which needs no secants, and allocate its shot budget by downstream impact. A locally linearized propagation model combining measurement covariance, readout, and state dynamics assigns each iteration a terminal impact coefficient, and calibration-averaged coefficients yield a shared schedule that is optimized offline, spends the budget exactly, and stays frozen at test time. On image classification benchmarks, the schedule reduces measurement-induced terminal squared error by 25.2%–33.9% relative to uniform allocation at equal budgets; downstream dynamics, rather than full covariance structure, account for most of the gain, and the preferred solver shifts from Picard to tuned Broyden as budgets grow. These results turn the measurement budget of quantum implicit models into an analyzable and controllable resource.